Every parallelohedron is a shadow of a cube

Five parallelohedra and no others enumerates the convex bodies that fill space by translation alone: the cube, the hexagonal prism, the rhombic dodecahedron, the elongated dodecahedron and the truncated octahedron. The enumeration is an argument about symmetry, and it produces a list of five names.

A list of five names is a poor description of a family. This essay gives the same five a construction, and the construction turns out to be one sentence long.

The construction is worth stating in the reader’s own terms first. A crystal’s Wigner–Seitz cell is normally built by cutting: take the lattice, draw the perpendicular bisector of every vector to a neighbour, and keep what is inside all of them. That is a subtractive recipe — start with everything and remove half-spaces — and it makes the cell look like the residue of a process rather than an object with a description of its own. The recipe below is additive, and the shapes come out the same.

Add up some vectors #

Take n vectors and form every point Σ tᵢ vᵢ with each coefficient tᵢ between zero and one. The set of all such points is convex, and it is called a zonotope. Three vectors give a parallelepiped; more give something with more faces.

That is the whole classification restated: the five parallelohedra are the zonotopes of three, four, four, five and six vectors. No shape is described and no picture is consulted; the input is a list of vectors and the body is what the recipe makes of them.

Two things about the recipe are worth fixing before it is used. The generators are vectors rather than points, so the body has a position only up to translation — which is the right level of description for something meant to tile by translation, and it is why the construction and the classification are about the same objects. And the order of the generators is irrelevant: summing the same segments in a different order gives the same set, so a zonotope is a property of an unordered list.

The second matters more than it looks. It means the family of zonotopes with n generators is parameterised by n vectors up to permutation and up to sign — reversing a generator translates the body and changes nothing — so the space of three-generator zonotopes in three dimensions is nine numbers, and the cube is one point of it. Everything in this essay is a statement about which points of that space tile.

Why the name mentions a cube #

The recipe is the image of a cube.

Send the i-th edge of the n-dimensional unit cube to the vector vᵢ and extend linearly. A point of the cube is a list of n coefficients between zero and one, and its image is Σ tᵢ vᵢ — which is a point of the zonotope, and every point of the zonotope arises this way. So a zonotope with n generators is literally the shadow of an n-cube under a linear map, and a shadow is exactly what a projection is.

The dimension is not a flourish. A body with n generators genuinely needs n degrees of freedom to describe the walk that traces it: getting from one corner of the truncated octahedron to the opposite one means taking six independent steps, and the order they are taken in picks out one of the sixty-four paths. Six independent binary choices is a six-cube, and the body is what those choices look like after they are added up in three dimensions.

So the truncated octahedron — the cell of the body-centred cubic lattice, the shape of a soap froth’s ideal bubble, the Wigner–Seitz cell every solid-state textbook draws — is a picture of a six-dimensional cube seen from an angle. That is not a figure of speech; the map is written down and the sixty-four corners are pushed through it.

The count of distinct images is worth reading. Sixty-four cube vertices give thirty-eight distinct points, and twenty-four of those are corners of the shadow. The rest are interior: pairs of cube corners that project onto each other, and points that fall inside the outline. A shadow loses information, and the loss is countable.

There is a small pleasure in checking the arithmetic in that table against something already computed. The cell nobody chose builds the truncated octahedron as the Voronoi cell of the body-centred cubic lattice and reports fourteen faces, twenty-four vertices and thirty-six edges; the zonotope of six vectors reports the same three numbers. The two constructions have nothing in common — one takes the perpendicular bisectors of lattice vectors and intersects half-spaces, the other adds segments — so the agreement is a check on both rather than a restatement of either.

The five are the degenerate ones #

Here the construction says something the list of names cannot.

The formulas are worth a sentence of derivation because they explain where the degeneracy has to go. Each unordered pair of generators spans a plane, and a generic pair gives two faces — one at each end of the body — so n(n − 1) faces in all. Each face is a parallelogram because only the two generators spanning it can lie in its plane; if a third does, the two parallelograms it would have contributed merge with the first into a hexagon, and three faces become one. So every hexagon costs two faces, and the shortfalls in the previous table are even numbers for that reason.

Six generators in general position give thirty faces. The truncated octahedron has fourteen. Its six generators are therefore very far from general position: every hexagonal face is a plane that three of them share, and each such coincidence merges three parallelograms into one hexagon.

The same is true of the hexagonal prism, which loses four faces of the twelve a generic four-generator zonotope has, and of the elongated dodecahedron, which loses eight of twenty. Only the cube and the rhombic dodecahedron are generic.

So the parallelohedra are the degenerate zonotopes, and the degeneracies are not incidental. A face of a tiling by translation has to be shared with exactly one neighbour, which forces faces to come in parallel pairs of equal shape; hexagonal faces arise exactly where three generators conspire, and the conspiracy is what the tiling condition demands. A list of five names carries none of that.

The degeneracy has a converse worth stating, because it makes the classification feel less like a coincidence. A generic zonotope has only parallelogram faces, and a body whose faces are all parallelograms tiles space only in the trivial way — the parallelepiped case. Hexagonal faces are what let a body interlock with its neighbours in more than one direction at once, and a hexagonal face is exactly three coplanar generators. So the tiling condition wants degeneracy, and the five are the ways of arranging it in three dimensions.

There is a reading of the two generic cases which is worth having, because it says which of the five are the “ordinary” ones. The cube and the rhombic dodecahedron lose nothing to coplanarity — they are what three and four vectors give when nothing conspires — and they are also the two cells belonging to the two cubic lattices a metal actually adopts. The three degenerate ones are the cells of lattices with something special about them: a hexagonal axis, a stretched direction, a body centring. So the degeneracy of the shape and the specialisation of the lattice are the same fact, which is what six integers and the lattice that holds them says in the language of parameters.

Belts #

There is one more structure the construction supplies for nothing.

That is how the five relate to each other. Remove a generator from the truncated octahedron’s six and the belt it carried collapses; what is left is a five-generator zonotope, and with the right choice it is the elongated dodecahedron. Remove another and the rhombic dodecahedron or the hexagonal prism appears, depending on which. The five are not five unrelated shapes but a chain, and the chain is the sequence of generator counts.

The belt structure also explains a fact about the tiling that is otherwise mysterious. In a tiling by a parallelohedron, the bodies meeting along an edge form a ring — four or six of them — and the ring’s size is decided by the belt the edge belongs to. So the local combinatorics of the tiling are readable off the generator list, without building any tiling at all, which is what five parallelohedra and no others has to do the long way.

One more consequence of the belt reading, and it is the one that makes the chain of five feel inevitable. A belt cannot be shorter than four faces, because a generator’s plane must be crossed by at least two others going each way round. So a body with n generators has at least n belts of at least four faces, and the face count cannot fall below what those belts require — which is the lower end of the range the five occupy. The tiling condition pushes the count down and the belt structure stops it, and the five are where the two meet.

What a tetrahedron fails #

A zonotope is a sum of segments, and a sum of segments is symmetric about its own centre. So is every one of its faces, for the same reason one dimension down.

That is a second proof that the regular tetrahedron does not tile space, and it is worth setting beside the first. The angle that is not a fraction of a turn rules it out by a conserved quantity: the Dehn invariant is non-zero and a space-filler’s must vanish. This rules it out by a symmetry the shape does not have. Neither argument mentions the other’s object, they reach the same conclusion, and only one of them extends: Minkowski’s condition applies to translation only, so it says nothing about tilings that use rotations, while the Dehn invariant rules those out too.

One more reading of the belts, and it is the one that connects to the reduction machinery elsewhere in this collection. The generators of the Voronoi cell of a lattice are its relevant vectors, and a reduction with one rule computes exactly those. So the generator count of a lattice’s cell is a quantity Selling reduction hands over directly, and the classification into five is a classification of how many relevant vectors survive — which is what that essay’s pattern of vanishing parameters is measuring from the other side.

The tetrahedron’s failure has a positive version worth stating alongside it. Central symmetry is necessary and it is not sufficient: an octahedron is centrally symmetric with centrally symmetric faces and does not tile space by translation either. So Minkowski’s condition rules things out and never rules anything in, which is the usual shape of a symmetry argument — the crystallographic restriction has the same character, forbidding five-fold rotation without ever certifying that a given lattice exists.

There is a converse to Minkowski’s condition that is worth knowing about even though it is not proved here, because it is what makes the five a complete list rather than a shortlist. Venkov and McMullen showed that a convex body tiles space by translation exactly when it is centrally symmetric, its faces are centrally symmetric, and each of its belts has four or six faces — three conditions, all local, all checkable on the body alone. The five satisfy them and nothing else in three dimensions does. So the belt count above is not a curiosity: it is the third clause of the criterion, and it is why a belt of five or seven faces never appears in the table.

What this is for #

Three things, and the third is the one that reaches beyond the five.

The first is that it makes the classification computable rather than merely quotable. Building a parallelohedron from a generator list is a dozen lines; identifying a body as one of the five is a matter of counting faces and vertices. The cell nobody chose builds the same bodies as Voronoi cells of lattices and gets the same face counts, and the two constructions share no step — one intersects half-spaces, the other adds segments.

The second is that the generator count is a genuine invariant. Two parallelohedra with the same number of faces are the same body; two with different generator counts cannot be deformed into each other without a face appearing or disappearing. That is a discrete quantity attached to a shape that otherwise varies continuously, and it is the kind of thing a classification is made of.

The third is dimension. Every statement here holds in any dimension: a zonotope in d dimensions with n generators is the shadow of an n-cube, and the parallelohedra of d-dimensional space that are zonotopes are classified the same way. In four dimensions there are fifty-two parallelohedra rather than five, and the zonotopes among them are still generator lists. Where five-fold becomes legal is where this collection goes up a dimension for a different reason; the construction here travels with no changes at all.

The fourth of those is the one that keeps the degeneracy claim honest. Fourteen against thirty is a large gap and it would be easy to produce by a bug — a face-finding routine that missed most of them would report fourteen for everything. Running the same routine on generators in general position and getting exactly n(n − 1), for every n from three to eight, is what shows the fourteen is the body’s and not the code’s.

A last observation about what the construction buys that the enumeration does not. Given a lattice, its Voronoi cell is one of five shapes and the enumeration says which; the zonotope description additionally says how — it hands over a set of vectors, and those vectors are the lattice’s own relevant vectors rather than an abstract label. So identifying a cell and describing it become the same operation, and a body arriving with the wrong number of faces for its generator count is a body whose generators were computed wrongly. That is a check the name “truncated octahedron” cannot supply.

One number in the shadow table deserves a closer look, because it is the only place the projection loses something interesting. The truncated octahedron’s six-cube has sixty-four vertices and thirty-eight distinct images — so twenty-six pairs of cube corners land on the same point, and of the thirty-eight only twenty-four are corners of the shadow. The fourteen interior images are the cube’s own structure folded inside the body, and eight of them sit exactly at the centres of the eight hexagonal faces — one per hexagon, none on a square. That is not a coincidence the construction was told about; it falls out of pushing sixty-four points through a map, and it is the same fact as a hexagon being three coplanar generators seen edge-on.

Where this stops #

Not every parallelohedron in every dimension is a zonotope. In three dimensions all five are, which is why this essay can be short; in four the count of parallelohedra is fifty-two and only some of them are zonotopes, so the construction stops being a classification and becomes a family within one. What survives is the direction of the implication: every zonotope that tiles is centrally symmetric with centrally symmetric faces, and Minkowski’s condition is necessary in every dimension.

And the vertex enumeration here is a sampling of directions rather than an exact convex-hull computation. That is safe because the answer is checked — Euler’s formula holds on every body computed, and the generic counts match closed formulas — but it is not a proof procedure, and a body whose vertices were too close together to separate would be found by the Euler check failing rather than by anything more graceful. The check is the guarantee; the sampling is only how the candidates are proposed.