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Polyhedra in (inorganic) chemistry

Identifieur interne : 000536 ( Istex/Corpus ); précédent : 000535; suivant : 000537

Polyhedra in (inorganic) chemistry

Auteurs : Santiago Alvarez

Source :

Dalton Transactions [ 1477-9226 ] ; 2005.

RBID : ISTEX:20B3F2DEF644932FB0FD0C75EFCA819E651BF020

English descriptors

Teeft :
- Angew, Antiprisms, Archimedean, Archimedean polyhedra, Atwood, Augmentation, Avnir, Barcelona, Bipyramid, Bipyramids, Calixarene, Catalan, Chem, Circumscribed, Commun, Compound polyhedra, Coordination polyhedra, Cshm, Cube, Cuboctahedron, Cupola, Dalton trans, Dodecahedron, Duality, Duals, Edge augmentation, English translation, Face augmentation, Fullerene, Gure, Gures, Icosahedral, Icosahedral symmetry, Icosahedron, Icosidodecahedron, Inorg, Johnson polyhedra, Ligand, Metal atoms, Metal clusters, Nanoparticles, Octahedral, Octahedron, Orthobicupola, Panelling, Pentagonal, Pet3, Platonic, Platonic polyhedra, Polygon, Polyhedral, Polyhedron, Polyoxometallates, Prism, Rhombic, Rhombic dodecahedron, Rhombicosidodecahedron, Rhombicuboctahedron, Stereochemical, Stoichiometry, Subgroup, Supertetrahedron, Supramolecular, Tetrahedral, Tetrahedron, Trans, Trapezohedron, Triakis, Truncation.

Abstract

A systematic description of polyhedra with varying degrees of regularity is illustrated with examples of chemical structures, mostly from different fields of Inorganic Chemistry. Also the geometrical relationships between different polyhedra are highlighted and their application to the analysis of complex structures is discussed.

Url:

https://api.istex.fr/ark:/67375/QHD-ZJLJ1NWV-0/fulltext.pdf

DOI: 10.1039/b503582c

Links to Exploration step

ISTEX:20B3F2DEF644932FB0FD0C75EFCA819E651BF020

Le document en format XML

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<note>Electronic supplementary information (ESI) available: Table S1: Complete list of the Johnson polyhedra, ordered according to the number of vertices (V), giving the number of edges (E) and faces (F) in each case. Table S2: Nested polyhedra that appear as successive shells in prototypical solid state structures. The shells are ordered according to increasing distance to the center. Table S3: Nested polyhedra of icosahedral symmetry in the molecular structure of [Pd145(CO)x(PEt3)30]. Fig. S1: Polyhedra generated from a dodecahedron through augmentation and truncation operations. Fig. S2: Polyhedra generated from a cube through augmentation and truncation operations. Fig. S3: Diamond shells 2 and 6 around centroid of adamantanoid unit, forming an octahedron and a truncated octahedron. Fig. S4: Generation of an approximate icosahedron of bridging selenide ions in an Ag8 cube in the molecular structure of Ag8Cl2[Se2P(OEt)2]6. See http://www.rsc.org/suppdata/dt/b5/b503582c/</note>
<note>Reflections on the use of different kinds of polyhedra in Chemistry and elsewhere, and on the geometrical and chemical relationships between nested polyhedra, such as the seven Platonic and Archimedean polyhedra with octahedral symmetry found at around each atom in the bcc structure illustrated in the cover [b503582c-ga.tif]</note>
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<note type="biography">Santiago Alvarez was born in Panamá, República de Panamá, in 1950 and studied Chemistry at the University of Barcelona, where he obtained a PhD working on inorganic vibrational spectroscopy under the supervision of Prof. J. Casabó. After doing experimental research on one-dimensional conductors in Barcelona, he carried out theoretical research with Prof. R. Hoffmann at Cornell University. He was appointed as Professor Titular in the University of Barcelona in 1984 and has held an inorganic chemistry chair since 1987. He has been a visiting scientist in the USA, France, Chile and Israel, was elected as a Distinguished Researcher by the Generalitat de Catalunya in 2000 and has been awarded the prize for research in Inorganic Chemistry of the Real Sociedad Española de Química and the Solvay prize for research in Chemical Science in 2003. His research interests include bonding and structure in molecular and solid state transition metal compounds, structural and structure–property correlations and the application of continuous symmetry and shape measures to the stereochemical description of transition metal compounds. [b503582c-p1.tif]</note>
<affiliation>Santiago Alvarez was born in Panamá, República de Panamá, in 1950 and studied Chemistry at the University of Barcelona, where he obtained a PhD working on inorganic vibrational spectroscopy under the supervision of Prof. J. Casabó. After doing experimental research on one-dimensional conductors in Barcelona, he carried out theoretical research with Prof. R. Hoffmann at Cornell University. He was appointed as Professor Titular in the University of Barcelona in 1984 and has held an inorganic chemistry chair since 1987. He has been a visiting scientist in the USA, France, Chile and Israel, was elected as a Distinguished Researcher by the Generalitat de Catalunya in 2000 and has been awarded the prize for research in Inorganic Chemistry of the Real Sociedad Española de Química and the Solvay prize for research in Chemical Science in 2003. His research interests include bonding and structure in molecular and solid state transition metal compounds, structural and structure–property correlations and the application of continuous symmetry and shape measures to the stereochemical description of transition metal compounds. [b503582c-p1.tif]</affiliation>
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<volumeref><link>Unassigned</link>
</volumeref>
<issueref><link>Advance Articles</link>
</issueref>
<pubfront><fpage></fpage>
<lpage></lpage>
<no-of-pages></no-of-pages>
<date><year>2005</year>
<month>June</month>
<day>9</day>
</date>
</pubfront>
</published>
<published type="print"><journalref><link>DT</link>
</journalref>
<volumeref><link>0</link>
</volumeref>
<issueref><link>13</link>
</issueref>
<pubfront><fpage>2209</fpage>
<lpage>2233</lpage>
<no-of-pages>25</no-of-pages>
<date><year>2005</year>
<month>6</month>
<day>16</day>
</date>
</pubfront>
</published>
<published type="subsyear"><journalref><title type="abbreviated">Dalton Trans.</title>
<title type="full">Dalton Transactions</title>
<title type="journal">Dalton Transactions</title>
<title type="display">Dalton Transactions</title>
<title type="pubmed">Dalton Trans</title>
<sercode>DT</sercode>
<publisher><orgname><nameelt>The Royal Society of Chemistry</nameelt>
</orgname>
</publisher>
<issn type="print">1477-9226</issn>
<issn type="online">1477-9234</issn>
<coden>ICHBD9</coden>
<cpyrt>This journal is © The Royal Society of Chemistry</cpyrt>
</journalref>
<volumeref><link>5</link>
</volumeref>
<issueref><link>13</link>
</issueref>
<pubfront><fpage>2209</fpage>
<lpage>2233</lpage>
<no-of-pages>25</no-of-pages>
<date><year>2005</year>
<month>Unassigned</month>
<day>Unassigned</day>
</date>
</pubfront>
</published>
<art-links><suppinf><link>INFO</link>
</suppinf>
</art-links>
<art-front><titlegrp><title>Polyhedra in (inorganic) chemistry<fnoteref idrefs="fn1"></fnoteref>
<footnote id="fn1">Electronic supplementary information (ESI) available: Table S1: Complete list of the Johnson polyhedra, ordered according to the number of vertices (<it>V</it>
), giving the number of edges (<it>E</it>
) and faces (<it>F</it>
) in each case. Table S2: Nested polyhedra that appear as successive shells in prototypical solid state structures. The shells are ordered according to increasing distance to the center. Table S3: Nested polyhedra of icosahedral symmetry in the molecular structure of [Pd<inf>145</inf>
(CO)<inf><it>x</it>
</inf>
(PEt<inf>3</inf>
)<inf>30</inf>
]. Fig. S1: Polyhedra generated from a dodecahedron through augmentation and truncation operations. Fig. S2: Polyhedra generated from a cube through augmentation and truncation operations. Fig. S3: Diamond shells 2 and 6 around centroid of adamantanoid unit, forming an octahedron and a truncated octahedron. Fig. S4: Generation of an approximate icosahedron of bridging selenide ions in an Ag<inf>8</inf>
 cube in the molecular structure of Ag<inf>8</inf>
Cl<inf>2</inf>
[Se<inf>2</inf>
P(OEt)<inf>2</inf>
]<inf>6</inf>
. See <url>http://www.rsc.org/suppdata/dt/b5/b503582c/</url>
</footnote>
</title>
</titlegrp>
<authgrp><author aff="affa"><person><persname><fname>Santiago</fname>
<surname>Alvarez</surname>
</persname>
<biography><section><plate xsrc="b503582c-p1.tif" id="plt1"></plate>
<p>Santiago Alvarez was born in Panamá, República de Panamá, in 1950 and studied Chemistry at the University of Barcelona, where he obtained a PhD working on inorganic vibrational spectroscopy under the supervision of Prof. J. Casabó. After doing experimental research on one-dimensional conductors in Barcelona, he carried out theoretical research with Prof. R. Hoffmann at Cornell University. He was appointed as Professor Titular in the University of Barcelona in 1984 and has held an inorganic chemistry chair since 1987. He has been a visiting scientist in the USA, France, Chile and Israel, was elected as a Distinguished Researcher by the <it>Generalitat de Catalunya</it>
 in 2000 and has been awarded the prize for research in Inorganic Chemistry of the <it>Real Sociedad Española de Química</it>
 and the Solvay prize for research in Chemical Science in 2003. His research interests include bonding and structure in molecular and solid state transition metal compounds, structural and structure–property correlations and the application of continuous symmetry and shape measures to the stereochemical description of transition metal compounds.</p>
</section>
</biography>
</person>
</author>
<aff id="affa"><org><orgname><nameelt>Departament de Química Inorgànica and Centre de Recerca en Química Teòrica</nameelt>
<nameelt>Universitat de Barcelona</nameelt>
</orgname>
</org>
<address><addrelt>Martí i Franquès 1-11</addrelt>
<postcode>08028</postcode>
<state>Barcelona</state>
</address>
</aff>
</authgrp>
<art-toc-entry><ictext>Reflections on the use of different kinds of polyhedra in Chemistry and elsewhere, and on the geometrical and chemical relationships between nested polyhedra, such as the seven Platonic and Archimedean polyhedra with octahedral symmetry found at around each atom in the bcc structure illustrated in the cover</ictext>
<icgraphic xsrc="b503582c-ga.tif" id="ga"></icgraphic>
</art-toc-entry>
<abstract><p>A systematic description of polyhedra with varying degrees of regularity is illustrated with examples of chemical structures, mostly from different fields of Inorganic Chemistry. Also the geometrical relationships between different polyhedra are highlighted and their application to the analysis of complex structures is discussed.</p>
</abstract>
</art-front>
<art-body><section><title>Introduction</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>No vull parlar-vos d'obsidianes tallants</it>
</entry>
</row>
<row valign="top"><entry><it>ni de piràmides asteques…</it>
</entry>
</row>
<row><entry><it>puríssims octàedres, secrets tetràedres</it>
</entry>
</row>
<row valign="top"><entry><it>li ofrenen la proporció encesa de les simetries.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>Angel Terrón, <it>El Elegido</it>
<citref idrefs="cit1">1</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>[I don't want to speak about cutting obsidians, </entry>
</row>
<row valign="top"><entry>Nor about Aztec pyramids…
							</entry>
</row>
<row valign="top"><entry>Sheer octahedra, secret tetrahedra </entry>
</row>
<row valign="top"><entry>Offer him a glowing proportion of all symmetries.]<citref idrefs="cit2">2</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>Probably the most common and useful idealization of molecular structures consists in associating the positions of a set of atoms to the vertices of a polyhedron. As Chemists, we benefit from the knowledge of the geometry of polyhedra accumulated since ancient times. Early on, Plato used the regular polyhedra to describe the four <it>elements</it>
: cube, octahedron, tetrahedron and icosahedron for earth, air, fire and water, respectively, with the fifth regular polyhedron, the dodecahedron, used to represent the universe. The current application of Platonic polyhedra in stereochemistry, though, had to wait for quite a few centuries, until van’t Hoff and Le Bel, in 1874, proposed that the spatial arrangement of atoms around a carbon atom in organic molecules should be described as occupying the vertices of a tetrahedron. Shortly after, Werner introduced the octahedron to describe the stereochemistry of transition metal atoms in coordination complexes. These (tetrahedron and octahedron) are the most common structural motifs in molecular transition metal chemistry and are also considered as the basic structural units of most inorganic solids. The icosahedron was incorporated later to the stereochemical toolbox with the development of the chemistry of boron, the first structural characterization of an icosahedron being probably that of the B<inf>12</inf>
 groups in the extended structure of B<inf>4</inf>
C,<citref idrefs="cit3">3</citref>
 followed a few years later by the report of an icosahedron of Cu atoms surrounding an Al one in Mg<inf>2</inf>
Cu<inf>6</inf>
Al<inf>5</inf>
.<citref idrefs="cit4">4</citref>
 In the molecular world, the icosahedron appeared first<citref idrefs="cit5">5</citref>
 as dodecaborane anions in K<inf>2</inf>
B<inf>12</inf>
H<inf>12</inf>
. However, the incorporation of regular or semiregular polyhedra to other fields of Chemistry continued at a slow pace. The history of regular polyhedral alkanes started in 1964, when Eaton and Cole<citref idrefs="cit6">6</citref>
 reported the synthesis of the elegantly symmetric cubane, C<inf>8</inf>
H<inf>8</inf>
. It took more than a decade for new Platonic alkanes to appear: a tetrahedrane<citref idrefs="cit7">7</citref>
 was reported in 1978, after a theoretical prediction and serious synthetic attempts, and the long sought for dodecahedrane was achieved shortly after.<citref idrefs="cit8">8</citref>
 Much more recent is the discovery of buckminsterfullerene, of formula C<inf>60</inf>
, with the shape of a truncated icosahedron. A well established area of Inorganic Chemistry now, that of the polyoxometallates, also provides a host of polyhedral structures, including polyhedra of polyhedra.</p>
<p>It is interesting to see how polyhedral structures appear at different length scales, from the electron and spin densities at the subatomic level up to everyday objects and probably beyond. Starting at the subatomic level, spin densities for different electron configurations in transition metal compounds may show cubic, octahedral or tetrahedral shapes.<citref idrefs="cit9">9</citref>
 Also representations of the d electron density around a metal atom in [Cr(CO)<inf>6</inf>
],<citref idrefs="cit10">10</citref>
 of its Laplacian in [Fe<inf>2</inf>
(CO)<inf>9</inf>
]<citref idrefs="cit11">11</citref>
 or in [Mn(CO)<inf>6</inf>
]<sup>+</sup>
,<citref idrefs="cit12">12</citref>
 and of the electron localization function in [Re<inf>2</inf>
(CO)<inf>10</inf>
]<citref idrefs="cit13">13</citref>
 reveal nice cubic shapes. At a polyatomic level, the coordination polyhedra around metal atoms have diameters of a few tenths of a nanometer, while typical metal clusters can approach 1 nm and large clusters can be up to 2 nm in diameter. Very large polynuclear complexes assembled through bridging ligands can reach sizes of about 2–3 nm, as in Pd<inf>145</inf>
 and Mo<inf>132</inf>
 compounds to be discussed later. Icosahedral quasicrystals, nanoclusters and nanoparticles are in the 10–20 nm size range, where we can also find a single-stranded DNA molecule folded into a hollow octahedron (with a diameter of approximately 22 nm), as well as other DNA polyhedra (including the tetrahedron, the cube and the truncated octahedron) reported in recent years,<citref idrefs="cit14">14</citref>
 or the major light harvesting complex of photosystem II, organized in a particle of practically that same size with icosahedral symmetry.<citref idrefs="cit15">15</citref>
 The capsids of viruses may reach a size an order of magnitude larger (between 10 and 100 nm), among which icosahedral structures have been widely reproduced. Radiolarians, protozoa that form part of the marine plankton present in all oceans, constitute the next step in the size scale with their elaborate external siliceous skeletons, often with polyhedral shapes, which appear in the size range of 10<sup>3</sup>
–10<sup>5</sup>
 nm. Not much different in size, Coccoliths are minute calcium carbonate platelets secreted by certain protozoans or algae in plankton that form colonies (coccolithophores) with a variety of shapes, among which one can find <it>Braarudosphaera bigelowii</it>
 with a neat dodecahedron shape of about 10<sup>3</sup>
 nm in diameter.<citref idrefs="cit16">16</citref>
 As artificial as many molecular polyhedra, man-made small clusters of cross-linked polystyrene microspheres (844 nm in diameter) are not that different in size from the natural shapes just mentioned.<citref idrefs="cit17">17</citref>
 Somewhat larger, we can find in nature beautiful polyhedra both in the biological and mineral worlds, represented by pollen grains (∼10<sup>5</sup>
 nm) and crystals of, <it>e.g.</it>
, pyrite and magnetite (sizes of the order of centimeters, or 10<sup>7</sup>
 nm). I will not mention larger, man-made polyhedral objects that may come easily to the mind of the reader, but to expand even more the scale covered by fascinating polyhedra, let me just mention a recent report proposing a dodecahedral topology of the universe.<citref idrefs="cit18">18</citref>
</p>
<p>Going back to a chemical scale, the assignment of an ideal polyhedron to a set of atoms from a crystal structure is not always straightforward, and for discussing polyhedral structures in Chemistry we have found it useful to employ the continuous symmetry or shape measures (CSM or CShM) proposed by Avnir and co-workers,<citref idrefs="cit19">19</citref>
 in particular in their application to polyhedral shapes.<citref idrefs="cit20">20</citref>
 More detailed accounts of this type of studies as applied to transition metal chemistry can be found elsewhere,<citref idrefs="cit21">21</citref>
 and we will summarize here only the main concepts for those readers who are unfamiliar with those measures, since shape measures will be occasionally used in this paper. In short, the CShM of a set of atoms relative to an ideal polyhedron, S(polyhedron), gives a quantitative indication of how far is the fragment from having that ideal shape. Thus, a zero value for S(polyhedron) means that the fragment has exactly the reference polyhedral shape (regardless of size and orientation in space) and larger values (CShMs are always positive) measure increasing degrees of distortion. According to our experience, we can roughly say that CShM values larger than 0.1 units correspond to significant but small distortions and values larger than 1.0 correspond to moderate to severe distortions. As a rule of thumb, a CShM value larger than 3 units represents such an important distortion that the reference polyhedron can only be taken as a rather crude description of the real structure. One of the advantages of the CShMs is that they allow us to compare on the same scale the deviation of a given structure from several polyhedra, or of different structures from the same reference polyhedron. Another interesting feature of such measures lies in the possibility of establishing the deviation of molecular structures from a polyhedral interconversion path.<citref idrefs="cit22">22</citref>
 If instead of measuring the distance to a reference shape (<it>e.g.</it>
, a polyhedron), we measure the distance to the closest structure that posseses a given symmetry, we talk then of continuous symmetry measures (CSMs), or of continuous chirality measures (CCMs) if we refer to the closest achiral sructure.<citref idrefs="cit23">23</citref>
</p>
<p>The fast development of complex molecular and supramolecular architectures, combined with an enhanced ability to solve their structures by diffraction techniques, has increased the number of polyhedra that are relevant to Chemistry, although our ability to recognize and name them has not increased to the same extent. The aim of this Perspective is to offer an overview of the variety of polyhedra that are (or can be) useful to chemists, from historical, geometrical and chemical perspectives. We will discuss not only those polyhedra well known by chemists, such as the Platonic and Archimedean solids, but also those that are familiar only in specific fields of Chemistry, in particular the large and varied set of Johnson polyhedra. Special emphasis will be made on different types of relationships between polyhedra, trying to establish a connection between geometrical relationships and chemical bonding relationships in molecules or solids. Thus, examples of some of the less well known polyhedra and of nested polyhedra are taken from crystal structures coming from different fields of Chemistry, mostly but not exclusively from an Inorganic perspective. I would suggest the readers, while or after reading this paper, to explore by themselves the wide and visually pleasing world of polyhedra by playing with tridimensional versions of the ideal polyhedra, or with their computer representations as those obtained with the PolyPro<citref idrefs="cit24">24</citref>
 program or available in some excellent Web pages,<citref idrefs="cit24">24</citref>
 as well as with molecular and crystal structures.</p>
</section>
<section><title>Families of polyhedra</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>…aperçut avec horreur l'horizon de sa pensée s'élargir confusément en cercles concentriques, à l'apparition matinale du rhythmique pétrissage d'un sac icosaèdre, contre son parapet calcaire!</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Isidore Ducasse, <it>Les Chants de Maldoror</it>
								(Chant VI)</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[…witnessed with horror the matinal sight of an icosahedron sack being rythmically kneaded against its limestone parapet, and its notion of the possible had been confusedly enlarged in ever widening concentric circles.]<citref idrefs="cit25">25</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>The best known family of polyhedra, the <it>Platonic</it>
 ones, are characterized by having all their faces formed by identical regular polygons, all their vertices equivalent and all their edges equivalent, as summarized in <tableref idrefs="tab1">Table 1</tableref>
. These polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosahedron) belong to the most symmetric point groups: tetrahedral, octahedral or icosahedral. The first systematic studies of these figures that we know about come from classical Greece.<citref idrefs="cit26">26</citref>
 Three of these regular shapes, the tetrahedron, the octahedron and the cube, were certainly known by the Pitagoreans around 480 BC. Furthermore, the legend goes that one of them, Hippasus of Metapontus, perished in the sea as a punishment for having discovered the secrets of the dodecahedron. Plato described in detail these five polyhedra in <it>Timaeus</it>
				(one of his <it>Dialogues</it>
, dated about 370 BC). The philosopher apparently acquired such knowledge from Theaetetus, but the only news we have come from Plato himself, in the text with the same title. Euclid (300 BC), in book XIII of <it>The Elements</it>
, completed the task, establishing the basic geometrical principles of these five polyhedra. However, interest in the symmetry of the Platonic polyhedra appeared much earlier,<citref idrefs="cit26 cit27">26,27</citref>
 probably as a consequence of the observation by men of such regular figures in minerals. Examples of objects with polyhedral forms dated in 2000 BC found in Scotland, are well documented (<figref idrefs="fig1">Fig. 1</figref>
).<citref idrefs="cit28">28</citref>
 The cubic and octahedral shapes can be also recognized as markings in approximately spherical man-made objects from the Iberian (<it>ca.</it>
 1000 BC, <it>Museo Arquelógico Nacional</it>
, Madrid) and Etruscan (V–VIth century, <it>Museo Civico Archeologico</it>
, Bologna) cultures, not to mention cubic dice made of bones or ivory found in ancient Greek and Roman excavations. Also hollow bronze dodecahedra apparently were quite popular in the Roman empire (dated between the II and IVth centuries AC) and can be seen, <it>e.g.</it>
, in the <it>Rheinisches Landesmuseum</it>
				(Bonn) and in the <it>Musée des antiquités nationales</it>
				(Saint Germain en Laye, near Paris). What was the use given to those dodecahedra is presently unclear.</p>
<table-entry id="tab1"><title>Classification of families of convex polyhedra according to the regularity of their faces and to the equivalence of their components: all faces equivalent (isohedral), all vertices equivalent (isogonal) and all edges equivalent (isotoxal)</title>
<table><tgroup cols="4"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<thead><row><entry>Family</entry>
<entry>Faces</entry>
<entry>Vertices</entry>
<entry>Edges<fnoteref idrefs="tab1fna"></fnoteref>
</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="4"><footnote id="tab1fna">In polyhedra with regular faces, even when the edges are not all equivalent, they all have the same length.</footnote>
<footnote id="tab1fnb">Among the Archimedean polyhedra, only the cuboctahedron and the icosidodecahedron have all the edges equivalent.</footnote>
<footnote id="tab1fnc">Only the trigonal and pentagonal bipyramids can have regular faces (the regular square bipyramid is in fact the octahedron, included among the Platonic polyhedra).</footnote>
<footnote id="tab1fnd">Only the square and pentagonal pyramids may have regular polygons in all their faces.</footnote>
<footnote id="tab1fne">Among the Johnson polyhedra only two are isohedral: the trigonal bipyramid (J12) and the pentagonal bipyramid (J13).</footnote>
<footnote id="tab1fnf">Among the Catalan polyhedra only the rhombic dodecahedron and the rhombic triacontahedron have all their edges equivalent.</footnote>
<footnote id="tab1fng">Twelve pentagonal faces, the rest are hexagonal. The fullerene with only pentagonal faces is the Platonic dodecahedron and that with 20 hexagonal faces and 60 vertices is an Archimedean polyhedron, the truncated icosahedron.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Platonic</entry>
<entry>Regular, equivalent</entry>
<entry>Equivalent</entry>
<entry>Equivalent</entry>
</row>
<row><entry>Archimedean</entry>
<entry>Regular</entry>
<entry>Equivalent</entry>
<entry>—<fnoteref idrefs="tab1fnb"></fnoteref>
</entry>
</row>
<row><entry>Prisms, antiprisms</entry>
<entry>Regular</entry>
<entry>Equivalent</entry>
<entry>—</entry>
</row>
<row><entry>Metaprisms</entry>
<entry>—</entry>
<entry>Equivalent</entry>
<entry>—</entry>
</row>
<row><entry>Pyramids</entry>
<entry>—<fnoteref idrefs="tab1fnd"></fnoteref>
</entry>
<entry>—</entry>
<entry>—</entry>
</row>
<row><entry>Bipyramids</entry>
<entry>Equivalent<fnoteref idrefs="tab1fnc"></fnoteref>
</entry>
<entry>—</entry>
<entry>—</entry>
</row>
<row><entry>Trapezohedra</entry>
<entry>Equivalent</entry>
<entry>—</entry>
<entry>—</entry>
</row>
<row><entry>Johnson</entry>
<entry>Regular</entry>
<entry>—<fnoteref idrefs="tab1fne"></fnoteref>
</entry>
<entry>—</entry>
</row>
<row><entry>Catalan</entry>
<entry>Equivalent</entry>
<entry>—</entry>
<entry>—<fnoteref idrefs="tab1fnf"></fnoteref>
</entry>
</row>
<row><entry>Simplicial Frank–Kasper</entry>
<entry>Triangles</entry>
<entry>—</entry>
<entry>—</entry>
</row>
<row><entry>Fullerenes</entry>
<entry>Regular<fnoteref idrefs="tab1fng"></fnoteref>
</entry>
<entry>—</entry>
<entry>—</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f1.tif" id="fig1"><title>Carved-stone Platonic polyhedra found in Scotland and dated 2000 BC. Reproduced with authorization from the Ashmolean Museum, Oxford.</title>
</figure>
<p>A second family of high symmetry polyhedra is formed by the <it>Archimedean</it>
 solids.<citref idrefs="cit29 cit30">29,30</citref>
 This set of 15 shapes is characterized by having also regular polygons as faces and all the vertices equivalent, although more than one type of faces are present, at difference with the Platonic solids. A list of these polyhedra and their characteristics can be found in <tableref idrefs="tab2">Table 2</tableref>
. Only in two of them are all edges equivalent, while in general there is more than one type of edges. Due to the inequivalence of faces and edges (<tableref idrefs="tab1">Table 1</tableref>
), they are often called <it>semiregular</it>
 solids. These polyhedra belong to the same point groups as the Platonic solids, with which they bear close relationships. As an example, truncated Archimedean polyhedra can be generated from the Platonic ones by substituting each vertex by an equilateral triangle. An interesting property of two of the Archimedean polyhedra, the <it>snub cube</it>
 and the <it>snub dodecahedron</it>
 is that they are chiral and have therefore two enantiomers each. These polyhedra are called Archimedean because they were described by Archimedes (∼200 BC), even if we have only second hand references to his writings on this topic from Heron of Alexandria and Pappus of Alexandria.</p>
<table-entry id="tab2"><title>Characteristics of the Archimedean and Catalan polyhedra</title>
<table><tgroup cols="5"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<colspec colname="5"></colspec>
<thead><row><entry>Archimedean</entry>
<entry>Vertices</entry>
<entry>Faces<fnoteref idrefs="tab2fna"></fnoteref>
</entry>
<entry>Edges</entry>
<entry>Symmetry</entry>
</row>
</thead>
<tbody><row><entry>Truncated tetrahedron</entry>
<entry align="char" char=".">12</entry>
<entry>4<inf>3</inf>
									+ 4<inf>6</inf>
									= 8</entry>
<entry align="char" char=".">18</entry>
<entry><it>T</it>
<inf>d</inf>
</entry>
</row>
<row><entry>Cuboctahedron</entry>
<entry align="char" char=".">12</entry>
<entry>8<inf>3</inf>
									+ 6<inf>4</inf>
									= 14</entry>
<entry align="char" char=".">24</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Truncated cube</entry>
<entry align="char" char=".">24</entry>
<entry>8<inf>3</inf>
									+ 6<inf>8</inf>
									= 14</entry>
<entry align="char" char=".">36</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Truncated octahedron</entry>
<entry align="char" char=".">24</entry>
<entry>6<inf>4</inf>
									+ 8<inf>6</inf>
									= 14</entry>
<entry align="char" char=".">36</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Rhombicuboctahedron</entry>
<entry align="char" char=".">24</entry>
<entry>8<inf>3</inf>
									+ 6<inf>4</inf>
									+ 12<inf>4</inf>
									= 26</entry>
<entry align="char" char=".">48</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Snub cube<fnoteref idrefs="tab2fnb"></fnoteref>
</entry>
<entry align="char" char=".">24</entry>
<entry>24<inf>3</inf>
									+ 8<inf>3</inf>
									+ 6<inf>4</inf>
									= 38</entry>
<entry align="char" char=".">60</entry>
<entry><it>O</it>
</entry>
</row>
<row><entry>Icosidodecahedron</entry>
<entry align="char" char=".">30</entry>
<entry>20<inf>3</inf>
									+ 12<inf>5</inf>
									= 32</entry>
<entry align="char" char=".">60</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Truncated cuboctahedron</entry>
<entry align="char" char=".">48</entry>
<entry>12<inf>4</inf>
									+ 8<inf>6</inf>
									+ 6<inf>8</inf>
									= 26</entry>
<entry align="char" char=".">72</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Truncated dodecahedron</entry>
<entry align="char" char=".">60</entry>
<entry>20<inf>3</inf>
									+ 12<inf>10</inf>
									= 32</entry>
<entry align="char" char=".">90</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Truncated icosahedron</entry>
<entry align="char" char=".">60</entry>
<entry>12<inf>5</inf>
									+ 20<inf>6</inf>
									= 32</entry>
<entry align="char" char=".">90</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Rhombicosidodecahedron</entry>
<entry align="char" char=".">60</entry>
<entry>20<inf>3</inf>
									+ 30<inf>4</inf>
									+ 12<inf>5</inf>
									= 62</entry>
<entry align="char" char=".">120</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Snub dodecahedron<fnoteref idrefs="tab2fnb"></fnoteref>
</entry>
<entry align="char" char=".">60</entry>
<entry>60<inf>3</inf>
									+ 20<inf>3</inf>
									+ 12<inf>5</inf>
									= 92</entry>
<entry align="char" char=".">150</entry>
<entry><it>I</it>
</entry>
</row>
<row><entry>Truncated icosidodecahedron</entry>
<entry align="char" char=".">120</entry>
<entry>30<inf>4</inf>
									+ 20<inf>6</inf>
									+ 12<inf>10</inf>
									= 62</entry>
<entry align="char" char=".">180</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
</tbody>
</tgroup>
<tgroup cols="5"><colspec colname="1" colwidth="1.00*"></colspec>
<colspec colname="2" colwidth="1.00*"></colspec>
<colspec colname="3" colwidth="1.00*"></colspec>
<colspec colname="4" colwidth="1.00*"></colspec>
<colspec colname="5" colwidth="1.00*"></colspec>
<thead><row><entry valign="top">Catalan</entry>
<entry valign="top">Vertices<fnoteref idrefs="tab2fnc"></fnoteref>
</entry>
<entry valign="top">Faces</entry>
<entry valign="top">Edges</entry>
<entry valign="top">Symmetry</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="5" valign="top"><footnote id="tab2fna">The subindex indicates the order of the polygons that form the faces. For instance, 6<inf>4</inf>
										+ 8<inf>6</inf>
 stands for six tetragonal and eight hexagonal faces. Each <it>m</it>
<inf><it>n</it>
</inf>
 value corresponds to a set of symmetry-related faces.</footnote>
<footnote id="tab2fnb">There are two enantiomeric versions of this polyhedron.</footnote>
<footnote id="tab2fnc">Each number indicates a group of symmetry-related vertices.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Triakis tetrahedron</entry>
<entry>4 + 4 = 8</entry>
<entry>12<inf>3</inf>
</entry>
<entry align="char" char=".">18</entry>
<entry><it>T</it>
<inf>d</inf>
</entry>
</row>
<row><entry>Rhombic dodecahedron</entry>
<entry>8 + 6 = 14</entry>
<entry>12<inf>4</inf>
</entry>
<entry align="char" char=".">24</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Small triakis octahedron</entry>
<entry>8 + 6 = 14</entry>
<entry>24<inf>3</inf>
</entry>
<entry align="char" char=".">36</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Tetrakis hexahedron</entry>
<entry>6 + 8 = 14</entry>
<entry>24<inf>3</inf>
</entry>
<entry align="char" char=".">36</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Deltoidal icositetrahedron</entry>
<entry>8 + 6 + 12 = 26</entry>
<entry>24<inf>4</inf>
</entry>
<entry align="char" char=".">48</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Pentagonal icositetrahedron<fnoteref idrefs="tab2fnb"></fnoteref>
</entry>
<entry>24 + 8 + 6 = 38</entry>
<entry>24<inf>5</inf>
</entry>
<entry align="char" char=".">60</entry>
<entry><it>O</it>
</entry>
</row>
<row><entry>Rhombic triacontahedron</entry>
<entry>20 + 12 = 32</entry>
<entry>30<inf>4</inf>
</entry>
<entry align="char" char=".">60</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Disdyakis dodecahedron</entry>
<entry>12 + 8 + 6 = 26</entry>
<entry>48<inf>3</inf>
</entry>
<entry align="char" char=".">72</entry>
<entry><it>O</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Triakis icosahedron</entry>
<entry>20 + 12 = 32</entry>
<entry>60<inf>3</inf>
</entry>
<entry align="char" char=".">90</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Pentakis dodecahedron</entry>
<entry>12 + 20 = 32</entry>
<entry>60<inf>3</inf>
</entry>
<entry align="char" char=".">90</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Deltoidal hexecontahedron</entry>
<entry>20 + 30 + 12 = 62</entry>
<entry>60<inf>4</inf>
</entry>
<entry align="char" char=".">120</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Pentagonal hexecontahedron<fnoteref idrefs="tab2fnb"></fnoteref>
</entry>
<entry>20 + 12 = 32</entry>
<entry>60<inf>5</inf>
</entry>
<entry align="char" char=".">90</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
<row><entry>Disdyakis triacontahedron</entry>
<entry>30 + 20 + 12 = 62</entry>
<entry>120<inf>3</inf>
</entry>
<entry align="char" char=".">180</entry>
<entry><it>I</it>
<inf>h</inf>
</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<p>The knowledge of Platonic and Archimedean polyhedra was disseminated through the Arabic culture by means of translations made during the VIIIth and IXth centuries, among which the most outstanding one is the account of Abu'l–Wafa (Baghdad, 940–988).<citref idrefs="cit30">30</citref>
 Most of Archimedean polyhedra were incorporated into the world of Art during the Renaissance, except for the snub dodecahedron, that was described later by Kepler. Among them, we must mention Piero della Francesca and his opus <it>Libellus de quinque corporibus regularium</it>
, as well as his disciple Fra Luca Pacioli, who published <it>De Divina Proportione</it>
				(1509), a book in which a number of polyhedra and their mutual relationships are described in detail, accompanied by well known illustrations attributed to Leonardo (<figref idrefs="fig2">Fig. 2</figref>
). The revised edition of Albrecht Dürer's <it>Underweysung der Messung</it>
				(1538), contains the first known representation of the truncated cuboctahedron and of the snub cube, two Archimedean polyhedra. Even without being comprehensive, we cannot forget the work of Wenzel Jamnitzer, <it>Perspectiva Corporum Regularium</it>
				(1568), that presents a plethora of polyhedral shapes.<citref idrefs="cit31">31</citref>
 Somewhat later, at the beginning of the XVII century, appear the first known studies on polyhedra in China and Japan.<citref idrefs="cit32">32</citref>
</p>
<figure xsrc="b503582c-f2.tif" id="fig2"><title>Drawing of a truncated icosahedron attributed to Leonardo da Vinci that appears in <it>De Divina Proportione</it>
 of Luca Pacioli.</title>
</figure>
<p>Another family of semiregular polyhedra is that formed by the <it>n</it>
-gonal <it>prisms</it>
 and <it>antiprisms</it>
. While the prisms have 2<it>n</it>
 equivalent vertices and their faces are formed by two regular <it>n</it>
-gons and <it>n</it>
 squares, the corresponding antiprisms have 2<it>n</it>
 regular triangles instead of the squares. The so called <it>metaprisms</it>
 are figures formed by two parallel <it>n</it>
-gonal faces rotated with respect to each other by any angle that is not a multiple of π/<it>n</it>
				(even multiples would correspond to prisms, odd multiples to antiprisms). At difference with prisms and antiprisms, some faces of the metaprisms are non-regular polygons. All regular prisms belong to the <it>D</it>
<inf><it>n</it>
h</inf>
 point groups, the antiprisms to <it>D</it>
<inf><it>n</it>
d</inf>
 and the metaprisms to the disymmetric <it>D</it>
<inf><it>n</it>
</inf>
 point groups and are, therefore, chiral.<citref idrefs="cit33">33</citref>
</p>
<p>The families of polyhedra just discussed – Platonic, Archimedean, prisms, antiprisms and metaprisms – are all <it>isogonal</it>
, <it>i.e.</it>
, all the vertices of a given polyhedron are equivalent. However, there are no Platonic or Archimedean polyhedra having between 13 and 19 vertices, neither are there polyhedra with an odd number of vertices in these families, as graphically shown in <figref idrefs="fig3">Fig. 3</figref>
, where the solid circles represent the number of Platonic and Archimedean polyhedra with a given number of vertices. Furthermore, for some even numbers we have only one shape available, which seems a rather limited choice and represents a drawback for the use of polyhedra as stereochemical descriptors. A wider variety of shapes can be at hand if we consider two additional families, the Johnson and the Catalan polyhedra, to be discussed below. Suffices it to note at this point that the inclusion of such geometries offers a much higher flexibility for the stereochemical description of structures, as illustrated in <figref idrefs="fig3">Fig. 3</figref>
				(dashed line). To these we can add the planar polygons, the pyramids and bipyramids, the deltahedra, the fullerenes and the simplicial polyhedra. Although some of these geometries are well known by chemists, we will briefly describe here their main characteristics to facilitate comparisons in a systematic way.</p>
<figure xsrc="b503582c-f3.tif" id="fig3"><title>Number of available polyhedra with a given number of vertices considering only Platonic and Archimedean solids (dashed line) and including also the Catalan and Johnson polyhedra (solid line).</title>
</figure>
<p>The <it>Johnson polyhedra</it>
,<citref idrefs="cit30 cit34">30,34</citref>
 often referred to in short as J1–J92, are those polyhedra whose faces are all regular polygons and whose edges have all the same length, with the exception of the Platonic, Archimedean, prismatic and antiprismatic polyhedra. Zalgaller showed that there are only 92 such polyhedra, of which some chemical examples are listed in <tableref idrefs="tab3">Table 3</tableref>
, with a few of them schematically represented in <figref idrefs="fig4">Fig. 4</figref>
. A full list of the Johnson polyhedra and their characteristics (number of vertices, number and type of faces) is given as ESI.<fnoteref idrefs="fn1"></fnoteref>
 Among them we can find five couples of enantiomers of chiral polyhedra. Let us note here that there are two pyramids (tetragonal and pentagonal pyramids) and two bipyramids (trigonal and pentagonal bipyramids) among the Johnson polyhedra, the only ones that can have equilateral triangles as faces (the regular trigonal pyramid and tetragonal bipyramid are, in fact, the tetrahedron and the octahedron). In our small catalog of polyhedra (<tableref idrefs="tab1">Table 1</tableref>
) we might include a wider family of pyramids and bipyramids whose faces are isosceles triangles.</p>
<table-entry id="tab3"><title>Some Johnson polyhedra that describe the coordination sphere of a metal atom, the structure of a metallic cluster, a ligand polyhedron in a cluster, the metal framework in a supramolecular assembly or coordination polyhedra in alloys, ordered by increasing number of vertices</title>
<table><tgroup cols="5"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<colspec colname="5"></colspec>
<thead><row><entry colname="1"><it>V</it>
</entry>
<entry namest="2" nameend="3">Polyhedron</entry>
<entry colname="4">Example<fnoteref idrefs="tab3fna"></fnoteref>
</entry>
<entry colname="5">Ref.</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="5"><footnote id="tab3fna">In boldface the group of atoms that forms the corresponding polyhedron.</footnote>
<footnote id="tab3fnb">Also known as <it>anticuboctahedron</it>
.</footnote>
<footnote id="tab3fnc">L<sup>4</sup>
										= 2,6-diacetylpyridine-bis(2-pyridylhydrazone).</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>5</entry>
<entry>J1</entry>
<entry>Square pyramid</entry>
<entry>[Co<bo>H</bo>
<inf><bo>5</bo>
</inf>
]<sup>4−</sup>
</entry>
<entry><citref idrefs="cit35" position="baseline">35</citref>
</entry>
</row>
<row><entry></entry>
<entry>J12</entry>
<entry>Trigonal bipyramid</entry>
<entry>[Mn(<bo>C</bo>
O)<inf>5</inf>
]<sup>−</sup>
</entry>
<entry><citref idrefs="cit36" position="baseline">36</citref>
</entry>
</row>
<row><entry>6</entry>
<entry>J2</entry>
<entry>Pentagonal pyramid</entry>
<entry>[HgCl(L<sup>4</sup>
)]<sup>+</sup>
<fnoteref idrefs="tab3fnb"></fnoteref>
</entry>
<entry><citref idrefs="cit37" position="baseline">37</citref>
</entry>
</row>
<row><entry>7</entry>
<entry>J13</entry>
<entry>Pentagonal bipyramid</entry>
<entry>[Ni(<bo>N</bo>
<inf><bo>5</bo>
</inf>
-macrocycle)<bo>L</bo>
<inf><bo>2</bo>
</inf>
]</entry>
<entry><citref idrefs="cit38" position="baseline">38</citref>
</entry>
</row>
<row><entry></entry>
<entry>J49</entry>
<entry>Augmented trigonal prism</entry>
<entry>[W(<bo>C</bo>
NBu)<inf><bo>7</bo>
</inf>
]<sup>2+</sup>
</entry>
<entry><citref idrefs="cit39" position="baseline">39</citref>
</entry>
</row>
<row><entry>8</entry>
<entry>J14</entry>
<entry>Elongated trigonal bipyramid</entry>
<entry>MoFe<inf>7</inf>
 in nitrogenase cofactor</entry>
<entry><citref idrefs="cit40" position="baseline">40</citref>
</entry>
</row>
<row><entry></entry>
<entry>J26</entry>
<entry>Gyrobifastigium (<figref idrefs="fig8">Fig. 8</figref>
)</entry>
<entry>[<bo>Mn</bo>
<inf><bo>8</bo>
</inf>
Sb<inf>4</inf>
(μ-O)<inf>4</inf>
(μ-EtO)<inf>20</inf>
]</entry>
<entry><citref idrefs="cit41" position="baseline">41</citref>
</entry>
</row>
<row><entry></entry>
<entry>J50</entry>
<entry>Biaugmented trigonal prism</entry>
<entry>C in Fe<inf>3</inf>
C (cementite)</entry>
<entry></entry>
</row>
<row><entry></entry>
<entry>J84</entry>
<entry>Snub disphenoid</entry>
<entry>C in Sc<inf>4</inf>
C<inf>3</inf>
</entry>
<entry><citref idrefs="cit42" position="baseline">42</citref>
</entry>
</row>
<row><entry>9</entry>
<entry>J3</entry>
<entry>Triangular cupola (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>[P<bo>W</bo>
<inf><bo>9</bo>
</inf>
O<inf>28</inf>
Br<inf>6</inf>
]<sup>3−</sup>
, [<bo>Au</bo>
<inf><bo>9</bo>
</inf>
Te<inf>7</inf>
]<sup>5−</sup>
</entry>
<entry><citref idrefs="cit43 cit44" position="baseline">43, 44</citref>
</entry>
</row>
<row><entry></entry>
<entry>J10</entry>
<entry>Gyroelongated square pyramid (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>[<bo>Rh</bo>
<inf><bo>9</bo>
</inf>
P(CO)<inf>21</inf>
]<sup>2−</sup>
</entry>
<entry><citref idrefs="cit45" position="baseline">45</citref>
</entry>
</row>
<row><entry></entry>
<entry>J51</entry>
<entry>Triaugmented trigonal prism</entry>
<entry>[Ln(H<inf>2</inf>
<bo>O</bo>
)<inf><bo>9</bo>
</inf>
]<sup>3+</sup>
</entry>
<entry></entry>
</row>
<row><entry>10</entry>
<entry>J15</entry>
<entry>Elongated square bipyramid</entry>
<entry>[<bo>W</bo>
<inf><bo>10</bo>
</inf>
O<inf>32</inf>
]<sup>4−</sup>
</entry>
<entry><citref idrefs="cit46" position="baseline">46</citref>
</entry>
</row>
<row><entry></entry>
<entry>J17</entry>
<entry>Gyroelongated square bipyramid</entry>
<entry>[<bo>Rh</bo>
<inf><bo>10</bo>
</inf>
P(CO)<inf>22</inf>
]<sup>3−</sup>
</entry>
<entry><citref idrefs="cit47" position="baseline">47</citref>
</entry>
</row>
<row><entry></entry>
<entry>J86</entry>
<entry>Sphenocorona (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>[Nd(NO<inf>3</inf>
)<inf>3</inf>
(H<inf>2</inf>
O)<inf>2</inf>
(OPR<inf>3</inf>
)<inf>2</inf>
]; Mn in MnAl<inf>6</inf>
</entry>
<entry></entry>
</row>
<row><entry>11</entry>
<entry>J11</entry>
<entry>Gyroelongated pentagonal pyramid</entry>
<entry><it>nido</it>
-(<bo>B</bo>
<inf><bo>11</bo>
</inf>
H<inf>13</inf>
)<sup>2−</sup>
</entry>
<entry></entry>
</row>
<row><entry>12</entry>
<entry>J4</entry>
<entry>Square cupola</entry>
<entry>[<bo>V</bo>
<inf><bo>12</bo>
</inf>
O<inf>32</inf>
]<sup>4−</sup>
</entry>
<entry><citref idrefs="cit48" position="baseline">48</citref>
</entry>
</row>
<row><entry></entry>
<entry>J27</entry>
<entry>Trigonal orthobicupola<fnoteref idrefs="tab3fnc"></fnoteref>
									(<figref idrefs="fig18">Fig. 18</figref>
)</entry>
<entry>X<bo>M</bo>
<inf><bo>12</bo>
</inf>
O<inf>40</inf>
									(β-Keggin structure)</entry>
<entry></entry>
</row>
<row><entry>14</entry>
<entry>J91</entry>
<entry>Biluna birotunda (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>V<inf>14</inf>
 in [V<inf>18</inf>
O<inf>44</inf>
H<inf>2</inf>
]<sup>4−</sup>
</entry>
<entry><citref idrefs="cit49" position="baseline">49</citref>
</entry>
</row>
<row><entry>15</entry>
<entry>J22</entry>
<entry>Gyroelongated triangular cupola</entry>
<entry>(μ<inf>2</inf>
-O)<inf>15</inf>
 in [PW<inf>9</inf>
O<inf>28</inf>
Br<inf>6</inf>
]<sup>3−</sup>
</entry>
<entry><citref idrefs="cit43" position="baseline">43</citref>
</entry>
</row>
<row><entry></entry>
<entry>J57</entry>
<entry>Triaugmented hexagonal prism</entry>
<entry>S<inf>15</inf>
 in wurtzite</entry>
<entry></entry>
</row>
<row><entry>16</entry>
<entry>J28</entry>
<entry>Square orthobicupola</entry>
<entry>[Re<inf>4</inf>
H<inf>4</inf>
(<bo>C</bo>
O)<inf><bo>16</bo>
</inf>
]</entry>
<entry><citref idrefs="cit50" position="baseline">50</citref>
</entry>
</row>
<row><entry></entry>
<entry>J85</entry>
<entry>Snub square antiprism (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>C<inf>16</inf>
 in [Ni<inf>8</inf>
(μ<inf>6</inf>
-C)(μ<inf>3</inf>
-CO)<inf>8</inf>
(CO)<inf>8</inf>
]<sup>2−</sup>
</entry>
<entry><citref idrefs="cit51" position="baseline">51</citref>
</entry>
</row>
<row><entry>18</entry>
<entry>J35</entry>
<entry>Elongated trigonal orthobicupola</entry>
<entry>X<inf>2</inf>
<bo>M</bo>
<inf><bo>18</bo>
</inf>
O<inf>62</inf>
; [Os<inf>6</inf>
P(<bo>C</bo>
O)<inf>18</inf>
]<sup>−</sup>
</entry>
<entry><citref idrefs="cit52" position="baseline">52</citref>
</entry>
</row>
<row><entry></entry>
<entry>J36</entry>
<entry>Elongated trigonal gyrobicupola</entry>
<entry>X<inf>2</inf>
<bo>M</bo>
<inf><bo>18</bo>
</inf>
O<inf>62</inf>
									(Dawson structure)</entry>
<entry></entry>
</row>
<row><entry>20</entry>
<entry>J31</entry>
<entry>Pentagonal gyrobicupola (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>[<bo>Ni</bo>
<inf><bo>20</bo>
</inf>
(SeMe)<inf>10</inf>
Se<inf>12</inf>
]<sup>2−</sup>
</entry>
<entry><citref idrefs="cit53" position="baseline">53</citref>
</entry>
</row>
<row><entry>24</entry>
<entry>J37</entry>
<entry>Elongated tetragonal gyrobicuopla</entry>
<entry>(μ<inf>3</inf>
-<bo>O</bo>
)<inf>24</inf>
 in [V<inf>18</inf>
O<inf>42</inf>
]<sup>4−</sup>
</entry>
<entry><citref idrefs="cit54" position="baseline">54</citref>
</entry>
</row>
<row><entry>30</entry>
<entry>J38</entry>
<entry>Elongated pentagonal orthobicupola (<figref idrefs="fig4">Fig. 4</figref>
)</entry>
<entry>[P<inf>5</inf>
<bo>W</bo>
<inf><bo>30</bo>
</inf>
O<inf>110</inf>
]<sup>15−</sup>
									(Preyssler structure)</entry>
<entry></entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f4.tif" id="fig4"><title>Some metal skeletons with Johnson polyhedra (see <tableref idrefs="tab3">Table 3</tableref>
 for names of polyhedra and references): [PW<inf>9</inf>
O<inf>28</inf>
Br<inf>6</inf>
]<sup>3−</sup>
, [Rh<inf>9</inf>
P(CO)<inf>21</inf>
]<sup>2−</sup>
, Mn environment in MnAl<inf>6</inf>
, [V<inf>18</inf>
O<inf>44</inf>
H<inf>2</inf>
]<sup>4−</sup>
, C<inf>16</inf>
 in [Ni<inf>8</inf>
(μ<inf>6</inf>
-C)(μ<inf>3</inf>
-CO)<inf>8</inf>
(CO)<inf>8</inf>
]<sup>2−</sup>
, [Ni<inf>20</inf>
(SeMe)<inf>10</inf>
Se<inf>12</inf>
]<sup>2−</sup>
 and [P<inf>5</inf>
W<inf>30</inf>
O<inf>110</inf>
]<sup>15−</sup>
.</title>
</figure>
<p>The <it>Catalan polyhedra</it>
,<citref idrefs="cit55">55</citref>
 defined as duals of the Archimedean solids (see below for a definition of duality), have all their faces identical and more than one type of symmetry-related vertices (<tableref idrefs="tab1">Table 1</tableref>
). A distinct characteristic of the Catalan solids is that their faces are non-regular polygons. A classification criterion for polyhedra often used in Chemistry includes in the family of the <it>deltahedra</it>
 all those figures whose faces are formed by equilateral triangles. That family includes regular polyhedra such as the tetrahedron, the octahedron and the icosahedron, but also irregular solids such as the <it>snub disphenoid</it>
				(J84, also named <it>siamese dodecahedron</it>
) that has eight vertices and is very useful as a stereochemical descriptor for many transition metal octacoordinate compounds.<citref idrefs="cit56">56</citref>
 Since this classification criterion is ambiguous and shows some overlap with those families described above, we should refer to deltahedra only when there is no alternative description corresponding to one of the families in <tableref idrefs="tab1">Table 1</tableref>
.<ugraphic xsrc="b503582c-u1.tif" id="ugr1"></ugraphic>
</p>
<p>If we allow for non-regular faces, a wider variety of polyhedra can appear. Consider, for instance, those having triangular faces, often called <it>simplicial</it>
 polyhedra.<citref idrefs="cit57">57</citref>
 Special attention has been paid to simplicial polyhedra in which either five or six triangles meet at each vertex. If <it>V</it>
<inf><it>n</it>
</inf>
 is the number of vertices linking <it>n</it>
 edges, it can be shown that <it>V</it>
<inf>5</inf>
				= 12 in every case, hence the total number of vertices is <it>V</it>
				=
				<it>V</it>
<inf>6</inf>
				+ 12. The smaller members of this family (those with <it>V</it>
<inf>6</inf>
				= 2, 3 or 4) are known as <it>Frank</it>
–<it>Kasper</it>
 polyhedra and are commonly found as coordination polyhedra in intermetallic phases (<compoundref idrefs="chem1">1</compoundref>
).</p>
<p>A family of polyhedra that has attracted much attention from chemists in recent years is that of the <it>fullerenes</it>
, which have in common the presence of only hexagonal and pentagonal faces. It can be easily deduced from Euler's formula (<it>F</it>
				+
				<it>V</it>
				=
				<it>E</it>
				+ 2, where <it>F</it>
, <it>V</it>
 and <it>E</it>
 are the numbers of faces, vertices and edges, respectively) that a hypothetical polyhedron formed only by hexagons is not feasible. A nice application of such a principle to the analysis of the shape of radiolaria was given by D'Arcy Thompson in his classical book <it>On Growth and Form</it>
,<citref idrefs="cit58">58</citref>
</p>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>“No system of hexagons can enclose space; whether the hexagons be equal or unequal, regular or irregular, it is still under all circumstances mathematically impossible. Neither our reticulum plasmatique nor what seems to be the very perfection of hexagonal symmetry in Aulonia are as we are wont to conceive them; hexagons indeed predominate in both, but a certain number of facets are and must be other than hexagonal. If we look carefully at Carnoy's careful drawing we see that both pentagons and hexagons are shewn in his reticulum…”</it>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>A description of this family of polyhedra can be found in Wells’ classical treatise on Structural Inorganic Chemistry prior to the discovery of C<inf>60</inf>
 and the subsequent coinage of the name fullerene.<citref idrefs="cit59">59</citref>
 The application of Euler's formula to a polyhedron formed by <it>h</it>
 hexagonal and <it>p</it>
 pentagonal faces in which three polygons meet at each vertex tells us that there must be 12 pentagons but any number of hexagons may be present, the total number of faces being <it>F</it>
				= 12 +
				<it>h</it>
. Furthermore, the number of vertices and edges are fully determined by the number of hexagonal faces, according to the following relationships: <it>V</it>
				= 2<it>h</it>
				+ 20 and <it>E</it>
				= 30 + 3<it>h.</it>
 Thus, the family of fullerenes is formed by a variety of polyhedra with 12 pentagons and any number of hexagons, having even number of vertices, of 20 or more.<citref idrefs="cit59">59</citref>
 There are two particular cases among the fullerenes. One of them, the polyhedron with <it>h</it>
				= 0, formed only by pentagons, is nothing else than the dodecahedron. The other special case is the prototypic member of this family, a polyhedron with 60 vertices, that appears in the structure of C<inf>60</inf>
, which is the Archimedean truncated icosahedron. A few examples of fullerenes of increasing sizes are given in <tableref idrefs="tab4">Table 4</tableref>
, and the structures of some of them are shown in <figref idrefs="fig5">Fig. 5</figref>
, where the structure of C<inf>60</inf>
 is just the chemical realization of the truncated icosahedron drawn by Leonardo a few centuries earlier (<figref idrefs="fig2">Fig. 2</figref>
). Giant fullerenes (<it>e.g.</it>
, C<inf>1520</inf>
) and graphitic onions have been the object of a recent review.<citref idrefs="cit60">60</citref>
 In Nature, fullerenes can be found among radiolaria, as commented above and illustrated by the drawing of <it>Aulonia hexagona</it>
 in <figref idrefs="fig5">Fig. 5</figref>
, as well as in pollen grains of convolvullaceae. It is interesting to note also that the fullerene with two hexagons, found in the structure of A<inf>8</inf>
X<inf>46</inf>
 compounds (<tableref idrefs="tab4">Table 4</tableref>
), has been proposed by Weaire and Phelan to combine with dodecahedra to provide space-filling packing of bubbles.<citref idrefs="cit61">61</citref>
 Such a space-filling architecture, known as the clathrate II structure is presented by the NaZn<inf>13</inf>
 structure type, by the MTN zeotypes compounds,<citref idrefs="cit62">62</citref>
 and by the recently reported 3D assembly of tetrahedra of triads of Cr(<scp>iii</scp>
) octahedra.<citref idrefs="cit63">63</citref>
</p>
<table-entry id="tab4"><title>Some examples of fullerenes, listed according to their number of hexagonal faces (<it>h</it>
) and total vertices (<it>V</it>
)</title>
<table><tgroup cols="5"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<colspec colname="5"></colspec>
<thead><row><entry><it>h</it>
</entry>
<entry><it>V</it>
</entry>
<entry>Name</entry>
<entry>Example</entry>
<entry>Ref.</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="5"><footnote id="tab4fna">In order to fill out space the fullerenes are combined in these structures with pentagonal dodecahedra.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry align="char" char=".">0</entry>
<entry>20</entry>
<entry>Dodecahedron</entry>
<entry>Na@Si<inf>20</inf>
 in Na<inf>2</inf>
CsSi<inf>17</inf>
</entry>
<entry><citref idrefs="cit64" position="baseline">64</citref>
</entry>
</row>
<row><entry align="char" char=".">2</entry>
<entry>24</entry>
<entry>Clathrate I<fnoteref idrefs="tab4fna"></fnoteref>
</entry>
<entry>Cs1 in Cs<inf>8</inf>
Sn<inf>46</inf>
</entry>
<entry><citref idrefs="cit65" position="baseline">65</citref>
</entry>
</row>
<row><entry align="char" char="."></entry>
<entry></entry>
<entry></entry>
<entry>Na@Si<inf>24</inf>
 in Na<inf>8</inf>
Si<inf>46</inf>
</entry>
<entry><citref idrefs="cit66" position="baseline">66</citref>
</entry>
</row>
<row><entry align="char" char=".">4</entry>
<entry>28</entry>
<entry>Clathrate II<fnoteref idrefs="tab4fna"></fnoteref>
</entry>
<entry>Na@Si<inf>28</inf>
 in Na<inf>2</inf>
CsSi<inf>17</inf>
</entry>
<entry><citref idrefs="cit64" position="baseline">64</citref>
</entry>
</row>
<row><entry align="char" char=".">20</entry>
<entry>60</entry>
<entry>Truncated icosahedron</entry>
<entry>C<inf>60</inf>
; M<inf>60</inf>
 in Na<inf>13</inf>
(Cd<inf>1−<it>x</it>
</inf>
Tl<inf><it>x</it>
</inf>
)<inf>27</inf>
</entry>
<entry><citref idrefs="cit67 cit68">67,68</citref>
</entry>
</row>
<row><entry align="char" char=".">25</entry>
<entry>70</entry>
<entry></entry>
<entry>C<inf>70</inf>
; In<inf>70</inf>
 in Na<inf>172</inf>
In<inf>197</inf>
M<inf>2</inf>
</entry>
<entry><citref idrefs="cit69 cit70">69,70</citref>
</entry>
</row>
<row><entry align="char" char=".">27</entry>
<entry>74</entry>
<entry></entry>
<entry>Ba@C<inf>74</inf>
</entry>
<entry><citref idrefs="cit71" position="baseline">71</citref>
</entry>
</row>
<row><entry align="char" char=".">28</entry>
<entry>76</entry>
<entry></entry>
<entry>C<inf>76</inf>
</entry>
<entry><citref idrefs="cit72" position="baseline">72</citref>
</entry>
</row>
<row><entry align="char" char=".">29</entry>
<entry>78</entry>
<entry></entry>
<entry>In<inf>78</inf>
 in Na<inf>172</inf>
In<inf>197</inf>
M<inf>2</inf>
									(M = Ni, Pd, Pt)</entry>
<entry><citref idrefs="cit70" position="baseline">70</citref>
</entry>
</row>
<row><entry align="char" char=".">30</entry>
<entry>80</entry>
<entry></entry>
<entry>Sc<inf>3</inf>
N@C<inf>80</inf>
</entry>
<entry><citref idrefs="cit73" position="baseline">73</citref>
</entry>
</row>
<row><entry align="char" char=".">31</entry>
<entry>82</entry>
<entry></entry>
<entry>Er@C<inf>82</inf>
</entry>
<entry><citref idrefs="cit74" position="baseline">74</citref>
</entry>
</row>
<row><entry align="char" char=".">32</entry>
<entry>84</entry>
<entry></entry>
<entry>Sc<inf>3</inf>
@C<inf>84</inf>
; Al<inf>84</inf>
 in Sr<inf>3</inf>
Bi<inf>24+δ</inf>
Al<inf>48</inf>
O<inf>141+3δ/2</inf>
</entry>
<entry><citref idrefs="cit75" position="baseline">75</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f5.tif" id="fig5"><title>The structures of some fullerenes, from left to right: Si<inf>20</inf>
, Si<inf>24</inf>
, Ge<inf>28</inf>
, C<inf>60</inf>
, C<inf>70</inf>
, C<inf>80</inf>
, Al<inf>84</inf>
, together with the classical Häckel drawing of the radiolaria <it>Aulonia hexagona</it>
 and an electron micrograph of a pollen grain of a convulvullaceae with a diameter of about 120 μm (courtesy of L. Howard, Electron Microscope Facility, Dartmouth College).</title>
</figure>
<p>The different families of convex polyhedra described in this section, as well as their main geometrical characteristics, have been summarized in <tableref idrefs="tab1">Table 1</tableref>
. Although care have been taken to use a classification of polyhedra as systematic and unequivocal as possible, we must recall that some overlap exists. In those cases, we might consider the particular polyhedron to belong to the most regular family, <it>i.e.</it>
, that occupying the highest position in <tableref idrefs="tab1">Table 1</tableref>
.</p>
</section>
<section><title>Simple molecular polyhedra</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>A ti, mar de los sueños angulares,</it>
</entry>
</row>
<row valign="top"><entry><it>flor de las cinco formas regulares,</it>
</entry>
</row>
<row valign="top"><entry><it>dodecaedro azul, arco sonoro.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Rafael Alberti, <it>A la divina proporción</it>
<citref idrefs="cit76">76</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>[To you, sea of angulated dreams, </entry>
</row>
<row valign="top"><entry>flower of the five regular forms, </entry>
</row>
<row><entry>blue dodecahedron, sonorous arc.]<citref idrefs="cit77">77</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>There are different ways in which we can find a simple polyhedral arrangement of atoms within a molecule. We may distinguish (a)
				<it>bonded polyhedra</it>, in which each vertex is occupied by an atom and whose edges correspond to chemical bonds between them (eventually including electron deficient bonds, as found in boranes, carboranes or metal clusters); (b)
				<it>edge</it>
-<it>deficient polyhedra</it>, in which we associate the vertices to atoms with only part of the edges corresponding to chemical bonds, and (c)
				<it>coordination polyhedra</it>
, whose vertices are occupied by atoms bonded to a single central atom at the center of the polyhedron, but without chemical bonds directly joining the vertices.</p>
<subsect1><title>Bonded polyhedra</title>
<p>Among this family of polyhedral structures we find the boranes, the polyhedranes (or polyhedral hydrocarbons) and the metal clusters. The fascination for the simplicity associated to a high symmetry made the polyhedral hydrocarbons long-desired targets of synthetic strategies,<citref idrefs="cit78">78</citref>
 from which a handful have been structurally characterized today, as discussed above. Cubanes of other group 14 elements (<it>i.e.</it>
, with Si<inf>8</inf>
, Ge<inf>8</inf>
 or Sn<inf>8</inf>
 cores) are also well known. <it>Buckminsterfullerene</it>
					(C<inf>60</inf>
) presents the structure of an Archimedean polyhedron, the <it>truncated icosahedron</it>
, and the members of the continuously expanding family of fullerenes (see above and <figref idrefs="fig5">Fig. 5</figref>
) can also be considered as bonded polyhedra. The octahedron and the icosahedron are missing among the Platonic alkanes, but we do not expect them to be stable as neutral molecules, according to Wade's rules. The corresponding skeletons, though, are known among the anionic boranes, <it>i.e.</it>
, the octahedral (B<inf>6</inf>
H<inf>6</inf>
)<sup>2−</sup>
 and the icosahedral (B<inf>12</inf>
H<inf>12</inf>
)<sup>2−</sup>
. I will not insist on the structural chemistry of boranes and carboranes, one of the areas of Chemistry in which regular polyhedral structures are most common and have a widest literature. Many analogies can be established between boranes and transition metal clusters, the latter presenting numerous examples<citref idrefs="cit79">79</citref>
 of tetrahedral, octahedral, cubic, icosahedral and poly-icosahedral structures.<citref idrefs="cit80">80</citref>
 Some examples of clusters with metallic skeletons corresponding to Johnson polyhedra have been given above (<tableref idrefs="tab3">Table 3</tableref>
), and two of them, [Rh<inf>9</inf>
P(CO)<inf>21</inf>
]<sup>2−</sup>
 and [Ni<inf>20</inf>
(SeMe)<inf>10</inf>
Se<inf>12</inf>
]<sup>2−</sup>
, are schematically represented in <figref idrefs="fig4">Fig. 4</figref>
.</p>
<p>Among the bonded polyhedra we find also octahedrane, C<inf>12</inf>
H<inf>12</inf>
, which corresponds to a non-regular octahedron of ideal symmetry <it>D</it>
<inf>3d</inf>
 formed by two triangular and six pentagonal faces. Such a polyhedron (<figref idrefs="fig6">Fig. 6</figref>
) was represented by Albrecht Dürer in 1514 in his enigmatic and well known woodcut <it>Melancholia</it>
, although in its molecular counterpart the faces are not strictly planar. Other versions of this <it>melancholyhedron</it>
 are found for a shell of twelve As atoms around Ni (at a distance of 4.3 Å) in the solid state structure of NiAs, and in the lowest energy form predicted for hypothetical N<inf>12</inf>
 and C<inf>12</inf>
H<inf>12</inf>
 molecules.<citref idrefs="cit81">81</citref>
 There is still another hypothetical octahedron mentioned by Wells,<citref idrefs="cit59">59</citref>
 formed by four tetragonal and four pentagonal faces (<it>D</it>
<inf>2d</inf>
 symmetry), although it does not seem to have been realized in the molecular world so far. Other known polyhedranes are prismane (C<inf>6</inf>
H<inf>6</inf>
, an isomer of benzene, and its Si, Ge and Sn analogues), and pentaprismane (C<inf>10</inf>
H<inf>10</inf>
), with trigonal and pentagonal prismatic structures, respectively.</p>
<figure xsrc="b503582c-f6.tif" id="fig6"><title>Comparison of the polyhedron included by Dürer in <it>Melancholia</it>
						(reproduced with authorization, © Copyright The trustees of The British Museum) with the As<inf>12</inf>
 shell at 4.3 Å from Ni in the solid state structure of NiAs.</title>
</figure>
<p>The Frank–Kasper polyhedron with <it>V</it>
<inf>6</inf>
					= 2 can be found, <it>e.g.</it>
, in the Cr<inf>10</inf>
Si<inf>4</inf>
 coordination polyhedron of Cr in Cr<inf>3</inf>
Si, and can alternatively be described as a bicapped hexagonal antiprism (more precisely, a gyroelongated hexagonal bipyramid). The 16-vertex polyhedron (<it>V</it>
<inf>6</inf>
					= 4), also known as the <it>Friauf</it>
 polyhedron, appears in intermetallic phases, such as MgCu<inf>2</inf>
, where each Mg atom is surrounded by a Cu<inf>12</inf>
Mg<inf>4</inf>
 coordination sphere (<figref idrefs="fig7">Fig. 7</figref>
), as well as in Mn<inf>16</inf>
 clusters in the structure of α-Mn. Frank–Kasper polyhedra, though, are rarely found in the molecular literature, maybe because these are not among the panoply of polyhedra that chemists use to describe molecular shapes. However, the same 16-vertex Frank–Kasper polyhedron can be recognized in the oxo-bridged V<inf>16</inf>
 group found in a polyoxovanadate.<citref idrefs="cit82">82</citref>
 We will see in a subsequent section that this polyhedron can be described in an alternative, more symmetric way.</p>
<figure xsrc="b503582c-f7.tif" id="fig7"><title>The 16-vertex Frank–Kasper polyhedron (also known as the Friauf polyhedron) corresponding to the Cu<inf>12</inf>
Mg<inf>4</inf>
 group around a Mg atom in MgCu<inf>2</inf>
						(left) and its decomposition as a compound of a Cu<inf>12</inf>
 truncated tetrahedron and a Mg<inf>4</inf>
 tetrahedron (right).</title>
</figure>
</subsect1>
<subsect1><title>Edge-deficient polyhedra</title>
<p>From a chemical point of view, an interesting property of a polyhedron is the number of edges than can be omitted without it falling apart. Let us consider as an example a Johnson polyhedron so far neglected by chemists, the <it>gyrobifastigium</it>
					(J26, <figref idrefs="fig8">Fig. 8</figref>
), that can be described as an eight vertex octahedron with squares and triangles as faces.<citref idrefs="cit83">83</citref>
 This polyhedron can be found in the As<inf>4</inf>
S<inf>4</inf>
 molecules of realgar and related species, such as P<inf>4</inf>
(NR)<inf>4</inf>
, (Te<inf>4</inf>
S<inf>4</inf>
)<sup>2+</sup>
, Sb<inf>4</inf>
(SbR)<inf>4</inf>
, (RSi)<inf>4</inf>
S<inf>4</inf>
, (Sn<inf>8</inf>
)<sup>4−</sup>
, or the derivatives of tricyclo[3.3.0.0<sup>3,7</sup>
]octane, (RC)<inf>4</inf>
(R<inf>2</inf>
C)<inf>4</inf>
.<citref idrefs="cit84">84</citref>
 In all these molecules we can easily recognize the shape of the gyrobifastigium, even if only ten of the fourteen edges correspond to chemical bonds (<figref idrefs="fig7">Fig. 7</figref>
). The same polyhedron appears in <it>cunneane</it>
					(<figref idrefs="fig8">Fig. 8</figref>
), in which now twelve of the fourteen edges are spanned by carbon–carbon bonds.</p>
<figure xsrc="b503582c-f8.tif" id="fig8"><title>The gyrobifastigium (J26) formed by fusing together two trigonal prisms through square faces, together with two molecular representatives of such a polyhedron: the As<inf>4</inf>
S<inf>4</inf>
 molecule in realgar and the carbonaceous skeleton of octamethylcunnenane (C<inf>8</inf>
Me<inf>8</inf>
).</title>
</figure>
</subsect1>
<subsect1><title>Coordination polyhedra</title>
<p>Transition metals and rare earths are the elements that present a wider variety of coordination polyhedra, not only because of the variable number of metal–ligand bonds they can form, but also because of the variety of coordination geometries that a metal can yield with a given number of ligands.<citref idrefs="cit34">34</citref> In this context there are some key questions that we ask ourselves very often, and for which we have only partial responses: For a given metal ion and a certain ligand, (1) which is the preferred coordination number?
					(2) Which spatial arrangement will those ligands present around the metal?
					(3) What is the effect that the electron configuration of the metal has on the stereochemical preferences?
					(4) How and when can the ligands impose a certain stereochemistry around the metal?</p>
<p>The polyhedra needed to describe the coordination environment of metal atoms is limited by their coordination numbers, usually between four and eight, and rarely larger than 12. For that reason the Archimedean polyhedra are of little use in coordination chemistry. On the other hand, the Platonic polyhedra, the prisms and the bipyramids all have even number of vertices and therefore we must recourse to Johnson polyhedra for the stereochemical description of metals with odd coordination numbers. Even for some even coordination numbers (<it>e.g.</it>
, 8 and 10), the regular and semiregular polyhedra do not adequately describe a significant portion of the experimental structures and one has to refer to polyhedra such as the eight vertex <it>triangular dodecahedron</it>
					(also called <it>siamese dodecahedron</it>
 and related to the <it>snub disphenoid</it>
, a Johnson polyhedron, J84) or the ten vertex <it>sphenocorona</it>
					(J86) that describes the coordination sphere of, <it>e.g.</it>
, the As atom in KLi<inf>8</inf>
As or that of Nd in [Nd(NO<inf>3</inf>
)<inf>3</inf>
(H<inf>2</inf>
O)<inf>2</inf>
(OPR<inf>3</inf>
)<inf>2</inf>
].<citref idrefs="cit85">85</citref>
 Some Johnson polyhedra that can be used to describe the stereochemistry of coordination spheres of metal atoms are presented in <tableref idrefs="tab3">Table 3</tableref>
.</p>
</subsect1>
</section>
<section><title>Compound polyhedra</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>… e delectarassi cõ varie questione de secretissima scientia.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Luca Pacioli, <it>De divina proportione</it>
								(1509)</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>[… and you will be delighted with several issues of highly secret science.]</entry>
</row>
</tbody>
</tgroup>
</table>
<p>A <it>compound polyhedron</it>
 is a figure that is obtained by superimposing two simple polyhedra centered at the same point. For instance, if we superimpose two tetrahedra (in blue and red in <compoundref idrefs="chem2">2</compoundref>
) and join their vertices to form a single polyhedron, several figures arise, depending on the relative sizes of the composing tetrahedra. At some specific size ratios, compound polyhedra with some degree of regularity appear. In the case of the compound of two tetrahedra, we may obtain a 12-faced star deltahedron (whose faces are equilateral triangles, <compoundref idrefs="chem3a">3a</compoundref>
), a triakis tetrahedron (<compoundref idrefs="chem3b">3b</compoundref>
, a Catalan polyhedron) or a cube (when the two tetrahedra have the same size, <compoundref idrefs="chem3c">3c</compoundref>
). Note that other regular polyhedra can be described as compounds of simpler polyhedra, as in the case of a dodecahedron, which can be built up as a compound of five tetrahedra (<figref idrefs="fig9">Fig. 9</figref>
). Examples of compounds of two Platonic polyhedra are presented in <tableref idrefs="tab5">Table 5</tableref>
. There, it can be seen that the case of the cube just commented is an exceptional one, since it is the only Platonic polyhedron that can be formed as a compound of two Platonic polyhedra.</p>
<table-entry id="tab5"><title>Compound polyhedra built up from a Platonic solid and its dual, ordered according to increasing relative size of the dual</title>
<table><tgroup cols="3"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<thead><row><entry>Polyhedron (dual)</entry>
<entry>Compound</entry>
<entry>Family<fnoteref idrefs="tab5fna"></fnoteref>
</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="3"><footnote id="tab5fna">D = Deltahedron, C = Catalan, P = Platonic, KP = Kepler–Poinsot.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Tetrahedron (tetrahedron)</entry>
<entry>Triakis tetrahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>Cube</entry>
<entry>P</entry>
</row>
<row><entry></entry>
<entry>12-Faced star deltahedron</entry>
<entry>D</entry>
</row>
<row><entry namest="1" nameend="3"> </entry>
</row>
<row><entry>Octahedron (cube)</entry>
<entry>Triakis octahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>Stella octangula</entry>
<entry>D</entry>
</row>
<row><entry></entry>
<entry>Tetrakis hexahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>Rhombic dodecahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>24-Faced star deltahedron</entry>
<entry>D</entry>
</row>
<row><entry namest="1" nameend="3"> </entry>
</row>
<row><entry>Icosahedron (dodecahedron)</entry>
<entry>Small stellated dodecahedron</entry>
<entry>KP</entry>
</row>
<row><entry></entry>
<entry>60-Faced star deltahedron</entry>
<entry>D</entry>
</row>
<row><entry></entry>
<entry>Pentakis dodecahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>Great dodecahedron</entry>
<entry>KP</entry>
</row>
<row><entry></entry>
<entry>Triakis Icosahedron</entry>
<entry>C</entry>
</row>
<row><entry></entry>
<entry>60-Faced star deltahedron</entry>
<entry>D</entry>
</row>
<row><entry></entry>
<entry>Great stellated dodecahedron</entry>
<entry>KP</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f9.tif" id="fig9"><title>Icosahedron made of five interpenetrating tetrahedra.</title>
</figure>
<p>Compound polyhedra can be found in Architecture, <it>e.g.</it>
, in the towers of the <it>Sagrada Familia</it>
 in Barcelona, designed by Gaudí, in which a non-regular truncated octahedron has been obtained as a compound of a cube and an octahedron.<citref idrefs="cit87">87</citref>
 Also molecular and crystal architecture make use of compound polyhedra. To illustrate how the use of compound polyhedra can simplify the description of apparently complex shapes, let us focus on a Friauf polyhedron, the Frank–Kasper polyhedron with 16 vertices that represents the Cu<inf>12</inf>
Mg<inf>4</inf>
 coordination sphere of a Mg atom in MgCu<inf>2</inf>
				(<figref idrefs="fig7">Fig. 7</figref>
, left). Such a figure can be alternatively described as a compound of one Platonic and one Archimedean polyhedron of the same symmetry: a Mg<inf>4</inf>
 tetrahedron and a Cu<inf>12</inf>
 truncated tetrahedron, (<figref idrefs="fig7">Fig. 7</figref>
, right), a description that is substantiated by the corresponding shape measures that are zero within crystallographic accuracy. Another chemical example is furnished by the shell of 14 Re atoms at distances of 6.5–7.5 Å from a given Re atom in the ReO<inf>3</inf>
 structure: the eight atoms at 6.5 Å appear forming a cube, the six atoms at 7.5 Å form an octahedron, and combined they form a rhombic dodecahedron, a Catalan solid. A further chemical example discussed below is the rhombic dodecahedron (<figref idrefs="fig11">Fig. 11(c)</figref>
) that can be organized as a compound of a cube and an octahedron (<tableref idrefs="tab5">Table 5</tableref>
).<ugraphic xsrc="b503582c-u2.tif" id="ugr2"></ugraphic>
</p>
<p>The symmetry of a compound of two polyhedra is that of the common subgroup with maximum symmetry. Thus, the compound of a tetrahedron and an octahedron belongs to the <it>T</it>
<inf>d</inf>
 point group (a subgroup of the octahedral group <it>O</it>
<inf>h</inf>
) whereas the compound of an octahedron (or a cube) and an icosahedron (or a dodecahedron) reduces the symmetry of the compound to that of a subgroup common to <it>O</it>
<inf>h</inf>
 and <it>T</it>
<inf>d</inf>
, the <it>T</it>
<inf>h</inf>
 or <it>D</it>
<inf>3d</inf>
 point groups (the hierarchical group–subgroup relationships in these cases<citref idrefs="cit88">88</citref>
 are shown in <compoundref idrefs="chem4">4</compoundref>
). In a similar way, compounds of polyhedra with icosahedral and tetrahedral symmetries only retain the symmetry operations of a common subgroup <it>T</it>
 or <it>D</it>
<inf>2d</inf>
.<ugraphic xsrc="b503582c-u3.tif" id="ugr3"></ugraphic>
</p>
<p>Compound polyhedra can also be found in Organic Chemistry. Consider, as an example, the adamantane molecule. It can be considered as formed by two groups of carbon atoms, four in the shape of a tetrahedron and six arranged in an octahedron (<figref idrefs="fig10">Fig. 10</figref>
), as in the compound schematized in <compoundref idrefs="chem5">5</compoundref>
. Such a description is fully supported by the shape measures of those two sets of carbon atoms relative to the tetrahedron and the octahedron, taken from the experimental structural data. The reason why we do not usually think on adamantane that way is because the chemical bonds are the links between the two types of polyhedral vertices rather than their edges. Probably the coexistence of octahedron and tetrahedron in one and the same molecule has to do with the fact that the tetrahedral point group is a subgroup of the octahedral one (<compoundref idrefs="chem4">4</compoundref>
). A corollary of such a geometrical description of adamantane is that, adequately substituted, it can be used as a building block for tridimensional supramolecular lattices, as elegantly shown by O'Keeffe, Yaghi and co-workers,<citref idrefs="cit89">89</citref>
 by connecting the tetra-anion of the adamantane-1,3,5,7-tetracarboxylic acid with Cu<sup>II</sup>
 ions. This metallic ion tends to form dinuclear units with four carboxylate groups acting as bridges, as in the paradigmatic structure of copper acetate. Therefore, each adamantane tetrahedrally connects four such dinuclear blocks, each of which carries four adamantanes, resulting in a diamond like network with large voids. In contrast, we are not aware<citref idrefs="cit90">90</citref>
 of the use of adamantane substituted at its secondary carbon atoms as an octahedral supramolecular building block. An inorganic example of adamantanoid structures is provided by P<inf>4</inf>
O<inf>6</inf>
, a geometric compound of a P<inf>4</inf>
 tetrahedron and an O<inf>6</inf>
 octahedron. A similar relationship can be found in the T3 supertetrahedral structures to be discussed below.</p>
<figure xsrc="b503582c-f10.tif" id="fig10"><title>Adamantane as a compound of a tetrahedron and an octahedron: (a) carbonaceous skeleton (tertiary and secondary carbon atoms in red and black, respectively); (b) tetrahedral arrangement of the tertiary carbon atoms and (c) octahedral arrangement of the secondary carbon atoms.</title>
</figure>
</section>
<section><title>Relationships between polyhedra from geometry and chemistry</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>Octaedron cum cubo, Icosaedron cum Dodecaedro permutant numerum basium & angulorum. Nam Cubi bases & Octaedri anguli sunt sex, illius anguli & huius bases octo. Sic Dodecaedri bases & Icosaedri anguli sunt utrinq; duodecim: vicidim illius anguli & huius bases sunt viginti.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>J. Kepler, <it>Mysterium Cosmographicum</it>
								(1597)</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[The octahedron interchanges with the cube, the icosahedron with the dodecahedron, its number of bases and angles. For the cube has six bases and the icosahedron six vertices; the former eight vertices, the latter eight bases. Similarly the bases of the dodecahedron and the vertices of the icosahedron are twelve in each case: correspondingly the vertices of the former and the bases of the latter are twenty.]<citref idrefs="cit86">86</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>So far we have mostly considered families of polyhedra in which all vertices are occupied by the same type of atoms, even if in some cases they may not be all equivalent by symmetry. In this section we will consider how different sets of atoms can be organized in different polyhedra that can combine to give either a more complex polyhedron or nested polyhedra that are related by some clear cut geometrical and symmetry relationship. We will attempt also to show how those relationships can be chemically established through different types of chemical bonding arrangements, and also how they can be found in a variety of molecular and solid state structures. The ultimate goal is to provide the reader with systematic and efficient tools to describe complex structures in terms of combinations of the simplest possible polyhedra.</p>
<subsect1><title>Duality relationships and face augmentation</title>
<p>The figure generated by placing a point at the center of each face of a polyhedron is said to be its <it>dual</it>
					(or reciprocal), and its face centers have the same angular coordinates as the vertices of the original figure. Hence, duality is a reflexive operation that consists in exchanging faces and vertices. Among the Platonic solids, the cube and the octahedron are duals of each other (<compoundref idrefs="chem6">6</compoundref>
), so are the icosahedron and the dodecahedron (<compoundref idrefs="chem7">7</compoundref>
), while the dual of a tetrahedron is just another tetrahedron (<compoundref idrefs="chem8">8</compoundref>
). These relationships were clearly described by Luca Pacioli in his book <it>De divina proportione</it>
<citref idrefs="cit91">91</citref>
 and were known to Kepler, as can be deduced from the quotation that opens this section. An important property of a pair of dual polyhedra is that both possess the same symmetry and, according to Euler's formula, they must have the same number of edges. We will not go into much detail about the duals of all polyhedra of chemical interest, but let us just mention that duals of polyhedra of a given family (<it>e.g.</it>
, the Archimedean solids) all belong to the same family (the Catalan solids in this case). The Catalan solids are precisely defined as the duals of the Archimedean ones, whereas the trapezohedra are the duals of the antiprisms and other families of polyhedra are related through duality relationships as summarized in <tableref idrefs="tab6">Table 6</tableref>
.</p>
<table-entry id="tab6"><title>Some of the most relevant duality relationships grouped by families of polyhedra</title>
<table><tgroup cols="3"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<tbody><row><entry>Platonic</entry>
<entry align="char">↔</entry>
<entry>Platonic</entry>
</row>
<row><entry> 
										  Tetrahedron</entry>
<entry></entry>
<entry> 
										  Tetrahedron</entry>
</row>
<row><entry> 
										  Cube</entry>
<entry></entry>
<entry> 
										  Octahedron</entry>
</row>
<row><entry> 
										  Dodecahedron</entry>
<entry></entry>
<entry> 
										  Icosahedron</entry>
</row>
<row><entry> </entry>
<entry align="char"></entry>
<entry></entry>
</row>
<row><entry>Archimedean</entry>
<entry align="char">↔</entry>
<entry>Catalan</entry>
</row>
<row><entry> 
										  Truncated tetrahedron</entry>
<entry></entry>
<entry> 
										  Triakis tetrahedron</entry>
</row>
<row><entry> 
										  Cuboctahedron</entry>
<entry></entry>
<entry> 
										  Rhombic dodecahedron</entry>
</row>
<row><entry> 
										  Small triakis octahedron</entry>
<entry></entry>
<entry> 
										  Truncated cube</entry>
</row>
<row><entry> 
										  Truncated octahedron</entry>
<entry></entry>
<entry> 
										  Tetrakis hexahedron</entry>
</row>
<row><entry> 
										  Small rhombicuboctahedron</entry>
<entry></entry>
<entry> 
										  Deltoidal icositetrahedron</entry>
</row>
<row><entry> 
										  Truncated cuboctahedron</entry>
<entry></entry>
<entry> 
										  Disdyakis dodecahedron</entry>
</row>
<row><entry> 
										  Icosidodecahedron</entry>
<entry></entry>
<entry> 
										  Rhombic triacontahedron</entry>
</row>
<row><entry> 
										  Snub dodecahedron</entry>
<entry></entry>
<entry> 
										  Pentagonal hexecontahedron</entry>
</row>
<row><entry> 
										  Truncated dodecahedron</entry>
<entry></entry>
<entry> 
										  Triakis icosahedron</entry>
</row>
<row><entry> 
										  Truncated icosahedron</entry>
<entry></entry>
<entry> 
										  Pentakis dodecahedron</entry>
</row>
<row><entry> 
										  Snub cube</entry>
<entry></entry>
<entry> 
										  Pentagonal icositetrahedron</entry>
</row>
<row><entry> 
										  Truncated icosidodecahedron</entry>
<entry></entry>
<entry> 
										  Disdyakis triacontahedron</entry>
</row>
<row><entry> 
										  Small rhombicosidodecahedron</entry>
<entry></entry>
<entry> 
										  Deltoidal hexecontahedron</entry>
</row>
<row><entry> </entry>
<entry align="char"></entry>
<entry></entry>
</row>
<row><entry>Prisms</entry>
<entry align="char">↔</entry>
<entry>Bipyramids</entry>
</row>
<row><entry> </entry>
<entry align="char"></entry>
<entry></entry>
</row>
<row><entry>Antiprisms</entry>
<entry align="char">↔</entry>
<entry>Trapezohedra</entry>
</row>
<row><entry> </entry>
<entry align="char"></entry>
<entry></entry>
</row>
<row><entry>Pyramids</entry>
<entry align="char">↔</entry>
<entry>Pyramids</entry>
</row>
<row><entry> </entry>
<entry align="char"></entry>
<entry></entry>
</row>
<row><entry>Fullerenes</entry>
<entry align="char">↔</entry>
<entry>Simplicial (Frank–Kaspar)</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<p><ugraphic xsrc="b503582c-u4.tif" id="ugr4"></ugraphic>
<it>Face augmentation</it>
					(or face capping, or face cumulation) is the operation of adding a vertex on top of each face of a polyhedron. In general, the new vertex can be at any distance from the original face, but at specific distances the compound polyhedron that results belong to one of the special families characterized by a high degree of regularity (<tableref idrefs="tab5">Table 5</tableref>
): the Catalan or the Kepler–Poinsot solids, and some deltahedra. The polyhedra resulting from face-augmentation were termed by Kepler <it>echinus</it>
					(the Latin for hedgehog or sea urchin). The result of face augmentation is a compound formed by two nested polyhedra: the original (inscribed) polyhedron and its circumscribed dual. It is worth noting here that trigonal face augmentation and its reciprocal operation, trigonal truncation, generate close packed structures. Chemically, a polyhedron can be face-augmented by the addition of atoms bridging the <it>n</it>
-gonal faces in a μ<inf><it>n</it>
</inf>
 bonding mode. Two examples are shown in <figref idrefs="fig11">Fig. 11</figref>
. In one of them, [W<inf>6</inf>
S<inf>8</inf>
(PEt<inf>3</inf>
)<inf>6</inf>
], the μ<inf>3</inf>
-sulfido bridges are capping the faces of the W<inf>6</inf>
 cube, thus forming its dual, an octahedron. Combined, the W<inf>6</inf>
S<inf>8</inf>
 group forms approximately a Catalan solid, the triakis octahedron (<tableref idrefs="tab5">Table 5</tableref>
). The reverse situation appears in [Ni<inf>8</inf>
(μ-S<inf>4</inf>
)<inf>6</inf>
(PPh<inf>3</inf>
)<inf>8</inf>
], in which the faces of the cubic metal cluster are augmented by the octahedron of bridging sulfide anions, giving altogether a rhombic dodecahedron (<figref idrefs="fig11">Fig. 11</figref>
).</p>
<figure xsrc="b503582c-f11.tif" id="fig11"><title>(a) structure of the [W<inf>6</inf>
S<inf>8</inf>
(PEt<inf>3</inf>
)<inf>6</inf>
] cluster<citref idrefs="cit92">92</citref>
 showing a central W<inf>6</inf>
 octahedron circumscribed by an S<inf>8</inf>
 cube and an external octahedron of phosphine ligands (brown = W, yellow = S, light blue = P). (b) Cube formed by the Ni atoms of [Ni<inf>8</inf>
(μ<inf>4</inf>
-S)<inf>6</inf>
(PPh<inf>3</inf>
)<inf>8</inf>
] with the circumscribed octahedron formed by the bridging sulfido ligands, and (c) the rhombic dodecahedron formed by the combined Ni<inf>8</inf>
S<inf>6</inf>
 skeleton (Green = Ni, yellow = S). (d) Idealized structure of the Zr<inf>6</inf>
Br<inf>12</inf>
 cluster presenting a central Zr<inf>6</inf>
 octahedron (magenta) and a circumscribed cuboctahedron of the edge-bridging bromide ions (gray).</title>
</figure>
<p>In the case of the Archimedean polyhedra, in which more than one symmetry-related group of faces exist, we can choose to augment only one type of faces at a time, thus generating as many new polyhedra as there are types of faces. In each case the circumscribed polyhedron has the full symmetry of the inscribed one. The circumscribed polyhedra generated in this way can be found in <tableref idrefs="tab7">Table 7</tableref>
. As an example, consider the cuboctahedron, that generates a circumscribed cube by augmentation of its trigonal faces, as found in successive shells around a metal atom in the bcc and ReO<inf>3</inf>
 structures. The same relationship is found between the Au<inf>12</inf>
 and Se<inf>8</inf>
 units in the [Au<inf>12</inf>
NaSe<inf>8</inf>
]<sup>3−</sup>
 anion.<citref idrefs="cit93">93</citref>
 On the other hand, augmentation of the tetragonal faces of the cuboctahedron results in an octahedron (<figref idrefs="fig12">Fig. 12</figref>
), as happens, <it>e.g.</it>
, in [Cu<inf>12</inf>
(μ<inf>4</inf>
-PR)<inf>6</inf>
].<citref idrefs="cit94">94</citref>
 Another nice example of partial augmentation is provided by the [Rh<inf>14</inf>
(μ-CO)<inf>24</inf>
(CO)<inf>6</inf>
]<sup>3−</sup>
 anion.<citref idrefs="cit95">95</citref>
 Here, the bridging carbonyl ligands form a rhombicuboctahedron (<tableref idrefs="tab2">Table 2</tableref>
), whose sets of six square and eight triangular faces are augmented toward the inside by Rh atoms, whereas the set of twelve square faces remain empty. Platonic polyhedra can in principle be also partially face-augmented, although in this case the symmetry is lowered (see sections on Polyhedral Hulls and on Keplerates below for examples).</p>
<table-entry id="tab7"><title>Platonic and Archimedean polyhedra (second column) generated by augmentation (capping) of each set of symmetry-related faces<fnoteref idrefs="tab7fna"></fnoteref>
 of parent Archimedean solids (first column). Truncation of the polyhedron in the second column results in the polyhedron of the first column</title>
<table><tgroup cols="4"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<thead><row><entry>Parent</entry>
<entry>Augmented</entry>
<entry>Parent</entry>
<entry>Augmented</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="4"><footnote id="tab7fna"><it>m</it>
<inf><it>n</it>
</inf>
 indicates a set of <it>m</it>
<it>n</it>
-gonal faces.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Truncated tetrahedron</entry>
<entry>4<inf>3</inf>
: tetrahedron</entry>
<entry>Icosidodecahedron</entry>
<entry>20<inf>3</inf>
: dodecahedron</entry>
</row>
<row><entry></entry>
<entry>4<inf>6</inf>
: tetrahedron</entry>
<entry></entry>
<entry>12<inf>5</inf>
: icosahedron</entry>
</row>
<row><entry>Cuboctahedron</entry>
<entry>8<inf>3</inf>
: cube</entry>
<entry>Truncated dodecahedron</entry>
<entry>20<inf>3</inf>
: dodecahedron</entry>
</row>
<row><entry></entry>
<entry>6<inf>4</inf>
: octahedron</entry>
<entry></entry>
<entry>12<inf>10</inf>
: icosahedron</entry>
</row>
<row><entry>Truncated cube</entry>
<entry>8<inf>3</inf>
: cube</entry>
<entry>Truncated icosahedron</entry>
<entry>12<inf>5</inf>
: icosahedron</entry>
</row>
<row><entry></entry>
<entry>6<inf>8</inf>
: octahedron</entry>
<entry></entry>
<entry>20<inf>6</inf>
: dodecahedron</entry>
</row>
<row><entry>Truncated octahedron</entry>
<entry>6<inf>4</inf>
: octahedron</entry>
<entry>Rhombicosidodecahedron</entry>
<entry>20<inf>3</inf>
: dodecahedron</entry>
</row>
<row><entry></entry>
<entry>8<inf>6</inf>
: cube</entry>
<entry></entry>
<entry>30<inf>4</inf>
: icosidodecahedron</entry>
</row>
<row><entry>Rhombicuboctahedron</entry>
<entry>8<inf>3</inf>
: cube</entry>
<entry></entry>
<entry>12<inf>5</inf>
: icosahedron</entry>
</row>
<row><entry></entry>
<entry>6<inf>4</inf>
: octahedron</entry>
<entry>Snub dodecahedron</entry>
<entry>60<inf>3</inf>
: snub dodecahedron</entry>
</row>
<row><entry></entry>
<entry>12<inf>4</inf>
: cuboctahedron</entry>
<entry></entry>
<entry>20<inf>3</inf>
: icosahedron</entry>
</row>
<row><entry>Truncated cuboctahedron</entry>
<entry>12<inf>4</inf>
: cuboctahedron</entry>
<entry></entry>
<entry>12<inf>5</inf>
: dodecahedron</entry>
</row>
<row><entry></entry>
<entry>8<inf>6</inf>
: cube</entry>
<entry>Truncated icosidodecahedron</entry>
<entry>30<inf>4</inf>
: icosidodecahedron</entry>
</row>
<row><entry></entry>
<entry>6<inf>8</inf>
: octahedron</entry>
<entry></entry>
<entry>20<inf>6</inf>
: icosahedron</entry>
</row>
<row><entry>Snub cube</entry>
<entry>24<inf>3</inf>
: snub cube</entry>
<entry></entry>
<entry>12<inf>10</inf>
: dodecahedron</entry>
</row>
<row><entry></entry>
<entry>8<inf>3</inf>
: cube</entry>
<entry></entry>
<entry></entry>
</row>
<row><entry></entry>
<entry>6<inf>4</inf>
: octahedron</entry>
<entry></entry>
<entry></entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f12.tif" id="fig12"><title>The augmentation of the trigonal faces of the cuboctahedron (left, pink) yields a circumscribed cube (left, blue), whereas the augmentation of its square faces gives an octahedron (right, green).</title>
</figure>
</subsect1>
<subsect1><title>Edge augmentation</title>
<p>Another way of expanding a polyhedron is through <it>edge</it>
-<it>augmentation</it>
. This operation is known in geometry as one of the possible ways of <it>truncating</it>
 a polyhedron, and consists in replacing each vertex by the <it>vertex figure</it>
					(<it>i.e.</it>
, the polygon that results by joining the centers of the edges meeting at that vertex). Alternatively, it can be described as the replacement of each edge by a vertex placed at its center, and we call it edge augmentation (abbreviated as A<inf>e</inf>
) to avoid confusion with other truncation schemes to be discussed below. The polyhedra resulting from edge-augmentation were termed by Kepler <it>ostrea</it>
					(the latin for oyster). In Chemistry, edge-augmentation results from addition of bridging atoms bonded to each pair of contiguous vertices (spanning edges in a μ<inf>2</inf>
 bonding mode as halo or carbonyl bridges in clusters). Two examples are shown in <figref idrefs="fig13">Fig. 13</figref>
: the edge-augmentation of a tetrahedron to give an octahedron, and the augmentation of an octahedron, that yields a cuboctahedron. The new polyhedron thus generated is characterized by its numbers of vertices, edges and faces (<it>V</it>
′, <it>E</it>
′ and <it>F</it>
′, respectively), that are related to those of the original polyhedron (<it>V</it>
, <it>E</it>
 and <it>F</it>
, respectively) through the following relationships: <it>V</it>′
					=
					<it>E</it>
, <it>E</it>′
					= 2<it>E</it>
 and <it>F</it>′
					=
					<it>E</it>
					+ 2. With these relationships and the characteristics of the ideal polyhedra at hand (<tableref idrefs="tab2">Table 2</tableref>
), it is easy to deduce the shape of the resulting polyhedron when the original one is <it>isotoxal</it>
					(<it>i.e.</it>
, has all its edges equivalent), as indicated in <tableref idrefs="tab8">Table 8</tableref>
. There we can see that (i) edge augmentation of Platonic polyhedra produce Platonic or Archimedean polyhedra; (ii) two dual polyhedra give the same figure by edge augmentation and (iii) edge augmentation of isotoxal Archimedean polyhedra yields “distorted” Archimedean polyhedra, in which some square faces are replaced by rectangles. We note that, even if these polyhedra are not strictly Archimedean figures, they have the full symmetry of the parent Archimedean solid. This is illustrated in <figref idrefs="fig14">Fig. 14</figref>
, where we show the rhombicuboctahedron that results from edge-augmentation of a cuboctahedron in two successive shells of the fcc structure, a situation that also appears in a very different chemical system, the cubooctahedral Pd<inf>12</inf>
 core supported by a rhombicuboctahedral arrangement of 24 bridging ligands.<citref idrefs="cit96">96</citref>
 On the other hand, a Pd<inf>60</inf>
 rhombicosidodecahedron is built up by edge-augmentation of a Pd<inf>30</inf>
 icosidodecahedron (<figref idrefs="fig14">Fig. 14</figref>
, right) corresponding to the fifth and fourth shells around a central Pd atom in the wonderful structure of [Pd<inf>145</inf>
(CO)<inf>x</inf>
(PEt<inf>3</inf>
)<inf>30</inf>
] reported by Dahl and co-workers, that will be discussed in more detail below.<citref idrefs="cit97">97</citref>
</p>
<table-entry id="tab8"><title>Polyhedra generated by edge augmentation of isotoxal figures</title>
<table><tgroup cols="3"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<tfoot><row><entry namest="1" nameend="3"><footnote id="tab8fna"><it>V</it>
 faces are rectangles, where <it>V</it>
 is the number of vertices of the original polyhedron.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Tetrahedron</entry>
<entry align="char">→</entry>
<entry>Octahedron</entry>
</row>
<row><entry>Octahedron</entry>
<entry align="char">→</entry>
<entry>Cuboctahedron</entry>
</row>
<row><entry>Cube</entry>
<entry align="char">→</entry>
<entry>Cuboctahedron</entry>
</row>
<row><entry>Icosahedron</entry>
<entry align="char">→</entry>
<entry>Icosidodecahedron</entry>
</row>
<row><entry>Dodecahedron</entry>
<entry align="char">→</entry>
<entry>Icosidodecahedron</entry>
</row>
<row><entry>Cuboctahedron</entry>
<entry align="char">→</entry>
<entry>Distorted rhombicuboctahedron<fnoteref idrefs="tab8fna"></fnoteref>
</entry>
</row>
<row><entry>Icosidodecahedron</entry>
<entry align="char">→</entry>
<entry>Distorted rhombicosidodecahedron<fnoteref idrefs="tab8fna"></fnoteref>
</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f13.tif" id="fig13"><title>Augmentation of the edges of a tetrahedron (left, pink) gives an octahedron (left, gray), and the same operation on an octahedron (right, gray) gives a cuboctahedron (right, golden).</title>
</figure>
<figure xsrc="b503582c-f14.tif" id="fig14"><title>Left: Augmentation of the edges of a cuboctahedron (light blue) to give a pseudo-rhombicuboctahedron (dark red) in the fcc structure. Right: Edge augmentation of the Pd<inf>30</inf>
 icosidodecahedron (light blue) resulting in a Pd<inf>60</inf>
 rhombicosidodecahedron (dark blue) in [Pd<inf>165</inf>
(CO)<inf>x</inf>
(PEt<inf>3</inf>
)<inf>30</inf>
].</title>
</figure>
</subsect1>
<subsect1><title>Isotropic expansion</title>
<p>A trivial way to expand a polyhedron is through isotropic expansion, that is brought about in Chemistry when an atom is bonded to each vertex of a polyhedron in a radial way, thus generating an identical polyhedron of larger size. This happens, for instance, in some metal clusters, as in [W<inf>6</inf>
S<inf>8</inf>
(PEt<inf>3</inf>
)<inf>6</inf>]
					(<figref idrefs="fig11">Fig. 11(a)</figref>
), where the phosphorus atoms form an octahedron that circumscribes the W<inf>6</inf>
 octahedron.</p>
</subsect1>
<subsect1><title>Truncation</title>
<p><it>Truncation</it>
 is a geometrical operation that consists in cutting off the vertices of a polyhedron, thus generating a new polyhedron with more faces. As an example, we show in <compoundref idrefs="chem9">9</compoundref>
 a particular truncation of the tetrahedron, in which all the resulting edges have the same length, giving the Archimedean truncated tetrahedron. We may think of truncation as the replacement of each vertex by a polygon, perpendicular to the radial direction, with the restriction that such polygons must have as many sides as the number of edges meeting at the vertex, or twice that number. Whether the generated polygons are added inside or outside the parent polyhedron is irrelevant with regard to the shape of the polyhedron generated by truncation. Several new polyhedra can be generated by truncation of a given figure because (a) the generated polygons can be either independent or made to share edges or vertices and (b) there are different choices as to the orientation of the new polygons relative to the skeleton of the original polyhedron.<ugraphic xsrc="b503582c-u5.tif" id="ugr5"></ugraphic>
</p>
<p>To illustrate the variety of truncation and augmentation schemes that can be applied to a particular polyhedron, let us consider the case of the octahedron. Since truncation is the inverse operation of face augmentation, the different Archimedean polyhedra that can be generated by truncation of a Platonic one can be found in <tableref idrefs="tab7">Table 7</tableref>
. From that information it is easy to deduce the polyhedra that can be generated from the octahedron by means of the different operations outlined above, as schematically shown in <figref idrefs="fig15">Fig. 15</figref>
: (1) the cube, through face augmentation (A<inf>F</inf>
); (2) the cuboctahedron, <it>via</it>
 edge augmentation (A<inf>E</inf>
); (3) the rhombicuboctahedron, through face-aligned truncation (T<inf>f</inf>
); (4) the snub cube, through rotated truncation (T<inf>r</inf>
); (5) the truncated octahedron, <it>via</it>
 edge-aligned truncation (T<inf>e</inf>
), and double truncation generates either (6) independent (T<inf>2i</inf>
) polygons in the truncated cuboctahedron or (7) edge-sharing (T<inf>2e</inf>
) octagons in the truncated cube. In a later section the reader will find the corresponding scheme for the icosahedron (<figref idrefs="fig26">Fig. 26</figref>
), which will be needed there for the analysis of nested icosahedral clusters. The analogous schemes showing the augmentations of the cube and the dodecahedron are provided as ESI.<fnoteref idrefs="fn1"></fnoteref>
 It is noteworthy that all the Archimedean polyhedra of octahedral symmetry can be generated by the operations described above applied to either the octahedron or the cube, whereas all those with icosahedral symmetry can be generated from either the icosahedron or the dodecahedron. In contrast, the tetrahedron can only generate another tetrahedron <it>via</it>
 face augmentation, the octahedron through edge augmentation, and the truncated tetrahedron by means of different truncation schemes.</p>
<figure xsrc="b503582c-f15.tif" id="fig15"><title>Polyhedra generated from an octahedron through the following operations: face augmentation (A<inf>F</inf>
), edge augmentation (A<inf>E</inf>
), face-directed truncation (T<inf>f</inf>
), rotated truncation (T<inf>r</inf>
), edge-directed truncation (T<inf>e</inf>
), and two double truncations with independent octagons (T<inf>2i</inf>
) and with edge-sharing octagons (T<inf>2e</inf>
). The correspondence between some vertices of the octahedron and the faces of the generated polyhedra are noted with gray dots, that between edges of the octahedron and vertices of the cuboctahedron with green squares.</title>
</figure>
<p>From a chemical point of view, truncation is the relationship that exists between a polyhedral cluster of atoms and the polyhedron formed by the next shell of atoms, typically the ligands in transition metal clusters, but also a shell of ions of the opposite sign in close packed structures of ionic solids. Examples of the relationships of the octahedron with other polyhedra just discussed can be found in several chemical structures analyzed in this Perspective, as corresponds to the operations sketched in <figref idrefs="fig15">Fig. 15</figref>
					(clockwise, starting at 12 o'clock): (1) the faces of the W<inf>6</inf>
 octahedron in <figref idrefs="fig11">Fig. 11(a)</figref>
 are augmented by the sulfido bridges that form a cube; (2) edge augmentation of a Zr<inf>6</inf>
 octahedron is represented by a cuboctahedral arrangement of bromide bridges in <figref idrefs="fig11">Fig. 11(d)</figref>
 and by the cuboctahedron of calixarene molecules surrounding a Pr<inf>6</inf>
 group (<figref idrefs="fig23">Fig. 23</figref>
, below); (3) truncation of V<inf>6</inf>
 to yield a rhombicuboctahedron of oxygen atoms from phosphonato bridges will be seen below in <figref idrefs="fig19">Fig. 19(b)</figref>
; (4) truncation to give a snub cube can be represented by the phenyl groups of an octahedral assembly of calixarene molecules (<figref idrefs="fig23">Fig. 23</figref>
 below, left); (5) a truncated octahedron superimposed to an octahedron can be found in successive concentric shells of the diamond structure (see ESI<fnoteref idrefs="fn1"></fnoteref>
); (6) double truncation is not easy to find in molecular structures, but can be illustrated by two different shells of the body centered cubic structure (bcc) to be discussed below (<figref idrefs="fig25">Fig. 25</figref>
), and (7) the generation of a truncated cube can be illustrated by two successive shells of the same bcc structure (<figref idrefs="fig25">Fig. 25</figref>
).</p>
</subsect1>
</section>
<section><title>Polyhedral hulls: the panelling technique</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>lluita amb el temps i amb l'espai, desafia</it>
</entry>
</row>
<row><entry><it>tactes, mirades… Mes venç l'harmonia:</it>
</entry>
</row>
<row><entry><it>cares més vèrtexs, arestes més dos.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>David Jou, <it>Cristall</it>
<citref idrefs="cit98">98</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>[fights against time and space, defies </entry>
</row>
<row valign="top"><entry>touch, sights… But harmony prevails: </entry>
</row>
<row><entry>faces plus vertices, edges plus two.]<citref idrefs="cit99">99</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>A <it>panelling</it>
 technique has been employed by the groups of Fujita<citref idrefs="cit100 cit101">100,101</citref>
 and Raymond<citref idrefs="cit102">102</citref>
 to build supramolecular architectures, consistent in using multidentate ligands that play the role of the faces of the polyhedron, coordinating to metal atoms that occupy the vertices or the edges. Hence, ligand <compoundref idrefs="chem10">10</compoundref>
 is a triangular panel that can be bound as a bridging ligand to three metals through its vertices. Ligand <compoundref idrefs="chem11">11</compoundref>
, also triangular, coordinates two metal atoms through each edge. With <compoundref idrefs="chem10">10</compoundref>
 one can build an octahedron of M<inf>6</inf>
L<inf>4</inf>
 stoichiometry (<figref idrefs="fig16">Fig. 16(a)</figref>
), where M represents a Pd(en) fragment, whereas <compoundref idrefs="chem11">11</compoundref>
 can be used to make a complex with trigonal bipyramidal geometry and M<inf>18</inf>
L<inf>6</inf>
 stoichiometry. Four of the faces of the M<inf>6</inf>
L<inf>4</inf>
 octahedron in <figref idrefs="fig16">Fig. 16(a)</figref>
 are occupied by the ligand, whereas the other four are hollow, forming what could be called an <it>octahedral hull</it>
. The panelling strategy, therefore, permits us to assembly a polyhedron without the need to use as many ligands as the number of faces of the polyhedron. One of the interesting properties of polyhedral hulls, of which Fujita and co-workers have taken advantage, is that they leave open faces through which reversible enclathration of guest molecules can take place. An artist's view of a tetrahedral hull and a guest reaching to the tetrahedral cavity can be found in the sculpture of Gustavo Torner, “<it>The Princess and the Dragon</it>”
				(<figref idrefs="fig17">Fig. 17</figref>
). With the aim of building efficient nanostructured architectures, therefore, we would like to be able to make polyhedra as large as possible with the minimum possible number of faces (panelling ligands). Two additional examples of potential Platonic hulls are shown in <figref idrefs="fig16">Fig. 16(b) and (c))</figref>
, that should have the symmetry and stoichiometry indicated in <tableref idrefs="tab9">Table 9</tableref>
. Notice that the proposed dodecahedral hull (<figref idrefs="fig16">Fig. 16(c)</figref>
) is a chiral structure.</p>
<table-entry id="tab9"><title>Platonic and Archimedean polyhedral hulls, and polyhedra formed by the center of the faces occupied by the <it>n</it>
-topic panelling ligands</title>
<table><tgroup cols="4"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<thead><row><entry>Polyhedral hull<fnoteref idrefs="tab9fna"></fnoteref>
</entry>
<entry>Ligand polyhedron</entry>
<entry>Stoichiometry</entry>
<entry>Topicity</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="4"><footnote id="tab9fna">The names in italics correspond to distorted versions of the Archimedean polyhedra, in which square faces are replaced by rectangles.</footnote>
<footnote id="tab9fnb">The hollow faces correspond to the vertices of the cube, the ligand centers to the pairs decorating each face in <figref idrefs="fig20">Fig. 20c</figref>
.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>Tetrahedron</entry>
<entry>Triangle</entry>
<entry>M<inf>4</inf>
L<inf>3</inf>
</entry>
<entry>3</entry>
</row>
<row><entry>Octahedron</entry>
<entry>Tetrahedron</entry>
<entry>M<inf>6</inf>
L<inf>4</inf>
</entry>
<entry>3</entry>
</row>
<row><entry>Cube</entry>
<entry>Square</entry>
<entry>M<inf>8</inf>
L<inf>4</inf>
</entry>
<entry>4</entry>
</row>
<row><entry>Icosahedron</entry>
<entry>Irregular icosahedron</entry>
<entry>M<inf>12</inf>
L<inf>12</inf>
<fnoteref idrefs="tab9fnb"></fnoteref>
</entry>
<entry>3</entry>
</row>
<row><entry>Dodecahedron</entry>
<entry>Tridiminished icosahedron (J63)</entry>
<entry>M<inf>20</inf>
L<inf>9</inf>
</entry>
<entry>5</entry>
</row>
<row><entry>Cuboctahedron</entry>
<entry>Octahedron</entry>
<entry>M<inf>12</inf>
L<inf>6</inf>
</entry>
<entry>3</entry>
</row>
<row><entry>Cuboctahedron</entry>
<entry>Cube</entry>
<entry>M<inf>12</inf>
L<inf>8</inf>
</entry>
<entry>4</entry>
</row>
<row><entry>Icosidodecahedron</entry>
<entry>Dodecahedron</entry>
<entry>M<inf>30</inf>
L<inf>20</inf>
</entry>
<entry>5</entry>
</row>
<row><entry>Icosidodecahedron</entry>
<entry>Icosahedron</entry>
<entry>M<inf>30</inf>
L<inf>12</inf>
</entry>
<entry>3</entry>
</row>
<row><entry><it>Truncated cuboctahedron</it>
</entry>
<entry>Cuboctahedron</entry>
<entry>M<inf>48</inf>
L<inf>12</inf>
</entry>
<entry>4</entry>
</row>
<row><entry><it>Rhombicosidodecahedron</it>
</entry>
<entry>Icosidodecahedron</entry>
<entry>M<inf>60</inf>
L<inf>30</inf>
</entry>
<entry>4</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<figure xsrc="b503582c-f16.tif" id="fig16"><title>(a) M<inf>6</inf>
L<inf>4</inf>
 octahedral hull with tetrahedral symmetry, built according to the <it>panelling</it>
 technique of Fujita. Also a hypothetical <it>C</it>
<inf>3v</inf>
-M<inf>20</inf>
L<inf>9</inf>
 dodecahedral hull (b) and a <it>T</it>
<inf>h</inf>
-M<inf>12</inf>
L<inf>12</inf>
 icosahedral hull (c) are shown (see <tableref idrefs="tab10">Table 10</tableref>
).</title>
</figure>
<figure xsrc="b503582c-f17.tif" id="fig17"><title><it>La princesa y el dragón</it>
					(<it>The Princess and the Dragon</it>
), sculpture of Gustavo Torner (1989), showing a tetrahedral hull and a guest reaching its hollow face. Reproduced with permission of the <it>Museu d'Art Espanyol Contemporani</it>
, <it>Fundació Joan March</it>
, Palma de Mallorca.</title>
</figure>
<p><ugraphic xsrc="b503582c-u6.tif" id="ugr6"></ugraphic>
We can generalize the principles of polyhedral hull construction to the Archimedean solids, in which at least two different types of faces exist. If such a polyhedron has <it>v</it>
 vertices, <it>a m</it>
-gonal faces and <it>b n</it>
-gonal faces, its molecular realization requires <it>v</it>
 metal atoms, <it>a m</it>
-topic polygonal ligands (<it>i.e.</it>
, with <it>m</it>
 donor atoms that we may call X) and <it>b n</it>
-topic polygonal ligands (Y), and the resulting stoichiometry of the polyhedral compound should be M<inf><it>v</it>
</inf>
X<inf><it>a</it>
</inf>
Y<inf><it>b</it>
</inf>
. So far, triangular, square and rectangular panelling ligands have been used,<citref idrefs="cit100">100</citref>
 but this author is not aware of the existence of polyhedral structures with pentagonal or hexagonal panelling ligands. We must not forget, though, that each metal can coordinate more ligands at the periphery of the polyhedron, and that the stoichiometry mentioned here corresponds only to the polyhedral faces. In general, Archimedean hulls can be built in more than one way, depending on the set of equivalent faces (see <tableref idrefs="tab2">Table 2</tableref>
) used for panelling, as shown for several cases in <tableref idrefs="tab9">Table 9</tableref>
. As an example, cuboctahedral hulls can be alternatively obtained through skeletons of stoichiometries M<inf>12</inf>
X<inf>6</inf>
 or M<inf>12</inf>
Y<inf>8</inf>
, where X and Y represent tetra- and tri-topic ligands, respectively. An interesting difference between the Archimedean and Platonic hulls is that the former can be built without loss of symmetry, whereas the Platonic polyhedra can be made hollow only at the price of a significant loss of symmetry (<tableref idrefs="tab9">Table 9</tableref>
). It is worth noting also that the centers of the panelling ligands also form specific polyhedra (indicated for some cases in <tableref idrefs="tab9">Table 9</tableref>
), that correspond to those discussed above for partial face augmentation (<tableref idrefs="tab7">Table 7</tableref>
).</p>
<p>The concept of panelling can probably be applied also to mononuclear coordination complexes in which multidentate ligands span specific faces of the coordination polyhedron. A nice example is provided by the dodecacoordinate complex of Ba<sup>2+</sup>
 with cyclohexane-1,3,5-triol.<citref idrefs="cit103">103</citref>
 In this compound, the twelve oxygen donor atoms form an icosahedron, but the tridentate ligands cover only four faces of the icosahedron in a tetrahedral arrangement with the unusual chiral <it>T</it>
 symmetry (see the group–subgroup relationship in <compoundref idrefs="chem4">4</compoundref>
).</p>
</section>
<section><title>Polyhedral organization of some molecules</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>Infatti, accade anche in chimica, come in architettura, che gli edifici “belli”, e cioè simmetrici e semplici, siano anche i piú saldi: avviene insomma per le molecole come per le cupole delle cattedrali o per le arcate dei ponti.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Primo Levi, <it>Il sistema periodico</it>
<citref idrefs="cit104">104</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[In fact it happens also in chemistry as in architecture that “beautiful” edifices, that is, symmetrical and simple, are also the most sturdy: in short, the same thing happens with molecules as with the cupolas of cathedrals or the arches of bridges.]<citref idrefs="cit105">105</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<subsect1><title>Ligands and metals in clusters</title>
<p>Johnson and co-workers have noted that the carbonyl ligands in transition metal clusters also form polyhedra,<citref idrefs="cit106">106</citref>
 and a variety of clusters have been analyzed from that point of view, known as the <it>ligand polyhedral model</it>
					(LPM). In this section we wish to show the connection that exists between the shape of a metal cluster and the polyhedron of the ligands, taking advantage of the geometrical relationships discussed in the two previous sections. From them, it is easy to establish the ideal stoichiometry for a ligand cluster with a given central metallic polyhedron. As an example, if an M<inf>4</inf>
 tetrahedron presents four X bridges (<it>e.g.</it>
, a halide, a chalcogenide or an alkoxide) in its face centers (<it>i.e.</it>
, they act as μ<inf>3</inf>
 bridges), those ligands form the dual polyhedron, <it>i.e.</it>
, another tetrahedron. Since two interpenetrated tetrahedra of the same size form a cube (<compoundref idrefs="chem3c">3c</compoundref>
), such compounds of M<inf>4</inf>
X<inf>4</inf>
 stoichiometry are often called <it>cubanes</it>
. We have shown,<citref idrefs="cit56">56</citref>
 though, that in most cases the deviation of the M<inf>4</inf>
X<inf>4</inf>
 group from the cube is significant, while still retaining the full tetrahedral symmetry of a compound of two tetrahedra. Alternatively, if the same M<inf>4</inf>
 tetrahedron has six μ<inf>2</inf>
 bridges at the center of its edges, we will end up having a compound with the M<inf>4</inf>
X<inf>6</inf>
 stoichiometry, formed by an M<inf>4</inf>
 tetrahedron and an X<inf>6</inf>
 octahedron (<tableref idrefs="tab8">Table 8</tableref>
). By addition of an extra terminal ligand to each metal atom (in a radial direction), we may have M<inf>4</inf>
X<inf>4</inf>
L<inf>4</inf>
 and M<inf>4</inf>
X<inf>10</inf>
L<inf>4</inf>
 compounds, formed by inscribed polyhedra of the types [tetrahedron ⊂ tetrahedron ⊂ tetrahedron] and [tetrahedron ⊂ octahedron ⊂ tetrahedron], respectively, where the “⊂” symbol stands for “inscribed in”.</p>
<p>In previous sections on duality relationships and isotropic expansion we have seen how the octahedral metal cluster of [W<inf>6</inf>
S<inf>8</inf>
(PEt<inf>3</inf>
)<inf>6</inf>
] gives raise to a cube and an octahedron of ligands (<figref idrefs="fig11">Fig. 11(a)</figref>
). If we consider the same M<inf>6</inf>
 cluster, but with bridging ligands at the edge centers instead (μ<inf>2</inf>
 coordination), the resulting stoichiometry is M<inf>6</inf>
X<inf>12</inf>
 and the arrangement of the ligands (<tableref idrefs="tab8">Table 8</tableref>
) corresponds to a circumscribed cuboctahedron. Clusters of [M<inf>6</inf>
X<inf>12</inf>
] composition appear in many solid phases (where X is a halogen or a chalcogen and M = Sc, Y, Zr, Nb, Ta, Re, Tc, Pr, Gd, Er, Lu, Th),<citref idrefs="cit107">107</citref>
 that present precisely this type of structure, as illustrated in <figref idrefs="fig11">Fig. 11(d)</figref>
 for Zr<inf>6</inf>
Br<inf>12</inf>
. A last example of a duality relationship between the metal and ligand polyhedra is provided by the [AsNi<inf>12</inf>
As<inf>20</inf>
]<sup>3−</sup>
 anion, in which the Ni<inf>12</inf>
 group forms an icosahedron and the 20 μ<inf>3</inf>
-As bridges adopt the structure of its dual dodecahedron.<citref idrefs="cit108">108</citref>
 In <tableref idrefs="tab10">Table 10</tableref>
 are some examples of clusters of different nuclearities, together with their nesting and symmetries. In general, the ligand polyhedron reproduces the symmetry of the bonded metal polyhedron, as in [Ni<inf>8</inf>
(μ<inf>6</inf>
-C)(μ<inf>3</inf>
-CO)<inf>8</inf>
(CO)<inf>8</inf>
]<sup>2−</sup>
 anion, in which 16 carbonyl ligands form a snub square antiprism (<figref idrefs="fig4">Fig. 4</figref>
) that preserves the <it>D</it>
<inf>4d</inf>
 symmetry of the inner Ni<inf>8</inf>
 square antiprism. In contrast, the [Ir<inf>6</inf>
(CO)<inf>16</inf>
] cluster offers an example in which the two types of carbonyl ligands present form polyhedra with lower symmetry than that of the metallic skeleton.</p>
<table-entry id="tab10"><title>Some examples of nesting of bonded and ligand polyhedra in transition metal clusters</title>
<table><tgroup cols="3"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<thead><row><entry>Compound</entry>
<entry>Polyhedral nesting<fnoteref idrefs="tab10fna"></fnoteref>
</entry>
<entry>Ref.</entry>
</row>
</thead>
<tfoot><row><entry namest="1" nameend="3"><footnote id="tab10fna">The symbol “⊂” stands for “inscribed in”.</footnote>
</entry>
</row>
</tfoot>
<tbody><row><entry>[W<inf>6</inf>
(μ<inf>3</inf>
-S)<inf>8</inf>
(PEt<inf>3</inf>
)<inf>6</inf>
]<sup>+</sup>
</entry>
<entry>W<inf>6</inf>
 octahedron ⊂ S<inf>8</inf>
 cube ⊂ P<inf>6</inf>
 octahedron</entry>
<entry><citref idrefs="cit109" position="baseline">109</citref>
</entry>
</row>
<row><entry>[Ni<inf>8</inf>
(μ<inf>4</inf>
-PPh)<inf>6</inf>
(CO)<inf>8</inf>
]</entry>
<entry>Ni<inf>8</inf>
 cube ⊂ P<inf>6</inf>
 octahedron</entry>
<entry><citref idrefs="cit110" position="baseline">110</citref>
</entry>
</row>
<row><entry>[M<inf>4</inf>
(μ<inf>3</inf>
-X)<inf>4</inf>
Y<inf>4</inf>
]</entry>
<entry>M<inf>4</inf>
 tetrahedron ⊂ X<inf>4</inf>
 tetrahedron</entry>
<entry></entry>
</row>
<row><entry>[M<inf>4</inf>
(μ<inf>2</inf>
-X)<inf>6</inf>
]</entry>
<entry>M<inf>4</inf>
 tetrahedron ⊂ X<inf>6</inf>
 octahedron</entry>
<entry></entry>
</row>
<row><entry>[AsNi<inf>12</inf>
(μ<inf>3</inf>
-As)<inf>20</inf>
]<sup>3−</sup>
</entry>
<entry>Ni<inf>12</inf>
 icosahedron ⊂ As<inf>20</inf>
 dodecahedron</entry>
<entry><citref idrefs="cit108" position="baseline">108</citref>
</entry>
</row>
<row><entry>[Zr<inf>6</inf>
(μ<inf>2</inf>
-Br)<inf>12</inf>
]</entry>
<entry>Zr<inf>6</inf>
 octahedron ⊂ Br<inf>12</inf>
 cuboctahedron</entry>
<entry><citref idrefs="cit111" position="baseline">111</citref>
</entry>
</row>
<row><entry>[Ru<inf>3</inf>
(CO)<inf>12</inf>
]</entry>
<entry>Ru<inf>3</inf>
 triangle ⊂ C<inf>12</inf>
 anticuboctahedron</entry>
<entry><citref idrefs="cit112" position="baseline">112</citref>
</entry>
</row>
<row><entry>[Os<inf>6</inf>
P(CO)<inf>18</inf>
]<sup>−</sup>
</entry>
<entry>Os<inf>6</inf>
 trigonal prism ⊂ C<inf>12</inf>
 elongated trigonal orthobicupola</entry>
<entry><citref idrefs="cit52" position="baseline">52</citref>
</entry>
</row>
<row><entry>[Ni<inf>8</inf>
C(CO)<inf>16</inf>
]<sup>2−</sup>
</entry>
<entry>Ni<inf>8</inf>
 tetragonal antiprism ⊂ C<inf>22</inf>
 snub square antiprism</entry>
<entry><citref idrefs="cit51" position="baseline">51</citref>
</entry>
</row>
<row><entry>[Re<inf>4</inf>
H<inf>4</inf>
(CO)<inf>16</inf>
]</entry>
<entry>Re<inf>4</inf>
 square ⊂ C<inf>16</inf>
 square orthobicupola</entry>
<entry><citref idrefs="cit50" position="baseline">50</citref>
</entry>
</row>
<row><entry>[Rh<inf>6</inf>
N(CO)<inf>6</inf>
(μ-CO)<inf>9</inf>
]<sup>−</sup>
</entry>
<entry>Rh<inf>6</inf> trigonal prism ⊂
										(μ-CO)<inf>9</inf>
 triaugmented trigonal prism</entry>
<entry><citref idrefs="cit113" position="baseline">113</citref>
</entry>
</row>
<row><entry>[Ir<inf>6</inf>
(CO)<inf>16</inf>
]</entry>
<entry>Ir<inf>6</inf>
 octahedron ⊂(μ-CO)<inf>4</inf>
 tetrahedron ⊂(CO)<inf>12</inf>
 truncated tetrahedron</entry>
<entry><citref idrefs="cit114" position="baseline">114</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
</subsect1>
<subsect1><title>Polyoxometallates and Keplerates</title>
<p>It is probably among the supramolecular architectures than one can most benefit from knowing the relationships between polyhedra. These systems are organized through autoassembly of a certain number of molecules or groups, generally of two or more different types. The different components can be associated to vertices, edges or faces of one or more polyhedra, in such a way that there is a direct relationship between the stoichiometry with which they participate in the autoassembly process and the polyhedra that they can conform. A proposal for a systematic description of the fragments that can form a given supramolecular polyhedron from a geometrical perspective is provided by the <it>molecular library</it>
 of Stang,<citref idrefs="cit115">115</citref>
 that will not be reproduced here for brevity. Among supramolecular architectures we could include polyoxometallates, whose rich structural variety results from the condensation of coordination polyhedra of transition metal atoms through oxo bridges.<citref idrefs="cit116">116</citref>
 Some of the most characteristic polyoxometallate structures are usually described by using proper names, that give credit to their discoverers, but do not convey all the information associated to the polyhedral organization of the metal atoms. Also the representation of the molecular structure most often emphasizes the coordination polyhedra of the metal atoms (<figref idrefs="fig18">Fig. 18(a) and (b)</figref>
), whereas the polyhedron formed by the metallic core (<figref idrefs="fig18">Fig. 18(c) and (d)</figref>
) is rarely represented. Thus, the <it>Keggin</it>
 structures, with XM<inf>12</inf>
O<inf>40</inf>
 stoichiometry, present two isomeric forms, α and β, that correspond to a cuboctahedron and a trigonal orthobicupola (J27), respectively. Likewise, the <it>Dawson</it>
 structures, of formula X<inf>2</inf>
M<inf>18</inf>
O<inf>62</inf>
, have their metal atoms arranged as in an elongated orthobicupola or in an elongated trigonal gyrobicupola (Johnson polyhedra J35 and J36). Among other polyhedra formed by the metal core in polyoxometallates (<tableref idrefs="tab3">Table 3</tableref>
), we can find the triangular cupola, the elongated square bipyramid, the biluna birotunda (<figref idrefs="fig4">Fig. 4</figref>
) found in the [V<inf>18</inf>
O<inf>44</inf>
H<inf>2</inf>
]<sup>4−</sup>
 anion templated by an azide anion,<citref idrefs="cit49">49</citref>
 and the elongated pentagonal orthobicupola.</p>
<figure xsrc="b503582c-f18.tif" id="fig18"><title>The α- (a) and β-Keggin (b) structures of the M<inf>12</inf>
O<inf>40</inf>
 polyoxometallates in the common view showing the metal coordination polyhedra and in the corresponding views that stress the polyhedral arrangement of the twelve metal atoms (c and d). The structures shown correspond to NaH<inf>2</inf>
(PW<inf>12</inf>
O<inf>40</inf>
)(H<inf>2</inf>
O)<inf>12</inf>
						(a and c) and β-K<inf>4</inf>
SiW<inf>12</inf>
O<inf>40</inf>
(H<inf>2</inf>
O)<inf>9</inf>
						(b and d).</title>
</figure>
<p>Polyoxometallates in which several polyhedra are nested have been named <it>keplerates</it>
 by Müller<citref idrefs="cit117">117</citref>
 because of their analogy with the cosmogony proposed by Kepler in his <it>Mysterium Cosmographicum</it>
					(1596), in which the orbits of the planets were associated to circumscribed Platonic polyhedra. An excellent example of how several polyhedra can be nested in a single supramolecular structure is provided by the compound of formula [(VO)<inf>6</inf>
(μ<inf>3</inf>
-<sup>t</sup>
BuPO<inf>3</inf>
)<inf>8</inf>
]Cl reported by Zubieta and co-workers.<citref idrefs="cit118">118</citref>
 This relatively simple architecture (<figref idrefs="fig19">Fig. 19</figref>
) can be described by six different polyhedra, five of which are Platonic or Archimedean. On one hand we have the local coordination polyhedra: tetrahedra for P and C, an octahedron for Cl and a square pyramid for V. On the other hand, the different groups that constitute the whole edifice are arranged forming a V<inf>6</inf>
 octahedron, P<inf>8</inf>
 and C<inf>8</inf>
 cubes (corresponding to the phosphate groups and to the quaternary carbon atoms, respectively), an O<inf>24</inf>
 rhombicuboctahedron of the phosphonato groups and a C<inf>24</inf>
 truncated cube of the primary carbon atoms. In this supramolecular assembly we note the coexistence of four circumscribed polyhedra, all belonging to the <it>O</it>
<inf>h</inf>
 symmetry point group.</p>
<figure xsrc="b503582c-f19.tif" id="fig19"><title>(a) Perspective view of the molecular structure of [(VO)<inf>6</inf>
(μ<inf>3</inf>
-<sup>t</sup>
BuPO<inf>3</inf>
)<inf>8</inf>
]Cl (magenta spheres, V; light blue, P; red, O; yellow, Cl; gray, C; the hydrogen atoms are omitted for clarity). (b) Rhombicuboctahedron of O atoms superimposed to the V octahedron. (c) Truncated cube of primary C atoms containing the rhombicuboctahedron of oxygen atoms.</title>
</figure>
<p>There are some interesting relationships between polyhedra with icosahedral and octahedral symmetries. For instance, by augmentation of only eight faces of an icosahedron, we can obtain a cube (<figref idrefs="fig20">Fig. 20(a)</figref>
). The ensemble of icosahedron and cube has the symmetry corresponding to the common subgroup of <it>I</it>
<inf>h</inf>
 and <it>O</it>
<inf>h</inf>
, <it>i.e.</it>
, the unusual <it>T</it>
<inf>h</inf>
 point group (see <compoundref idrefs="chem4">4</compoundref>
). The inverse operation that generates an icosahedron from a cube consists in truncation of the cube in such a way that the generated trigonal faces share vertices. An example of such a relationship can be found in<citref idrefs="cit119">119</citref>
 Ag<inf>8</inf>
Cl<inf>2</inf>
[Se<inf>2</inf>
P(OEt)<inf>2</inf>
]<inf>6</inf>
, where the cubic skeleton of Ag ions is surrounded by an icosahedron of bridging selenide ions (figure provided as ESI<fnoteref idrefs="fn1"></fnoteref>
). Alternatively, the icosahedron can be formed by decorating the faces of a cube with pairs of atoms (<figref idrefs="fig20">Fig. 20(b)</figref>
): for a specific ratio between the interatomic distance within the pair and their distance to the center of the cube, a perfect icosahedron can be obtained. At a shorter atom–atom distance, the combination of the vertices of the cube and the decorating pairs form a perfect dodecahedron (<figref idrefs="fig20">Fig. 20(c) and (d)</figref>
). Such is the architecture that appears<citref idrefs="cit120">120</citref>
 in the tetrapropylammonium salt of [(VO)<inf>12</inf>
(μ<inf>12</inf>
-OH)<inf>8</inf>
(μ<inf>3</inf>
-PhPO<inf>3</inf>
)<inf>8</inf>
]<sup>4−</sup>
. Each of the eight phosphonato groups acts as a bridge between three V atoms, giving a cubic arrangement of the P atoms, while the square pyramids centered at the V atoms form pairs that share a basal edge, in such a way that every pair of vanadyls is decorating one of the faces of the P<inf>8</inf>
 cube, again with the unusual symmetry corresponding to the <it>T</it>
<inf>h</inf>
 point group. The combination of P and V atoms forms a beautiful dodecahedron, as noted earlier,<citref idrefs="cit121">121</citref>
 built up through the same procedure originally proposed by Euclid for the construction of a dodecahedron. Extended structures of A<inf>2</inf>
M<inf>6</inf>
E<inf>8</inf>
 stoichiometry based on the same topology, in which M<inf>12</inf>
E<inf>8</inf>
 dodecahedra extend in three dimensions through edge-sharing, have been theoretically predicted by Hoffmann and co-workers.<citref idrefs="cit122">122</citref>
</p>
<figure xsrc="b503582c-f20.tif" id="fig20"><title>(a) Relationship between an icosahedron (dark red) and a cube (blue), as found in the Ag<inf>8</inf>
 and Se<inf>12</inf>
 groups of Ag<inf>8</inf>
Cl<inf>2</inf>
[Se<inf>2</inf>
P(OEt)<inf>2</inf>
]<inf>6</inf>
. (b) Alternative description of the icosahedron as the result of decorating the cube faces with atom pairs, where the tetrahedral symmetry of the ensemble can be appreciated. In a particular case, the compound formed by (c) the cube and its decoration is a dodecahedron (d), as found in the [(VO)<inf>12</inf>
(μ<inf>2</inf>
-OH)<inf>12</inf>
(μ<inf>3</inf>
-PhPO<inf>3</inf>
)<inf>8</inf>
]<sup>4−</sup>
 anion.</title>
</figure>
<p>Among the wealth of examples of supramolecular buildings that can be found in the chemistry of molybdenum, we comment here briefly on a last example, that of the Mo<inf>132</inf>
 group in [Mo<sup>VI</sup>
<inf>72</inf>
Mo<sup>V</sup>
<inf>60</inf>
O<inf>372</inf>
(MeCO<inf>2</inf>
)<inf>30</inf>
(H<inf>2</inf>
O)<inf>72</inf>
]<sup>42−</sup>
, prepared by A. Müller and co-workers.<citref idrefs="cit123">123</citref>
 Even if the spheroidal structure of these compounds is complex, its components are assembled in a surprisingly simple way (<figref idrefs="fig21">Fig. 21</figref>
). Hence, the Mo<sup>VI</sup>
 ions form pentagonal Mo<inf>6</inf>
O<inf>21</inf>
 units, with a central Mo coordinated by seven oxygen atoms in a pentagonal bipyramidal arrangement. Those twelve Mo atoms form a icosahedron, whereas the rest of the molybdenums of the Mo<inf>6</inf>
O<inf>21</inf>
 groups form a distorted truncated rhombicosahedron, as a result of the presence of the intervening Mo<sup>V</sup>
<inf>2</inf>
 pairs that separate the pentagons. The Mo<sup>V</sup>
 atoms, on their side, appear forming pairs along the edges of the icosahedron (60 metal atoms in total for the 30 icosahedral edges). Given the relationship between edges of the icosahedron and vertices of the circumscribed polyhedron (<tableref idrefs="tab8">Table 8</tableref>
), the centers of the 30 Mo<inf>2</inf>
 units form an icosidodecahedron, the same figure introduced recently for footballs by some manufacturer in replacement of the classical truncated icosahedral shape. In summary, the arrangement of the metal atoms can be described by a compound of a Mo<sup>VI</sup>
<inf>12</inf>
 icosahedron, an Mo<sup>VI</sup>
<inf>60</inf>
 rhombicosidodecahedron and a (Mo<sup>V</sup>
<inf>2</inf>
)<inf>30</inf>
 icosidodecahedron.</p>
<figure xsrc="b503582c-f21.tif" id="fig21"><title>Coordination polyhedra of the Mo atoms in the [Mo<sup>VI</sup>
<inf>72</inf>
Mo<sup>V</sup>
<inf>60</inf>
O<inf>372</inf>
(MeCO<inf>2</inf>
)<inf>30</inf>
(H<inf>2</inf>
O)<inf>72</inf>
]<sup>42−</sup>
 anion (left), together with the icosahedral arrangement of the 12 heptacoordinate (center) and the pseudo-truncated rhombicosahedral architecture of the hexacoordinate Mo<sup>VI</sup>
 atoms (right).</title>
</figure>
<p>Nesting of Johnson polyhedra can also be found among the polyoxometallates. Hence, [PW<inf>9</inf>
O<inf>28</inf>
Br<inf>6</inf>
]<sup>3−</sup>
 has a W<inf>9</inf>
 core with the shape of a triangular cupola, while 15 μ<inf>2</inf>
-oxo bridges capping the W⋯W edges outside the polyhedron form a gyroelongated triangular cupola (<figref idrefs="fig4">Fig. 4</figref>
) and nine terminal oxo ligands form a circumscribed triangular cupola.</p>
</subsect1>
<subsect1><title>Other strategies</title>
<p>Another approach for building supramolecular polyhedra, successfully explored by several authors,<citref idrefs="cit102 cit124">102,124</citref>
 consists in using ditopic bischelating ligands of the topology shown in <compoundref idrefs="chem12">12</compoundref>
, in such a way that each ligand spans an edge of the supramolecular polyhedron and each bidentate group coordinates to a metal atom at a vertex, as shown in <figref idrefs="fig22">Fig. 22</figref>
 for a [Co<inf>4</inf>
(ditopic ligand)<inf>6</inf>
] complex.<citref idrefs="cit125">125</citref>
 Since three edges meet at each vertex, the metals are hexacoordinated and four metal atoms and six ligand molecules are assembled forming a tetrahedron and an octahedron, respectively, as corresponds to edge augmentation of a tetrahedron (<tableref idrefs="tab8">Table 8</tableref>
).</p>
<figure xsrc="b503582c-f22.tif" id="fig22"><title>Tetrahedral skeleton of [Co<inf>4</inf>
(ditopic ligand)<inf>6</inf>
] with the ligands aligned along the edges of the Co<inf>4</inf>
 tetrahedron. Substituents at the ligands have been omitted for clarity.</title>
</figure>
<p><ugraphic xsrc="b503582c-u7.tif" id="ugr7"></ugraphic>
Supramolecular polyhedral systems can also be assembled without metal atoms, replacing the coordinate bonds by hydrogen bonds. This is the approach used by Atwood and co-workers to make structures in which six or twelve calixarene molecules are held together by hydrogen bonds, in such ways that their centers can form octahedra<citref idrefs="cit126 cit127">126,127</citref>
					(<figref idrefs="fig23">Fig. 23</figref>
), a icosahedron<citref idrefs="cit128">128</citref>
 or a cuboctahedron.<citref idrefs="cit129">129</citref>
 In addition, since each calixarene unit has four phenyl groups joined by four methylene bridges, in the octahedra of calixarenes the phenyl groups are arranged at the vertices of a snub cube<citref idrefs="cit126">126</citref>
 or a rhombicuboctahedron<citref idrefs="cit127">127</citref>
					(<figref idrefs="fig23">Fig. 23</figref>
), whereas the cuboctahedron of calixarenes results in a truncated cuboctahedron of phenyl rings,<citref idrefs="cit129">129</citref>
 thus providing us with nice examples of the capping/expansion relationships between polyhedra discussed above (<tableref idrefs="tab7">Table 7</tableref>
). It is remarkable that the same chiral polyhedral structure, the snub cube, is formed in a supramolecular architecture assembled through hydrogen bonds<citref idrefs="cit126">126</citref>
 and in solid state structures as NaZn<inf>13</inf>
 or CeBe<inf>13</inf>
.<citref idrefs="cit130">130</citref>
</p>
<figure xsrc="b503582c-f23.tif" id="fig23"><title>Above: Spatial disposition of six calixarene molecules<citref idrefs="cit126">126</citref>
 and schematic depiction of the snub cube that describes its geometry, in its two possible enantiomeric forms (each calixarene molecule corresponds to one square in the ideal polyhedron). Below: Octahedral arrangement of six calixarene units showing the rhombicuboctahedron formed by the centers of the phenyl ring,<citref idrefs="cit127">127</citref>
 and central Pr<inf>6</inf>
 octahedron (orange) surrounded by a cuboctahedron (dashed stripes) of 12 calixarene molecules,<citref idrefs="cit129">129</citref>
 whose phenyl rings are arranged at the vertices of a truncated cuboctahedron.</title>
</figure>
</subsect1>
</section>
<section><title>Nested polyhedra in clusters and in the solid state</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>Quand le soleil, débarrassé de nuages, commença d'éclairer ma machine, cet icosaèdre transparent, qui recevait à travers ses facettes les trésors du soleil, en répandait par le bocal la lumière dans ma cellule; et, comme cette splendeur s'affaiblissait à cause des rayons qui ne pouvaient se rejeter jusqu'à moi sans se rompre beaucoup de fois, cette vigueur de clarté tempérée convertissait ma châsse en un petit ciel de pourpre émaillé d'or.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Cyrano de Bergerac, <it>Histoire comique des états et empires de la Lune et du Soleil</it>
								(1886)</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[When the Sun emerged from the clouds and began to shine on my machine the transparent icosahedron received the treasures of the sun through its facets and transmitted the light through the globe into my cell; and since this splendour was weakened, because the rays could not reach me without being several times broken, this strength of tempered light converted my shrine into a little sky of purple enamelled with gold.]<citref idrefs="cit131">131</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>We have already seen some examples of molecules in which two successive atomic shells form two polyhedra connected through some geometrical relationship (<tableref idrefs="tab10">Table 10</tableref>
), induced by the existence of specific chemical bonding patterns between the two shells. As we move to more complex molecules with higher number of concentric shells, eventually reaching a nanocluster or a nanocrystal, we may find it useful to exploit the polyhedral relationships discussed above. In particular, a description of a complex structure in terms of nested polyhedra can be of great use in expressing in a simple way what can hardly be represented by a two-dimensional structural formula.</p>
<p>Let us start by looking at a relatively simple system, that of Nb<inf>6</inf>
I<inf>11</inf>
. This compound presents a spin crossover from a low temperature doublet to a high temperature quartet state and presents a structure in which [Nb<inf>6</inf>
I<inf>14</inf>
]<sup>3−</sup>
 clusters (similar to that shown in <figref idrefs="fig11">Fig. 11(a)</figref>
) are connected through iodide bridges to generate an extended structure.<citref idrefs="cit132">132</citref>
 The Nb<inf>6</inf>
 octahedron is seen to distort as the temperature is lowered, and such a distortion is transmitted to the coordinated iodides (<figref idrefs="fig24">Fig. 24</figref>
). It is interesting to note that the intracluster triply bridging iodides that form a cube distort slightly less than the Nb core, whereas the iodides that are terminal to one cluster and bridging to a neighboring one seem to amplify the distortion of the metal octahedron. It is also worthy of note that extrapolation to the ideal case of a perfect Nb<inf>6</inf>
 octahedron predicts perfect cubic and octahedral shapes for the two sets of iodide ions. Although we must be careful not to draw conclusions from structural data for a single compound, they suggest that symmetry (or asymmetry) is transmitted radially along the concentric shells of nested polyhedra.</p>
<figure xsrc="b503582c-f24.tif" id="fig24"><title>Evolution of the shape measures of the I<inf>6</inf>
 octahedron and the I<inf>8</inf>
 cube in the Nb<inf>6</inf>
I<inf>14</inf>
 cluster of Nb<inf>6</inf>
I<inf>11</inf>
 as a function of the distortion of the Nb<inf>6</inf>
 octahedron brought about by changes in temperature.</title>
</figure>
<p>Instead of describing the solid state structure of an elementary solid or of a binary phase as resulting from translation of a small unit cell along three directions in space, we can imagine it as built up by concentric shells of atoms around a central atom or a central polyhedron in well defined arrangements that result from the augmentation and expansion operations that have been discussed above. Let us consider as an example the case of the body centered cubic (bcc) structure. Its unit cell is formed by an atom surrounded by eight identical neighbors in a cubic arrangement. The subsequent atomic shell consists of the centering atoms of the neighboring unit cells that have perforce the octahedral shape as corresponds to face augmentation of a cube (<tableref idrefs="tab11">Table 11</tableref>
). The third shell is formed by edge augmentation of such an octahedron, resulting in a cuboctahedron. The reader should be able to establish the connection between the polyhedron of a given shell and that of the circumscribed subshell (shown in <figref idrefs="fig25">Fig. 25</figref>
) with the help of <tableref idrefs="tab7 tab8">Tables 7 and 8</tableref>
. It is remarkable that after 12 shells, all seven Platonic and Archimedean polyhedra with octahedral symmetry have been generated (<figref idrefs="fig25">Fig. 25</figref>). The 12th shell, with the shape of a truncated cuboctahedron, encloses a total of 229 atoms with a radius of about 5.2 Å
				(in the bcc structure of α-iron), forming 13 nested polyhedra. Notice that several Archimedean polyhedra (rhombicuboctahedron, truncated cube, truncated cuboctahedron and truncated octahedron) appear in their “distorted” versions discussed above, in which some rectangular faces replace the squares of the regular polyhedra. Remember that such polyhedra are distorted in the sense that they are no longer Archimedean, since some of the faces are not regular polygons, but they still have the full octahedral symmetry of the corresponding Archimedean solids.</p>
<figure xsrc="b503582c-f25.tif" id="fig25"><title>Polyhedral expansion around an atom in the body centered cubic structure. Each polyhedron corresponds to a shell of atoms at a certain distance from a reference atom (values given here correspond to the structure of α-iron) and is circumscribed around the preceeding one. The polyhedra shown, corresponding to the <it>n</it>
-th shell as indicated, are as follows: (1) cube, (2) octahedron, (3) cuboctahedron, (4) rhombicuboctahedron, (5) cube, (6) octahedron, (7) truncated cube, (8) truncated octahedron and (12) truncated cuboctahedron. All the seven Platonic and Archimedean polyhedra of octahedral symmetry are generated in this way.</title>
</figure>
<table-entry id="tab11"><title>Results of polyhedral expansion of the most common Platonic solids</title>
<table><tgroup cols="4"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<colspec colname="4"></colspec>
<thead><row><entry colname="1"></entry>
<entry namest="2" nameend="4">Parent polyhedron</entry>
</row>
<row><entry>Operation</entry>
<entry>Tetrahedron</entry>
<entry>Octahedron</entry>
<entry>Cube</entry>
</row>
</thead>
<tbody><row><entry>Truncation (face)</entry>
<entry>Cuboctahedron</entry>
<entry>Rhombicuboctahedron</entry>
<entry>Rhombicuboctahedron</entry>
</row>
<row><entry>Truncation (edge)</entry>
<entry>Truncated tetrahedron</entry>
<entry>Truncated octahedron</entry>
<entry>Truncated cube</entry>
</row>
<row><entry>Truncation (rotated)</entry>
<entry>Icosahedron</entry>
<entry>Snub cube</entry>
<entry></entry>
</row>
<row><entry>Truncation (mixed)</entry>
<entry>Anticuboctahedron J27</entry>
<entry></entry>
<entry></entry>
</row>
<row><entry>2<it>n</it>
-truncation</entry>
<entry>Truncated octahedron</entry>
<entry>Truncated cube</entry>
<entry>Truncated octahedron</entry>
</row>
<row><entry></entry>
<entry></entry>
<entry>Truncated cuboctahedron</entry>
<entry>Truncated cuboctahedron</entry>
</row>
<row><entry>Edge augmentation</entry>
<entry>Octahedron</entry>
<entry>Cuboctahedron</entry>
<entry>Cuboctahedron</entry>
</row>
<row><entry>Face augmentation</entry>
<entry>Tetrahedron</entry>
<entry>Cube</entry>
<entry>Octahedron</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<p>A similar analysis can be carried out for other solid state structures. In the case of the fcc structure, that can be grown from an octahedron, either centered or empty, also a variety of polyhedra with octahedral symmetry are generated (a full listing of the polyhedra formed by successive shells is given as ESI<fnoteref idrefs="fn1"></fnoteref>
). In the former case, all the nested polyhedra generated are Platonic or Archimedean, whereas in the latter case “distorted” versions of some Archimedean polyhedra appear again. In the diamond structure, that can be grown from the empty tetrahedron of an adamantanoid unit (see <figref idrefs="fig10">Fig. 10</figref>
), polyhedra having either tetrahedral or octahedral symmetry are generated. A different pattern of nested polyhedra are found in diamond if the subsequent shells are considered around a central carbon atom. The wurtzite structure of ZnS circumscribes around a ZnS<inf>4</inf>
 tetrahedron two Johnson polyhedra, a Zn<inf>12</inf>
 anticuboctahedron (J27) and an S<inf>15</inf>
 triaugmented hexagonal prism (J57), as well as a Zn<inf>6</inf>
 trigonal prism.</p>
<p>As pointed out above, all the Archimedean polyhedra with icosahedral symmetry can be generated starting from either the icosahedron or the dodecahedron, as summarized in <figref idrefs="fig26">Fig. 26</figref>
. The relationships represented there can help us understand the sequence of nested clusters found in the rather complex structure of Na<inf>13</inf>
(Cd<inf>1−<it>x</it>
</inf>
Tl<inf><it>x</it>
</inf>
)<inf>27</inf>
,<citref idrefs="cit68">68</citref>
 which has been elegantly described by Li and Corbett. Hence, the large M<inf>128</inf>
 cluster present in the repeat unit is described as M<inf>12</inf>
-icosahedron ⊂ Na<inf>20</inf>
-dodecahedron ⊂ Cd<inf>12</inf>
-icosahedron ⊂ M<inf>60</inf>
-truncated icosahedron ⊂ Na<inf>24</inf>
-truncated octahedron, where M represents the disordered sites with partial occupancies of Cd and Tl. The icosahedral shape measure of the central M<inf>12</inf>
 group (0.0001) indicates a perfect icosahedron beyond crystallographic accuracy, whereas the third shell has an icosahedral measure of 0.16, suggesting that the icosahedral symmetry is slightly perturbed from the center to the borders of the unit cell. Notice that the icosahedral symmetry of the four inner shells is incompatible with translational symmetry, hence the outermost shell in the unit cell is distorted toward the pseudo-octahedral symmetry (common subgroup <it>T</it>
<inf>h</inf>
) required for a cubic space group. Even if the crystallographic symmetry of the central shell is <it>T</it>
<inf>h</inf>
, its real shape is that of a perfect icosahedron within crystallographic accuracy.</p>
<figure xsrc="b503582c-f26.tif" id="fig26"><title>Polyhedra generated from an icosahedron through the following operations: face augmentation (A<inf>F</inf>
), edge augmentation (A<inf>E</inf>
), face-directed truncation (T<inf>f</inf>
), rotated truncation (T<inf>r</inf>
), face-directed truncation (T<inf>f</inf>
), and two double truncations with independent decagons (T<inf>2i</inf>
) and with edge-sharing decagons (T<inf>2e</inf>
). The correspondence between some vertices of the icosahedron and the faces of the generated polyhedra are noted with gray dots, that between edges of the icosahedron and vertices of the cuboctahedron with green squares.</title>
</figure>
<p>Another fascinating example of nesting of clusters with icosahedral symmetry is provided by the structure of [Pd<inf>145</inf>
(CO)<inf>x</inf>
(PEt<inf>3</inf>
)<inf>30</inf>
], reported by Dahl and co-workers.<citref idrefs="cit97">97</citref>
 The composition and shape of the five Pd shells and of the capping P atoms correspond to the following nesting sequence: Pd<inf>12</inf>
-icosahedron ⊂ Pd<inf>30</inf>
-icosidodecahedron ⊂ Pd<inf>12</inf>
-icosahedron ⊂ Pd<inf>60</inf>
-rhombicosidodecahedron ⊂ Pd<inf>30</inf>
-icosidodecahedron ⊂ P<inf>30</inf>
-icosidodecahedron, and the relationship between two of the Pd shells, a Pd<inf>30</inf>
 icosidodecahedron and a Pd<inf>60</inf>
 rhombicosidodecahedron, has been shown above in <figref idrefs="fig14">Fig. 14</figref>
. Since all the shells of the Pd<inf>145</inf>
P<inf>30</inf>
 core have icosahedral symmetry, it can be thought of as an icosahedral nanoparticle with a diameter of about 2.2 nm covered with an hydrophobic coating of 90 ethyl groups. This makes us think that it is possible to grow crystalline materials without translational symmetry but a large number of atoms arranged concentrically with icosahedral symmetry. In fact, A. L. Mackay proposed in 1962 nesting of icosahedral shells as a non-crystallographic dense packing principle,<citref idrefs="cit133">133</citref>
 and the impact that such a proposal have had on the structural studies on nanometer-sized icosa-twins, atomic clusters, intermetallics and quasicrystals has been the object of a recent review.<citref idrefs="cit134">134</citref>
</p>
</section>
<section><title>Simplicity of complex systems: polyhedra of polyhedra</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row valign="top"><entry><it>El secret dels egipcis (un dels secrets dels egipcis) ha estat la descoberta, simple i portentosa, que tota la realitat és simbòlica i que tot símbol és real.</it>
</entry>
</row>
<row><entry><it>El secret del seu art (un dels secrets del seu art) ha consistit en la troballa, excepcional entre totes, de la simetria asimètrica i de la asimetria simètrica.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Josep Palau i Fabre, <it>Sonet sord sobre l'art egipci</it>
<citref idrefs="cit135">135</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row valign="top"><entry>[The secret of the Egyptians (one of the secrets of the Egyptians) has been the discovery, simple and wonderful, that all reality is symbolic and that every symbol is real.</entry>
</row>
<row><entry>The secret of their Art (one of the secrets of their Art) has consisted in the finding, exceptional among all, of the asymmetric symmetry and of the symmetric asymmetry.]<citref idrefs="cit99">99</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>An elegant way in which some complex structures are assembled consists in putting together small polyhedra (<it>e.g.</it>
, MX<inf>4</inf>
 tetrahedra) through vertex sharing in such a way as to form a larger tetrahedron. The resulting structure is thus a polyhedron of polyhedra, or a <it>superpolyhedron</it>
, among which the ones most intensely studied in recent years are the <it>supertetrahedra</it>
. In short, a T<it>n</it>
 supertetrahedron is obtained by an assembly of n vertex-sharing tetrahedra along the edges of a larger <it>supertetrahedron</it>
, typically forming units with M<inf>4</inf>
X<inf>10</inf>
, M<inf>10</inf>
X<inf>20</inf>
, M<inf>20</inf>
X<inf>35</inf>
 and M<inf>35</inf>
X<inf>56</inf>
 formulae for <it>n</it>
				= 2, 3, 4 and 5, respectively. These supertetrahedra can be seen as fragments of increasing size of the zinc blende structure.<citref idrefs="cit136">136</citref>
 In the T3 supertetrahedron, the edge centers of the supertetrahedron are occupied by an M atom. Since we have already seen that edge-augmentation of a tetrahedron results in an octahedron (<tableref idrefs="tab8">Table 8</tableref>
), it can be concluded that a T3 supertetrahedron is in fact a compound of a tetrahedron and an octahedron of M atoms, as confirmed by the corresponding shape measures of the In atoms in the In<inf>10</inf>
S<inf>20</inf>
<sup>10−</sup>
 clusters of the ASU-31 and ASU-32 indium sulfide open frameworks<citref idrefs="cit137">137</citref>
				(shown in <figref idrefs="fig27">Fig. 27(a) and (b)</figref>
). To point to other possibilities of supertetrahedra, we show also in <figref idrefs="fig27">Fig. 27(c)</figref>
 the T3 supertetrahedron decorated with thiolato groups of formula [Zn<inf>10</inf>
(μ<inf>2</inf>
-SPh)<inf>12</inf>
(μ<inf>3</inf>
-S)<inf>4</inf>
(SPh)<inf>4</inf>
]<sup>4−</sup>
.<citref idrefs="cit138">138</citref>
</p>
<figure xsrc="b503582c-f27.tif" id="fig27"><title>Above: T3 supertetrahedral (In<inf>10</inf>
S<inf>20</inf>
)<sup>10−</sup>
 units in the ASU-32 structure, showing (a) the ten coordination polyhedra of the In atoms and (b) the relationship between the In tetrahedron (blue) and octahedron (green) generated <it>via</it>
 edge-augmentation. Also shown is (c) a Zn T3 supertetrahedron decorated with phenylthiolato groups in [Zn<inf>10</inf>
(μ<inf>2</inf>
-SPh)<inf>12</inf>
(μ<inf>3</inf>
-S)<inf>4</inf>
(SPh)<inf>4</inf>
]<sup>4−</sup>
.</title>
</figure>
<p>As a further example, consider the T4 supertetrahedron of composition Cd<inf>4</inf>
In<inf>16</inf>
S<inf>35</inf>
, found in the crystal structure of the [Cd<inf>16</inf>
In<inf>64</inf>
S<inf>134</inf>
]<sup>44−</sup>
 anion.<citref idrefs="cit139">139</citref>
 Such a supertetrahedron is usually presented as an assembly of the 20 vertex-sharing coordination tetrahedra of the metal atoms (<figref idrefs="fig28">Fig. 28(a)</figref>
). However, we may notice that there are three sets of symmetry-related tetrahedra, occupying the vertices, edges and face-centers of the supertetrahedron, respectively. As a consequence, we can also describe the arrangement of these metal atoms by three isogonal polyhedra that must each preserve the full tetrahedral symmetry. In principle, according to <tableref idrefs="tab1">Table 1</tableref>
, those polyhedra can only be Platonic, Archimedean, prismatic or antiprismatic. However, only in the first two families can we find figures that possess the tetrahedral symmetry. The vertex metal atoms (blue in <figref idrefs="fig28">Fig. 28</figref>
), certainly form a tetrahedron; those placed at the face centers (violet) form perforce the dual polyhedron, <it>i.e.</it>
, an interpenetrated tetrahedron, and those at the edges (green) form a truncated tetrahedron. It is interesting to recognize an artistic depiction of a supertetrahedron by Jamnitzer (<figref idrefs="fig28">Fig. 28(c)</figref>
) in his <it>Perspectiva Corporum Regularium</it>
 long before the birth of Structural Chemistry.<citref idrefs="cit31">31</citref>
</p>
<figure xsrc="b503582c-f28.tif" id="fig28"><title>T4 supertetrahedron in [Cd<inf>16</inf>
In<inf>64</inf>
S<inf>134</inf>
]<sup>44−</sup>
 showing (a) the three types of MS<inf>4</inf>
 coordination tetrahedra that occupy the vertices (In, blue), edges (In, green) and face centers (Cd, violet) of the supertetrahedron, (b) the polyhedra that each type of metal atoms form, and (c) an upside-down Jamnitzer's representation of the same T4 supertetrahedron.</title>
</figure>
<p>Apparently less work has been devoted to superoctahedra, but superoctahedral structures do exist, as in a cluster of Pd<inf>8</inf>
Ni<inf>36</inf>
				(<it>i.e.</it>
, M<inf>44</inf>
) composition, in which four metal atoms are regularly placed along the external edges of the octahedral cluster, reported by Longoni and co-workers.<citref idrefs="cit140">140</citref>
 According to the relationships between polyhedra discussed above, such a superoctahedron is in fact much more than just an octahedron formed by octahedra: the inner Pt<inf>6</inf>
 shell is certainly octahedral; the Ni atoms placed at the vertices of the outer superoctahedron also form an octahedron, the Ni atoms occupying the centers of the faces form a cube, and the Ni atoms occupying the edges give raise to a truncated octahedron. A related structure, that of the (Pb<inf>18</inf>
I<inf>44</inf>
)<sup>8−</sup>
 anion,<citref idrefs="cit141">141</citref>
 can be described as a similar cluster of I<inf>44</inf>
 atoms, but consideration of the coordination octahedra of the Pb atoms shows an octahedron of octahedra, in which three Pb atoms are aligned along the edges of the large octahedron (<figref idrefs="fig29">Fig. 29(a)</figref>
). A completely different way of arranging coordination octahedra in an octahedral way has been reported by Robson and co-workers,<citref idrefs="cit142">142</citref>
 in a compound in which six octahedrally coordinated Cd atoms are held in place through the arms of a substituted benzene (<figref idrefs="fig29">Fig. 29(b)</figref>
), resulting in a nearly regular octahedron (CShM = 0.45).</p>
<figure xsrc="b503582c-f29.tif" id="fig29"><title>(a) Superoctahedron of [Pb<inf>18</inf>
I<inf>44</inf>
]<sup>8−</sup>
 and (b) octahedral arrangement of the coordination polyhedra of six Cd atoms organized through an organic assembly unit.<citref idrefs="cit142">142</citref>
</title>
</figure>
<p>To show just one example with icosahedral symmetry, let us focus on the solid state structure of B<inf>6</inf>
O.<citref idrefs="cit143">143</citref>
 In that compound, a B<inf>12</inf>
 icosahedron is bonded through six vertices to neighboring B<inf>12</inf>
 groups, whereas the remaining six vertices are connected to six neighboring B<inf>12</inf>
 icosahedra through bridging oxygen atoms. The outcome of such an arrangement is that the centroids of the twelve nearest neighbor B<inf>12</inf>
 groups form a nice cuboctahedron of icosahedra (<figref idrefs="fig30">Fig. 30</figref>
). The composition of polyhedra is the same in the structure of α-boron, even if in that case there are no direct bridges between the central icosahedron and six neighboring icosahedra. It is interesting that we have a combination of icosahedral and octahedral symmetries in the same structure. Strictly speaking, the site symmetry of the center of the icosahedra is <it>D</it>
<inf>3d</inf>
, which is a common subgroup of both <it>I</it>
<inf>h</inf>
 and <it>O</it>
<inf>h</inf>
				(see <compoundref idrefs="chem4">4</compoundref>
), but shape measures tell us that nearly perfect B<inf>12</inf>
 and B<inf>6</inf>
O<inf>6</inf>
 icosahedra (icosahedral CShMs of 0.19 and 0.13, respectively) coexist in good harmony with the cuboctahedron formed by the centroids of 13 icosahedra (one at the center and twelve at the vertices of the cuboctahedron, with a cuboctahedral CShM of 0.09, <it>cf.</it>
 icosahedral CShM of 5.36).</p>
<figure xsrc="b503582c-f30.tif" id="fig30"><title>Cuboctahedron of twelve B<inf>12</inf>
 icosahedra (blue) built around an icosahedral B<inf>12</inf>
 core (orange) in the extended structure of B<inf>6</inf>
O. The central icosahedron is connected in the <it>x</it>
 and <it>y</it>
 directions to six icosahedra through oxygen bridging atoms (edge augmentation) and to six icosahedra in the <it>z</it>
 direction through direct B–B bonds (not shown for clarity).</title>
</figure>
</section>
<section><title>What do nanoparticles and dice have in common</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>A partir de François II, la forme architecturale de l'édifice s'efface de plus en plus et laisse saillir la forme géométrique, comme la charpente osseuse d'un malade amaigri. Les belles lignes de l'art font place aux froides et inexorables lignes du géomètre. Un édifice n'est plus un édifice, c'est un polyèdre.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Victor Hugo, <it>Nôtre Dame de Paris</it>
, Livre V, ch. II (1837)</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[From François II on, the architectural form of buildings becomes steadily less noticeable, and allows the geometrical form to show through, like the bone-structure of some emaciated invalid. The beautiful lines of art give way to the cold and inexorable lines of the geometer. A building is no longer a building; it is a polyhedron.]<citref idrefs="cit144">144</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>A Platonic polyhedron, the cube, is the most common shape used for dice in a wide variety of games, as known already in the ancient Roman and Greek cultures. That specific polyhedron is used as a die just because it is Platonic. In other words, because all its faces are equivalent and therefore the probability of every number marked on a face to appear on top is the same. This is true if the die has a uniform density, but also from the Roman times manipulation of the inside of the cube to alter the probability of a given number has been a common way to cheat while preserving the external aspect of the Platonic symmetry.<citref idrefs="cit145">145</citref>
</p>
<p>Knowing that the requisite to use a polyhedron as a die is that all the faces are equivalent, we can look for a wider assortment of shapes with varying numbers of faces. In fact dice with all Platonic shapes are commonly being manufactured and sold with some games (<figref idrefs="fig31">Fig. 31</figref>
), and these have 4, 6, 8, 12 or 20 faces. In all these geometries, except for the tetrahedron, every face has a parallel one and, therefore, while laying on a table on one face there is another face on top that can be unequivocally identified as the lucky number. In the tetrahedron, with one face sitting on the table, the remaining three faces have similar orientations and we cannot tell which is the selected number. For that reason the numbering of the tetrahedron is not by faces but by edges, so we can identify the lucky number by looking at the numbering of the edges corresponding to the bottom face (<figref idrefs="fig31">Fig. 31</figref>
).</p>
<figure xsrc="b503582c-f31.tif" id="fig31"><title>Commercial dice with the shapes of the Platonic polyhedra, a rhombic dodecahedron and a pentagonal trapezohedron (photograph from the author).</title>
</figure>
<p>Can we make a wider variety of dice? If we want to be fair, we must restrict ourselves to polyhedra with all faces equivalent. This rules out the Archimedean polyhedra, that have more than one type of faces, and also the prisms and antiprisms, the pyramids and the Johnson polyhedra (although a commercial pentagonal prismatic die can be found on the Internet). But we could use the bipyramids, the trapezohedra and the Catalan solids. Since the Catalan solids are duals of the Archimedean polyhedra, the non-equivalent faces of the latter correspond to non-equivalent vertices in the former, and the equivalent Archimedean vertices correspond to equivalent Catalan faces. Similarly, the duals of the isogonal prisms are the trapezohedra that are, therefore, isohedral. Even if the Catalan solids have irregular polygons as faces, they make good dice with 12, 24, 30, 48, 60 or 120 faces, some of them in more than one version (<it>e.g.</it>
, we have three choices for 24 faces and four choices for 60 faces, as seen in <tableref idrefs="tab2">Table 2</tableref>
). To the best of our knowledge, only two bipyramids (pentagonal and octogonal), one Catalan solid (the rhombic triacontahedron, with 30 faces) and a pentagonal trapezohedron can be found in the market as dice (<figref idrefs="fig31">Fig. 31</figref>
, two rightmost dice).</p>
<p>The preparation of nanoparticles of controlled shape and size constitutes one of the most active fields of current research.<citref idrefs="cit146">146</citref>
 We can consider such nanoparticles as tiny dice which may rest on any of the equivalent faces if they have Platonic or Catalan shapes, or to present two or more alternative orientations if they correspond to Archimedean or Johnson polyhedra. These nanocrystals are generally inspected <it>via</it>
 scanning electron microscopy (SEM) or transmission electron microscopy (TEM), from which one obtains in many instances a two-dimensional image that shows up a polygonal profile, even if in some cases the resolution is enough to recognize a 3D polyhedral shape. If the nanoparticles have the shape of a regular polyhedron (Platonic or Catalan), we observe the same profile independent of the face on which it is resting. In contrast, two or more different profiles can be seen when the nanoparticles correspond to Archimedean or Johnson polyhedra, as shown in <tableref idrefs="tab12">Table 12</tableref>
 for a few cases.</p>
<table-entry id="tab12"><title>Degree of the polygon shown as a profile for some Platonic (P), Archimedean (A) and Catalan (C) polyhedra when sitting on a given face. The prefix <it>i</it>
 indicates an irregular polygonal profile</title>
<table><tgroup cols="3"><colspec colname="1"></colspec>
<colspec colname="2"></colspec>
<colspec colname="3"></colspec>
<thead><row><entry>Polyhedron</entry>
<entry>Face</entry>
<entry>Profile</entry>
</row>
</thead>
<tbody><row><entry>Tetrahedron (P)</entry>
<entry>3</entry>
<entry>3</entry>
</row>
<row><entry>Cube (P)</entry>
<entry>4</entry>
<entry>4</entry>
</row>
<row><entry>Octahedron (P)</entry>
<entry>3</entry>
<entry>6</entry>
</row>
<row><entry>Dodecahedron (P)</entry>
<entry>5</entry>
<entry>10</entry>
</row>
<row><entry>Icosahedron (P)</entry>
<entry>3</entry>
<entry>6</entry>
</row>
<row><entry>Truncated tetrahedron (A)</entry>
<entry>6</entry>
<entry><it>i</it>
-9</entry>
</row>
<row><entry></entry>
<entry>3</entry>
<entry><it>i</it>
-9</entry>
</row>
<row><entry>Cuboctahedron (A)</entry>
<entry>4</entry>
<entry>4</entry>
</row>
<row><entry></entry>
<entry>3</entry>
<entry>6</entry>
</row>
<row><entry>Truncated cube (A)</entry>
<entry>6</entry>
<entry>4</entry>
</row>
<row><entry></entry>
<entry>3</entry>
<entry><it>i</it>
-12</entry>
</row>
<row><entry>Truncated octahedron (A)</entry>
<entry>4</entry>
<entry>8</entry>
</row>
<row><entry></entry>
<entry>6</entry>
<entry><it>i</it>
-12</entry>
</row>
<row><entry>Rhombicuboctahedron (A)</entry>
<entry>4</entry>
<entry>8</entry>
</row>
<row><entry></entry>
<entry>3</entry>
<entry>6</entry>
</row>
<row><entry></entry>
<entry>4</entry>
<entry><it>i</it>
-8</entry>
</row>
<row><entry>Rhombic dodecahedron (C)</entry>
<entry></entry>
<entry><it>i</it>
-6</entry>
</row>
<row><entry>Tetrakis hexahedron (C)</entry>
<entry>3</entry>
<entry>6</entry>
</row>
<row><entry>Deltoidal icositetrahedron (C)</entry>
<entry>4</entry>
<entry><it>i</it>
-8</entry>
</row>
</tbody>
</tgroup>
</table>
</table-entry>
<p>Let us consider as an example a recent report of gold nanoparticles, in which varying the synthesis conditions seems to result in different shapes.<citref idrefs="cit147">147</citref>
 Hence, the appearance of only square profiles in a TEM image allows one to conclude that these most likely have a cubic shape, as found also for Ag<inf>2</inf>
S nanocrystals.<citref idrefs="cit148">148</citref>
 In contrast, the images corresponding to different synthetic conditions show the presence of only regular hexagonal profiles that is consistent with Platonic (octahedron or icosahedron) or Catalan shapes. Given the fcc structure of Au and our above analysis of the nested polyhedra in the fcc packing, we might conclude that the hexagonal profiles observed by Sau and Murphy correspond to octahedra, since no icosahedral cluster can appear for an fcc crystal. Similarly, the tetragonal profiles observed in Ca<inf>3</inf>
Al<inf>2</inf>
(OH)<inf>12</inf>
 nanocrystals<citref idrefs="cit149">149</citref>
 cannot correspond to an octahedron, unless we assume that it is sitting on one of its vertices rather than on a face. Some examples of profiles assigned to Catalan shapes can be found in the literature,<citref idrefs="cit149 cit150">149,150</citref>
 whereas an icosahedral shape with its hexagonal profile (<tableref idrefs="tab12">Table 12</tableref>
) can be seen in the SEM micrographs of B<inf>6</inf>
O particles (<figref idrefs="fig32">Fig. 32</figref>
) prepared under high pressure.<citref idrefs="cit151">151</citref>
</p>
<figure xsrc="b503582c-f32.tif" id="fig32"><title>SEM image of icosahedral nanoparticles of B<inf>6</inf>
O. Reproduced from <citref idrefs="cit151" position="baseline">ref. 151</citref>
 with permission of the Royal Society of Chemistry.</title>
</figure>
</section>
<section><title>Final remarks</title>
<table><tgroup cols="1"><colspec colname="col1"></colspec>
<tbody><row><entry><it>Ich hab aber sonderlich dise Körper in meiner neuer Perspectif gebrauchen wöllen/dieweyl sie mancherley und viel unterschiedliche Eckh/seyten/winckel und spitzen/einwartß und außwartß gekehrt haben den jungen ansachenden diser Kunst ursach zugeben und sie damit zu raißen/der Kunst mit fleyß nachzudenken und solchs zu allerley andern sachen ferner haben zu appliciren und zu gebrauchen.</it>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>Wenzel Jamnitzer, <it>Perspectiva corporum regularium</it>
								(1568)<citref idrefs="cit31">31</citref>
</entry>
</row>
<row valign="top"><entry> </entry>
</row>
<row><entry>[I have specially chosen to use these bodies in my new perspective, since they have numerous ridges, sides, angles and tips, turned to the inside or the outside, to incite the young people who take their first steps in this art to reflect with zeal, so these possibilities could be later employed and applied to all sorts of objects.]<citref idrefs="cit99">99</citref>
</entry>
</row>
</tbody>
</tgroup>
</table>
<p>Polyhedra have been and continue to be highly useful tools for the representation of chemical structures, but also idealizations that have had for human minds some relationship with perfection and with the structure of matter from ancient times, as in the early association of the four elements with four Platonic solids. A much more recent application of Platonic polyhedra as models for the intimate structure of matter can be found in the landmark paper of G. N. Lewis,<citref idrefs="cit152">152</citref>
 in which use is made of a relationship between the octahedron and the tetrahedron. Recognizing that atoms have the tendency to complete a valence shell of eight electrons, Lewis proposed a cubic model of the atom in which the valence electrons occupy the vertices. Then, single bonds would correspond to the cubes of two bonded atoms sharing an edge, double bonds to face-sharing. Lewis recognized, though, that “<it>With the cubical structure it is not only impossible to represent the triple bond</it>
, <it>but also to explain the phenomenon of free mobility about a single bond which must always be assumed in stereochemistry</it>
”. He thus went on to propose that electrons could be paired at four of the edges of the atomic cube (<compoundref idrefs="chem13">13</compoundref>
), resulting in “<it>a model of the tetrahedral carbon atom which has been of such signal utility throughout the whole of organic chemistry</it>
”. Although from a geometrical point of view the resulting figure is not strictly a tetrahedron, which could be obtained by placing the electron pairs at vertices rather than edges, this model is topologically sound and nicely illustrates the tendency to use such ideal polyhedra as mental constructs to visualize rather abstract concepts.<ugraphic xsrc="b503582c-u8.tif" id="ugr8"></ugraphic>
</p>
<p>While studying molecular and crystal structures from the literature in the last few years, I became increasingly interested in some geometrical aspects of polyhedra, regular and non-regular, and chemical structures have been of great help to me for such a study. I eventually organized some data in tables, collected interesting chemical structures and learnt about the use of polyhedra in other fields of human knowledge. In this Perspective I have tried to share with the reader my experiences and facilitate the analysis of complex structures in terms of simple polyhedra. I only hope that it may help the reader to view Chemistry from a different perspective and to enjoy the perception of our science as one full of art, the art of sculpture at the atomic scale.</p>
</section>
</art-body>
<art-back><ack><p>The author is indebted to D. Avnir, for providing him with the concepts and tools to start systematic studies on polyhedra in Chemistry, to M. Verdaguer for frequent discussions and encouragement, to M. Llunell, P. Alemany, D. Casanova and J. Cirera for their stimulating and supportive collaboration, to L. Duch for his help with the transcription and translation of Jamnitzer's quotation and to A. Terron for kindly providing an English version of his verses. The author also thanks the <it>Generalitat de Catalunya</it>
 for a <it>Distinció per a la Promoció de la Recerca Universitària</it>
. Financial support from the Dirección General de Investigación (MEC), project BQU2002-04033-C02-01, and from the Comissió Interdepartamental de Ciència i Tecnologia (CIRIT), grant 2001SGR-0044 are gratefully acknowledged. Polyhedral figures have been generated with the Poly Pro (Pedagogery Software, <url url="http://www.peda.com/polypro">http://www.peda.com/polypro</url>
) and Crystal Maker 6.3 (<url url="http://www.crystalmaker.co.uk">http://www.crystalmaker.co.uk</url>
) programs.</p>
</ack>
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<mods version="3.6"><titleInfo><title>Polyhedra in (inorganic) chemistry</title>
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<name type="personal"><namePart type="given">Santiago</namePart>
<namePart type="family">Alvarez</namePart>
<affiliation>Departament de Química Inorgànica and Centre de Recerca en Química Teòrica, Universitat de Barcelona, 08028Martí i Franquès 1-11, Barcelona</affiliation>
<description>Santiago Alvarez was born in Panamá, República de Panamá, in 1950 and studied Chemistry at the University of Barcelona, where he obtained a PhD working on inorganic vibrational spectroscopy under the supervision of Prof. J. Casabó. After doing experimental research on one-dimensional conductors in Barcelona, he carried out theoretical research with Prof. R. Hoffmann at Cornell University. He was appointed as Professor Titular in the University of Barcelona in 1984 and has held an inorganic chemistry chair since 1987. He has been a visiting scientist in the USA, France, Chile and Israel, was elected as a Distinguished Researcher by the Generalitat de Catalunya in 2000 and has been awarded the prize for research in Inorganic Chemistry of the Real Sociedad Española de Química and the Solvay prize for research in Chemical Science in 2003. His research interests include bonding and structure in molecular and solid state transition metal compounds, structural and structure–property correlations and the application of continuous symmetry and shape measures to the stereochemical description of transition metal compounds. [b503582c-p1.tif]</description>
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<originInfo><publisher>The Royal Society of Chemistry.</publisher>
<dateIssued encoding="w3cdtf">2005</dateIssued>
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<abstract>A systematic description of polyhedra with varying degrees of regularity is illustrated with examples of chemical structures, mostly from different fields of Inorganic Chemistry. Also the geometrical relationships between different polyhedra are highlighted and their application to the analysis of complex structures is discussed.</abstract>
<note type="footnote" displayLabel="fn1">Electronic supplementary information (ESI) available: Table S1: Complete list of the Johnson polyhedra, ordered according to the number of vertices (V), giving the number of edges (E) and faces (F) in each case. Table S2: Nested polyhedra that appear as successive shells in prototypical solid state structures. The shells are ordered according to increasing distance to the center. Table S3: Nested polyhedra of icosahedral symmetry in the molecular structure of [Pd145(CO)x(PEt3)30]. Fig. S1: Polyhedra generated from a dodecahedron through augmentation and truncation operations. Fig. S2: Polyhedra generated from a cube through augmentation and truncation operations. Fig. S3: Diamond shells 2 and 6 around centroid of adamantanoid unit, forming an octahedron and a truncated octahedron. Fig. S4: Generation of an approximate icosahedron of bridging selenide ions in an Ag8 cube in the molecular structure of Ag8Cl2[Se2P(OEt)2]6. See http://www.rsc.org/suppdata/dt/b5/b503582c/</note>
<note>Reflections on the use of different kinds of polyhedra in Chemistry and elsewhere, and on the geometrical and chemical relationships between nested polyhedra, such as the seven Platonic and Archimedean polyhedra with octahedral symmetry found at around each atom in the bcc structure illustrated in the cover [b503582c-ga.tif]</note>
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<identifier type="ISSN">1477-9226</identifier>
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<identifier type="DOI">10.1039/b503582c</identifier>
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Polyhedra in (inorganic) chemistry

Polyhedra in (inorganic) chemistry

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