## Christopher Hammond

Print publication date: 2015

Print ISBN-13: 9780198738671

Published to Oxford Scholarship Online: August 2015

DOI: 10.1093/acprof:oso/9780198738671.001.0001

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# (p.393) Appendix 2 Polyhedra in crystallography

Source:
The Basics of Crystallography and Diffraction
Publisher:
Oxford University Press

In this Appendix the crystallographic and geometrical properties of the polyhedra which are introduced and briefly described in this book are summarized. The word comes from the Greek meaning many (poly) faces or planes (hedra). The (simpler) polyhedra are simply described by their number of faces as indicated below. It is curious that only the hexahedron has a ‘simple’ name—the cube.

# Polyhedron names

For the simpler polyhedra the prefix indicates the number of faces. Those of importance in crystallography are: tetra (4), penta (5), hexa (6), octa (8), deca (10), dodeca (12), icosa (20), icosidodeca (32). Further prefixes indicate the shapes of the faces themselves and/or their symmetry, e.g. pentagonal or rhombic dodecahedra (polyhedra with twelve pentagonal or rhombic (diamond shaped) faces respectively), or trapezorhombic dodecahedra (polyhedra in which the twelve rhombic faces are arranged in hexagonal symmetry). Further prefixes such as triakis or pentakis indicate that plane triangular or pentagonal faces are replaced by a group of three or five faces respectively. Thus a triakis tetrahedron has twelve ($4×3$) faces, and so on.

## Truncation

Many polyhedra are described as ‘truncated’ which simply means ‘cutting off the corners’ of a polyhedron—and the truncated polyhedron thus created depends on ‘how much’ of the corners are cut off. As shown in Fig. A2.1(a)(c) the truncated cube, cubeoctahedron and truncated octahedron are created by cutting increasing amounts off the cube corners. Alternatively, we could ‘start off’ with an octahedron and obtain the same polyhedra by again cutting off the corners as shown in Fig. A2.1(d)(f). Truncation can also be applied to the newly-created corners and edges: the possibilities are almost endless and a whole series of polyhedra, with many faces of different shapes and sizes (and with rather long descriptive names) can be created. The process has its analogy in the beautiful patterns of faces which can develop in crystal growth.

Fig. A2.1. Alternative truncation procedures to create a truncated cube, cubeoctahedron and truncated octahedron. (a) $→$ (b) $→$ (c) successive truncation of a cube; (d) $→$ (e) $→$ (f) successive truncation of an octahedron. (From Polyhedra by Peter R. Cromwell, Cambridge University Press, Cambridge, 1999.)

## Duality

Polyhedra of identical symmetry are said to be dual if the faces of one correspond to the corners (or vertices) of the other and vice versa. Thus the cube and octahedron are duals: the eight vertices of the cube correspond to the eight faces of the octahedron and, conversely, the six vertices of the octahedron correspond to the six faces of the cube. (p.394) Similarly, the pentagonal dodecahedron and icosahedron are duals. The tetrahedron is self-dual: the four vertices of one correspond to the four faces of the other and vice versa.

## Classification of polyhedra

There is an enormous number of polyhedra, including beautiful star-shapes with re-entrant faces. However, those of most importance in crystallography are the Platonic and Archimedean polyhedra, deltahedra, coordination and space-filling (Voronoi) polyhedra and Brillouin zones. These groupings are of course not mutually exclusive.

### (A) The five perfect (regular) polyhedra or Platonic solids

These polyhedra (Fig. A2.2) each have identical faces and edges of equal length. The vertices and faces of each circumscribe and inscribe respectively the surface of a sphere.

Fig. A2.2. The five perfect polyhedra or Platonic solids: tetrahedron, octahedron, cube, icosahedron, pentagonal dodecahedron.

The tetrahedron has cubic point group symmetry $4ˉ3m$ and the octahedron and cube have the full (holosymmetric) cubic point group symmetry $4​/m3ˉ2​/m≡m3ˉm$. The icosahedron and pentagonal dodecahedron both have point group symmetry $2​/m3ˉ5ˉ$—the symmetry possessed by quasicrystals, buckminsterfullerene and some viruses. Of all the Platonic polyhedra, only the cube is space-filling.

These solids were known to the Ancient Greeks and were clearly identified and described by Theaetetus of Athens (d. 369 BC). They were considered by Plato to constitute the fundamental building-blocks of matter: fire (tetrahedron), solid (cube), air (octahedron), liquid (icosahedron) and the universe or cosmos itself (pentagonal (p.395) dodecahedron). Hence, they are attributed to him, instead of (more properly) to Theaetetus.

The notion that the Platonic solids had a cosmic significance was taken up by Johannes Kepler in his first book Mysterium Cosmographicum, 1596, in which he attempted to show that the ratios of the diameters of the (circular) orbits of the then-known six planets corresponded with the ratios of the inscribed and exscribed diameters of the spheres of the five perfect solids. Thus the ratios of the diameters of the inscribed and exscribed spheres for an octahedron correspond to the orbits of Mercury and Venus; that for an icosahedron the orbits of Venus and Earth, and so on. In short he attempted to explain or model the orbits of the planets in purely geometrical terms. The model failed of course, even in Kepler’s time, by observation and calculations of greater precision on the ellipticity of the planetary orbits.

### (B) The thirteen semi-regular or Archimedean Polyhedra

In these polyhedra, the discovery of which is attributed to Archimedes, every face is a regular polygon as in the Platonic polyhedra but the faces by be of different kinds—two kinds in the case of ten of the polyhedra and three kinds in the remaining three. Around each vertex the faces are arranged in the same order, the edges are of equal length and all the polyhedra are inscribable in a sphere. A complete list is given in Table A2.1 and those of particular crystallographic interest are shown in Fig. A2.3.

Table A2.1 The Archimedean polyhedra

No.

Name

No. and shapes of faces

Point group symmetry

1

Truncated tetrahedron

4 hexag, 4 triang

$4ˉ3m$

2

Truncated cube

8 triang, 6 oct

$m3ˉm$

3

Truncated octahedron

6 square, 8 hexag

$m3ˉm$

4

Truncated dodecahedron

20 triang, 12 decag

$m3ˉ5ˉ$

5

Truncated icosahedron

12 pentag, 20 hexag

$2​/m3ˉ5ˉ$

6

Cubeoctahedron

8 triang, 6 square

$m3ˉm$

7

Icosidodecahedron

20 triang, 12 pentag

$2​/m3ˉ5ˉ$

8

Truncated cubeoctahedron (great rhombicubeoctahedron)

12 square, 8 hexag, 6 oct

$m3ˉm$

9

(small) rhombicubeoctahedron

8 triang, 18 square

$m3ˉm$

10

Truncated icosidodecahedron

30 square, 20 hexag, 12 decag

$2​/m3ˉ5ˉ$

11

Rhombicosidodecahedron

20 triang, 30 square, 12 pentag

$2​/m3ˉ5ˉ$

12

Snub cubeoctahedron

32 triang, 6 square

432

13

Snub icosidodecahedron

80 triang, 12 pentag

235

Fig. A2.3. Six of the thirteen Archimedean polyhedra, Nos. 1, 5, 8, 9, 11, 12 (see Table A2.1). Figure A2.1 shows Nos. 2, 3, 6. (From Mathematical Models by H.M. Cundy and A.P. Rollett, 2nd edn, Clarendon Press, Oxford, 1961 reprinted by Tarquin Publications, St Albans, 2006.)

We owe the (rather long) names of these polyhedra to Kepler, and as indicated, they may be obtained by truncation of the corresponding Platonic solids—either a simple truncation of the vertices as shown for example in Fig. A2.1, or a more complex truncation of the newly created vertices or edges. For example the truncated cubeoctahedron (also known as the great rhombicubeoctahedron) and the (small) rhombicubeoctahedron are obtained by truncating the cubeoctahedron in two different ways. The last two polyhedra, described by the curious word ‘snub’, are not derived simply by a process (p.396) (p.397) of truncation, but also involve a ‘twist’ or rotation, which destroys mirror planes and centres of symmetry. The point groups of the Archimedean polyhedra are also of interest. Seven have crystallographic symmetry (cubic point groups $4ˉ$3m, $m3ˉm$, and 432) and six have icosahedral symmetry (the two icosahedral point groups 2/m$3ˉ5ˉ$ and 235). Note again that the point groups 432 and 235 are enantiomorphic.

Fig. A2.4. (a) Rhombic dodecahedron (dual of cubeoctahedron), and (b) rhombic triacontahedron (dual of icosidodecahedron). (a) is the space-filling polyhedron for the cubic F lattice and also the coordination polyhedron for the cubic I lattice.

Of these thirteen polyhedra, only the truncated octahedron, also described by Lord Kelvin as a tetrakaidecahedron (the Voronoi polyhedron for the cubic I lattice) is space-filling (Fig. 3.9). The pattern of faces in the truncated icosahedron corresponds to that of C60 (see Section 1.11.6). The duals of the Archimedean polyhedra have the same symmetry. They are perhaps of less interest, except for the dual of the cubeoctahedron which is a rhombic dodecahedron (the space-filling Voronoi polyhedron for the cubic F lattice (Fig. 3.8(a))) and that of the icosidodecahedron which is a rhombic triacontahedron (Fig. A2.41). The ratios of the lengths of the long and short diagonals of the rhombic or diamond-shaped faces are not the same: for the rhombic dodecahedron it is $2:1$ and for the rhombic triacontahedron it is $5+1/2:1$ or 1.618 : 1—the Golden ratio (see Section 2.9).

### (C) The eight deltahedra

In these polyhedra all the faces are equilateral triangles (Greek Δ‎). Three (the tetrahedron, octahedron and icosahedron) are also Platonic polyhedra. The remaining five have 6, 10, 12, 14 and 16 faces. Two, the triangular and pentagonal dipyramids with 6 and 10 faces, are shown in Fig. A2.5. Their duals are triangular and pentagonal prisms respectively.

Fig. A2.5. Two of the eight deltahedra: the triangular and pentagonal dipyramids.

### (p.398) (D) Coordination and space-filling polyhedra (Voronoi polyhedra or Dirichlet domains)

Coordination polyhedra, which represent the pattern of atoms or lattice points surrounding an atom or lattice point, should be clearly distinguished from the domains (Voronoi polyhedra or Dirichlet domains) which define the environment around an atom or lattice point and which are always space-filling (see Section 3.4). A vertex of a coordination polyhedron corresponds to the face of the corresponding domain, but the polyhedra are not necessarily duals.

They are duals, for example, in the cubic F lattice in which the coordination polyhedron is a cubeoctahedron (Fig. A2.1(b) and (e)) and the space-filling polyhedron is a rhombic dodecahedron (Fig. A2.4(a)) but not in the case of the cubic I lattice in which the coordination polyhedron is a rhombic dodecahedron2 and the space-filling polyhedron is a truncated octahedron.

Fig. A2.6. Space-filling polyhedra for (a) the NaCl (rock salt) structure, (b) the ZnS (sphalerite) structure, (c) the ZnS (wurtzite) structure, (d) the NiAs (niccolite) structure, and (e) the α‎-Al2O3 (corundum) structure.

The procedure for constructing the space-filling Voronoi polyhedra for lattices is described in Section 3.4. For crystal structures consisting of only one kind of atom3—e.g. the simple cubic, bcc, fcc and hcp metal structures the procedure is the same. For structures consisting of more than one kind of atom the procedure is to construct the space-filling polyhedron around one kind of atom and its nearest neighbours of the same kind.4 The atoms of other kinds are then shown as occupying the appropriate vertices of the space-filling polyhedron.

A few examples will make this clear. In the fcc NaCl and ZnS (zinc blende or sphalerite) structures, the space-filling polyhedron centred around a Cl or S anion is a rhombic dodecahedron (Fig. 3.8 (a)). In NaCl the Na cations, which occupy all six octahedral sites, are situated at the six vertices where four edges (or faces) meet (Fig. A2.6(a)). In sphalerite the Zn cations that occupy half the tetrahedral sites are situated at four of the eight vertices where three edges (or faces) meet (Fig. A2.6(b)). In the ZnS (wurtzite), (p.399) (p.400) NiAs (niccolite), and α‎-Al2O3 (corundum), structures where the anions are (approximately) hexagonal close-packed, the anion-centred space-filling polyhedron is a trapezorhombic dodecahedron (Fig. 3.8(b)). In wurtzite the Zn cations are again situated at four of the eight vertices where three edges (or faces) meet (Fig. A2.6(c)); in niccolite the Ni cations are situated at all six vertices where four edges (or faces) meet (Fig. A2.6(d)); and in α‎-Al2O3 the Al cations are situated at four such vertices (Fig. A2.6(e)).

In more complex crystal structures, such as the spinels (Section 1.11.3, p. 36), two rhombic dodecahedra are needed to represent the ordered arrangement of the cations in the appropriate octahedral and tetrahedral interstitial sites.

Table A2.2 Coordination polyhedra for interstitial sites and radius ratios

Structure type

Co-ordination polyhedron around interstitial site

Simple cubic

Cubic

0.732

Square

0.414

Cubic close packed

Octahedral

0.414

Square

0.414

Tetrahedral

0.225

Triangular

0.155

Hexagonal close packed

Octahedral

0.414

Tetrahedral

0.225

Triangular

0.155

Body centered cubic

‘Distorted’ octahedral

0.154

‘Distorted’ tetrahedral

0.291

Simple hexagonal

Triangular prism

0.528

Triangular

0.155

Table A2.3 Coordination and space-filling polyhedra

Structure/lattice type

Co-ordination polyhedron

Space-filling polyhedron

cubic P (simple cubic)

octahedron (square dipyramid)

cube (dual)

cubic F (fcc)

cubeoctahedron

rhombic dodecahedron (dual)

cubic I (bcc)

rhombic dodecahedron

truncated octahedron

hexagonal P (simple hexagonal)

hexagonal dipyramid

hexagonal prism (dual)

hexagonal close-packed (hcp)

hexagonal cubeoctahedron

trapezorhombic dodecahedron (dual)

NaCl (structure type)

octahedron, Na around Cl, or Cl around Na

rhombic dodecahedron, with Na or Cl at the 6 vertices where 4 edges meet, Fig. A2.6(a)

ZnS (zinc blende) (structure type)

tetrahedron, Zn around S, or S around Zn

rhombic dodecahedron with Zn at 4 of the 8 vertices where 3 edges meet, Fig. A2.6(b)

ZnS (wurtzite) (structure type)

tetrahedron, Zn around S, or S around Zn

trapezorhombic dodecahedron with Zn at 4 of the 8 vertices where 3 edges meet Fig. A2.6(c)

NiAs (niccolite) (structure type)

trigonal prism, Ni around As

trapezorhombic dodecahedron, with Ni atoms at the 6 vertices where 4 edges meet, Fig. A2.6(d)

α‎-Al2O3

tetrahedron (distorted), Al around O

trapezorhombic dodecahedron, with Al atoms at the 4 of the 6 vertices where 4 edges meet, Fig. A2.6(e)

CsCl (structure type)

cube, Cs around Cl, or Cl around Cs

cube, with Cs or Cl at the 8 vertices

μ‎ltirow464ptCaF2 and Li2O (fluorite and anti-fluorite structure types)

cube, F, or Li around Ca, or O

rhombic dodecahedron, with F or Li at the 8 vertices where 3 edges meet

tetrahedron, Ca, or O around F or Li

cube, with Ca or O situated at 4 of the 8 vertices forming a tetrahedron

Coordination polyhedra also determine the radius ratios of interstitial sites, e.g. between the atoms in metals as described in Section 1.6 or between close-packed oxygen anions. The data are summarized in Table A2.2 which includes the interstitial sites delineated by squares and triangles.

Fig. A2.7. Construction for the first Brillouin zone for a cubic P lattice (dashed lines). θ‎ is shown for reflection from the (100) plane. The second Brillouin zone is shown by the dotted lines.

Table A2.3 lists the coordination and space-filling polyhedra for some simple structures.

### (E) Brillouin zones

Brillouin zones are ‘nests’ of polyhedra in k-space (reciprocal space with the scale factor 2$π)$. Their surfaces represent the directions and wavelengths (or wavenumbers) of free electrons in crystals which are Bragg-reflected by the lattice planes. The first, innermost Brillouin zone corresponds to the reflections from the planes of largest d-spacing, the next Brillouin zone corresponds to reflections from the planes of next largest spacing, and so on.

The first Brillouin zone is equivalent to the Voronoi polyhedron or Wigner–Seitz cell for (p.401) the corresponding reciprocal lattice. For example, for the cubic F lattice, for which the corresponding reciprocal lattice is cubic I, it is a truncated octahedron and for the cubic I lattice, for which the corresponding reciprocal lattice is cubic F, it is a rhombic dodecahedron.

The constructional procedure for Brillouin zones may be illustrated for a cubic P (simple cubic) lattice with a cell edge length a and for which the corresponding reciprocal (k-space) lattice is also cubic P with a cell edge length $2π​/a$ (Fig. A2.7).

Consider electrons travelling in the x*-y* plane reflected from lattice planes parallel to the z* axis. In order of decreasing dhkl-spacing (or increasing $dhkl∗$ spacing) these are 100, 110, 200, 210, … Consider an electron with wavevector k as indicated:

$Display mathematics$

For reflection from the 100 planes, the corresponding value of k is found by substituting for λ‎ in Bragg’s law

$Display mathematics$

(p.402) which value is indicated by point P. The locus of P, for all values of θ‎, and for reflections from the 010, etc. planes, gives rise to the outline of the first Brillouin zone, as indicated by the dashed lines. In three dimensions the Brillouin zone is a cube.

The second Brillouin zone arises from reflections from the 110 planes and, proceeding as before, the corresponding values of k are indicated by the dotted lines. In three dimensions this second Brillouin zone is a rhombic dodecahedron (like the first Brillouin zone for the cubic I reciprocal lattice).

## Notes:

(1) Only the rhombic dodecahedron was known to Archimedes, the rhombic triacontahedron was known to Kepler and the remaining 11 duals of the Archimedean polyhedra were discovered by the French mathematician E.C. Catalan in 1865.

(2) The eight ‘obtuse’ vertices correspond to the lattice points at the corners of the cell and the six ‘acute’ vertices correspond to the lattice points at the centres of neighbouring cells as indicated in Fig. A2.4(a)

(3) The term ‘atom’ encompasses charged species, i.e. both cations and anions.

(4) E.W. Gorter (1970) Classification representation and prediction of crystal structures of ionic compounds. J. of Solid State Chem. 1, 279. See also R.J.D. Tilley, John Wiley & Sons, Ltd. (2006) Crystals and Crystal Structures.