(p.393) Appendix 2 Polyhedra in crystallography
(p.393) Appendix 2 Polyhedra in crystallography
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 ($4\times 3$) 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 newlycreated 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.
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 selfdual: 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 starshapes with reentrant faces. However, those of most importance in crystallography are the Platonic and Archimedean polyhedra, deltahedra, coordination and spacefilling (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.
The tetrahedron has cubic point group symmetry $\stackrel{\u02c9}{4}3m$ and the octahedron and cube have the full (holosymmetric) cubic point group symmetry $\text{4}\text{}/m\stackrel{\u02c9}{3}\text{2}\text{}/m\equiv m\stackrel{\u02c9}{3}m$. The icosahedron and pentagonal dodecahedron both have point group symmetry $\text{2}\text{}/m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$—the symmetry possessed by quasicrystals, buckminsterfullerene and some viruses. Of all the Platonic polyhedra, only the cube is spacefilling.
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 buildingblocks 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 thenknown 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 semiregular 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 
$\stackrel{\u02c9}{4}3m$ 
2 
Truncated cube 
8 triang, 6 oct 
$m\stackrel{\u02c9}{3}m$ 
3 
Truncated octahedron 
6 square, 8 hexag 
$m\stackrel{\u02c9}{3}m$ 
4 
Truncated dodecahedron 
20 triang, 12 decag 
$m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ 
5 
Truncated icosahedron 
12 pentag, 20 hexag 
$2\text{}/m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ 
6 
Cubeoctahedron 
8 triang, 6 square 
$m\stackrel{\u02c9}{3}m$ 
7 
Icosidodecahedron 
20 triang, 12 pentag 
$2\text{}/m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ 
8 
Truncated cubeoctahedron (great rhombicubeoctahedron) 
12 square, 8 hexag, 6 oct 
$m\stackrel{\u02c9}{3}m$ 
9 
(small) rhombicubeoctahedron 
8 triang, 18 square 
$m\stackrel{\u02c9}{3}m$ 
10 
Truncated icosidodecahedron 
30 square, 20 hexag, 12 decag 
$2\text{}/m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ 
11 
Rhombicosidodecahedron 
20 triang, 30 square, 12 pentag 
$2\text{}/m\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ 
12 
Snub cubeoctahedron 
32 triang, 6 square 
432 
13 
Snub icosidodecahedron 
80 triang, 12 pentag 
235 
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 $\stackrel{\u02c9}{4}$3m, $m\stackrel{\u02c9}{3}m$, and 432) and six have icosahedral symmetry (the two icosahedral point groups 2/m$\stackrel{\u02c9}{3}\stackrel{\u02c9}{5}$ and 235). Note again that the point groups 432 and 235 are enantiomorphic.
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 spacefilling (Fig. 3.9). The pattern of faces in the truncated icosahedron corresponds to that of C_{60} (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 spacefilling Voronoi polyhedron for the cubic F lattice (Fig. 3.8(a))) and that of the icosidodecahedron which is a rhombic triacontahedron (Fig. A2.4^{1}). The ratios of the lengths of the long and short diagonals of the rhombic or diamondshaped faces are not the same: for the rhombic dodecahedron it is $\sqrt{2}\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}1$ and for the rhombic triacontahedron it is $\left(\sqrt{5}+1\right)\phantom{\rule{negativethinmathspace}{0ex}}/2\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}1$ 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.
(p.398) (D) Coordination and spacefilling 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 spacefilling (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 spacefilling 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 dodecahedron^{2} and the spacefilling polyhedron is a truncated octahedron.
The procedure for constructing the spacefilling Voronoi polyhedra for lattices is described in Section 3.4. For crystal structures consisting of only one kind of atom^{3}—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 spacefilling 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 spacefilling polyhedron.
A few examples will make this clear. In the fcc NaCl and ZnS (zinc blende or sphalerite) structures, the spacefilling 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 αAl_{2}O_{3} (corundum), structures where the anions are (approximately) hexagonal closepacked, the anioncentred spacefilling 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 αAl_{2}O_{3} 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 
Coordination polyhedron around interstitial site 
Radius ratio 

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 spacefilling polyhedra
Structure/lattice type 
Coordination polyhedron 
Spacefilling 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 closepacked (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) 
αAl_{2}O_{3} 
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 
μltirow464ptCaF_{2} and Li_{2}O (fluorite and antifluorite 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 closepacked oxygen anions. The data are summarized in Table A2.2 which includes the interstitial sites delineated by squares and triangles.
Table A2.3 lists the coordination and spacefilling polyhedra for some simple structures.
(E) Brillouin zones
Brillouin zones are ‘nests’ of polyhedra in kspace (reciprocal space with the scale factor 2$\mathrm{\pi})$. Their surfaces represent the directions and wavelengths (or wavenumbers) of free electrons in crystals which are Braggreflected by the lattice planes. The first, innermost Brillouin zone corresponds to the reflections from the planes of largest dspacing, 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 (kspace) lattice is also cubic P with a cell edge length $2\text{\pi}\text{}/a$ (Fig. A2.7).
Consider electrons travelling in the x^{*}y_{*} plane reflected from lattice planes parallel to the z_{*} axis. In order of decreasing d_{hkl}spacing (or increasing ${\mathbf{\text{d}}}_{hkl}^{\ast}$ spacing) these are 100, 110, 200, 210, … Consider an electron with wavevector k as indicated:
For reflection from the 100 planes, the corresponding value of k is found by substituting for λ in Bragg’s law
(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.