Tiling Space with Regular and Semi-regular Polyhedra

Tiling Space with Regular and Semi-regular Polyhedra

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Andreini, A. Sulle reti di poliedri regolare e semiregolari e sulle correspondenti reti correlative. Memorie della Societa Italiane delle Scienze 14 (1907) 75-129.Kepler, J. Harmonice Mundi. Lincii Austriae, Sumptibus Godfrdi Tampachhii, Francof. (1519); German transl. Johannes Kepler, Gesammelte Werke (M. Caspar, ed). Beck, Munich (1990).

Andreini considered the possibilities for uninodal tiling of 3D Euclidean space with regular and semi-regular polyhedra. (Uninodal means that all vertices of the structure are to be identically surrounded.) To each of these structures there corresponds a packing of equal spheres, centered at the polyhedron vertices. The analogous two-dimensional problem was solved by Kepler: there are just eleven uninodal ways of tiling a Euclidean plane with regular polygons:

Eleven tilings of 3-space using stacked layers built from triangular, square, hexagonal, octagonal and dodecagonal prisms follow trivially from Kepler’s patterns

The cube is the only regular polyhedron that will produce a tiling of Euclidean 3-space, if all tiles are required to be identical.

There is also just one semi-regular polyhedron that will produce a tiling of Euclidean 3-space, if all tiles are required to be identical, namely, the truncated octahedron. Every vertex of the truncated octahedron belongs to one square face and two hexagonal faces. Accordingly, it is the semi-regular polyhedron 4.62 . The space group symmetry of the tiling is Im3m :

Im3m: 4.62

The truncated octahedron is the Voronoi cell of the body-centered cubic lattice.

The 4-connected net of edges of the configuration is the zeolite framework for the sodalite structure (zeolite framework SOD).

The truncated octahedron is sometimes referred to as the Kelvin polyhedron because of Kelvin’s conjecture that the cellular structure obtained by curving the faces so as to minimise the face areas of this space filling gives the cellular structure with congruent cells with minimum area per unit volume

Andreini found twelve further uninodal space-fillings by regular and semi-regular polyhedra.

Truncating the cubes of the regular packing produces voids in the shape of octahedra. We get a uninodal space-filling with semi-regular tiles of two kinds, truncated cubes and octahedra:

Pm3m: 3.82 + 34

Similarly, by further truncation, we get a space-filling of cuboctahedra and octahedra:

Pm3m: 3.4.3.4 + 34

In perovskite CaTiO3 and minerals of perovskite type ABO3 vertices of this tiling are occupied by O atoms. A and B atoms occupy the centers of the cuboctahedra and octahedra, respectively.

The primitive unit cell of the face centered cubic (fcc) lattice can be dissected into two tetrahedra and an octahedron. When repeated by translation, this gives a space-filling of tetrahedra and octahedra

Fm3m: 33 + 34

Kepler’s stella octangula (an octahedron with a tetrahedron on every face) in a cubic unit cell. (Notice how further octahedra of the space filling would be centered at mid-points of the cube edges.)

The partitioning of the primitive unit cell of the fcc lattice into two tetrahedra and two truncated tetrahedra gives rise to:

Fd3m: 33 + 3.62

The tetrahedra of the spacefilling of tetrahedra and truncated tetrahedra. This is the arrangement of SiO4 tetrahedra in the silicate beta crystobalite. The vertices of the configuration are ocupied by the oxygen atoms. Silicon atoms are at tetrahedron centers.

The Friauf-Laves phases of intermetallics can be described in terms of the same 3D tiling. The vertices of the structure are sites for one kind of atom. Larger atoms are at the centers of the truncated tetrahedra.

The truncated tetrahedron is also sometimes called the Friauf polyhedron.

The structure of the spinel group of minerals is quite difficult to visualise in terms of the arrangement of atoms within a cubic unit cell. A simple representation of the structure is based on the regular space filling Fm3m: 33 + 34 of tetrahedra and octahedra or on the space filling Fd3m: 33 + 3.62 of tetrahedra and truncated tetrahedra.

Right: note that a truncated tetrahedron can be built as a block of the regular tetrahedron/octahedron tiling, consisting of seven tetrahedra (blue) and four octahedra (yellow).

Left: A primitive unit cell of the spinel structure. All vertices of the tetrahedra and octahedra are occupied by oxygen atoms; centers of the blue tetrahedra and centers of the yellow octahedron are occupied by two kinds of atom, A and B respectively. Other tetrahedra and octahedra are empty. The composition is AB2O4.

A portion of the spinel structure indicating the pattern of filled and vacant tetrahedra and octahedra. The tetrahedral symmetry of the structure, that is not apparent in the simple primitive unit cell description, is revealed.

Truncating the octahedra and the two tetrahedra that fill a primitive fcc cell leaves voids in the shape of cuboctahedra. We thus arrive at the space filling arrangement

Fm3m: 3.62 + 4.62 + 3.4.3.4

Boron atoms and metal atoms can form a configuration like this, with the the metal atoms at the centers of truncated octahedra, coordinated to 24 boron atoms located at its vertices.

An array of cuboctahedra centered on a primitive cubic lattice can be connected by cubes linking their square faces. The voids in this structure can be filled by rhombicuboctahedra (4.33), and we get the space filling

Pm3m: 43 + 3.43 + 3.4.3.4

Similarly, linking an array of Kelvin polyhedra by

cubes leaves voids in the shape of truncated cuboctahedra, and we get

Pm3m: 43 + 4.62 + 4.6.8The 4-connected network of edges of this configuration is the zeolite framework LTA. The trincated cuboctahedra (white) correspond to the large pores in the structure, accessible through the octagonal rings.

An array of truncated cubes centered on a primitive cubic lattice can be linked by octagonal prisms. The resulting voids can be filled by a complementary array of rhombicuboctahedra linked by cubes.

Pm3m: 43 + 42.8 + 3.82 + 3.43

Truncated cuboctahedra in face contact on hexagonal faces, centered on a body centered cubic (bcc) lattice. The voids can be filled by octagonal prisms.

Im3m: 42.8 + 4.6.8

Truncated cubes, truncated cuboctahedra and truncated tetrahedra

Fm3m: 3.62 + 3.82 + 4.6.8

A face-centered cubic close packed array of rhombicuboctahedra has voids in the shape of cubes and tetrahedra, giving the polyhedron packing

Fm3m:33 + 43 + 3.43

The packing of tetrahedra and octahedra whose vertices constitute an fcc lattice can be built up in layers:

Changing the stacking arrangement so that columns of octahedra are produced perpendicular to the layers gives

P63/mmc: 33 + 34

Layers of triangular prisms can be inserted between the layers of octahedra and tetrahedra, without affecting the uninodal property:

P63/mmc: 33 + 34 + 3.42 R m: 33 + 34 + 3.42

Grünbaum, B. Uniform tilings of 3-space. Geombinatorics 4 (1994) 49-56.

This completes Andreini’s list of uninodal tilings of 3D Euclidean space by regular and semi-regular polyhedra. There are four other cases.

The packing P6/m: 3.42 (left) of triangular prisms can be changed by rotating alternate layers through 90 degrees, giving

I41/amd: 3.42

The layers in either of these two configurations can be alternated with layers of cubes, and we obtain, finally...

Cmmm: 43 + 3.42

This one is just one of the eleven space fillings obtained from Kepler’s 2D tilings

I41/amd: 43 + 3.42