Transcript No Slide Title

Tiling Space with Regular
and Semi-regular Polyhedra
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
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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).
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 :
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
Im3m: 4.62
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:
Similarly, by further truncation, we get a spacefilling of cuboctahedra and octahedra:
Pm3m:
3.4.3.4 + 34
Pm3m: 3.82 + 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
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.)
Fm3m: 33 + 34
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 + 4.6.8
The 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
This completes Andreini’s list of uninodal tilings of 3D Euclidean space by regular and semi-regular
polyhedra. There are four other cases.
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.
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