Crystallography & Symmetry
patterns derived from their groups
Concept

Parallelohedron — where it appears

A convex solid that tiles space by translations alone, with no rotation or reflection needed to fit copies together. Every lattice's Voronoi cell is one, there are five of them in three dimensions, and which one a lattice gives is decided by how many of its reduced superbasis products vanish.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear.

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

lattices · Wigner–Seitz cells Four vectors summing to zero, and six numbers on the edges. A superbasis is the three basis vectors together with their negated sum, so the four sum to nothing and their pairwise products sit on the six edges of a tetrahedron. Selling's rule is: while any edge is positive, apply one transformation. The right panel is the same lattice reduced, with the vanishing parameters marked — and a vanishing parameter is a face the Voronoi cell does not have.

A reduction with one rule

Niggli's reduction is eight numbered conditions with sub-cases, applied in order until none applies. Selling's is a single rule on four vectors that sum to zero: while any of six numbers is positive, do one thing. It terminates sooner, its termination is a quantity that visibly falls, and when it stops the six numbers are the Voronoi cell — the pattern of which ones vanish gives Fedorov's five solids and nothing else.

lattices · Wigner–Seitz cells Why a tetrahedron is not a cube cut up. The two invariants side by side. A cube's twelve right angles are each a rational part of a turn and contribute nothing; a regular tetrahedron's six edges each contribute one α, giving six. Cutting a polyhedron and rearranging the pieces cannot change the invariant, so no dissection takes one to the other however the volumes are matched. That is Hilbert's third problem, and the whole of it is one angle.

The angle that is not a fraction of a turn

Any two polygons of equal area can be cut into pieces that rearrange into each other. In space that fails, and the obstruction is a sum over edges of length against dihedral angle — zero for anything that fills space, and not zero for a regular tetrahedron. The whole argument reduces to one claim about one angle, and that claim is an integer computation: a sequence that is never divisible by three, when it would have to be.

classification · Tilings The five, as generator counts. Each of the five convex bodies that tile space by translation, built as the set of combinations of a handful of vectors with coefficients between zero and one. Three generators give a cube, four give either a hexagonal prism or a rhombic dodecahedron depending on whether three of them are coplanar, five give the elongated dodecahedron and six the truncated octahedron. The last column is what the same number of generators would give in general position, and the shortfall is the number of faces lost to coplanarity.

Every parallelohedron is a shadow of a cube

Take a few vectors and form every combination of them with coefficients between zero and one. All five of the convex bodies that tile space by translation come out of that recipe, from three vectors, four, four, five and six — and since the recipe is exactly the image of a cube of that many dimensions, the truncated octahedron is a three-dimensional shadow of a six-dimensional cube. The five are not the generic answers: they are the degenerate ones, and the degeneracy is what the tiling demands.

classification · Wigner–Seitz cells

Named alongside it

The objects these essays reach for when they reach for this one.

Wigner seitz cellBravais latticeBrillouin zoneConvexityDehn invariantDihedral angleDissectionThe Euler characteristicGram matrixLatticeLattice characterMinkowski bound

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