Five parallelohedra, and no others

The cell nobody chose is the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no convention: two people who agree about the lattice cannot disagree about it, which is exactly what a conventional cell fails to be.

In three dimensions the same construction has a much better answer, and it is a theorem.

There are exactly five shapes. Not five for the cubic lattices or five in some list of common cases: five in total, and no lattice in three-dimensional space has a Wigner–Seitz cell of any other combinatorial type.

How each one is built, and what says it is right #

The construction is the same in any dimension. Take every lattice vector, bisect it with a plane perpendicular to it, and keep the region containing the origin. Most of those planes never reach the surface — they are cut off by nearer ones — and the ones that do are the faces.

The check is the volume, and it is exact. A Wigner–Seitz cell tiles space by lattice translations, one cell per lattice point, so its volume must be exactly the volume of a primitive cell — the square root of the determinant of the metric. Every cell computed here is measured against that number and agrees to within a part in a million, which is floating-point noise rather than geometry.

That check is not a formality. A search that misses a relevant vector leaves a plane uncut, and the cell comes out too large; nothing about the drawing shows it. The failure is available on demand: describing a cubic lattice on a skew basis — the same lattice, a change of basis away — puts the relevant vectors out of reach of a small search, and the cell that comes back has three times the volume it should. Widening the search fixes it, and gives a cube, which is what the plain description gives too.

The two cubic centrings are worth taking one after the other, because they produce the two most-drawn solids in the subject and the reason they differ is a competition between two shells of neighbours. In the body-centred lattice the eight nearest neighbours sit along the body diagonals at √3/2 of the cube edge, and the six next-nearest sit along the axes at one edge. The eight are nearer, so their bisectors cut first and produce the eight hexagons; the six are close enough behind that their bisectors still reach the surface, and they truncate the octahedron’s six corners into squares. Fourteen faces, and both shells contribute.

In the face-centred lattice the twelve nearest neighbours are the face-diagonal ones, at √2/2 of the edge, and the next shell — the six axial neighbours at one edge — is far enough out that its bisectors are entirely cut off. So one shell decides everything, and the result has twelve identical faces.

Fourteen lattices, five shapes #

Euler’s formula is the second check. Faces minus edges plus vertices is two for every one of the fourteen — 6 − 12 + 8, 14 − 36 + 24, and so on. A solid whose face list had lost a face, or whose vertex enumeration had produced a spurious point, would fail it, and the counts here come from the solid rather than from a table of expected values.

Lattices of very different symmetry share a shape. A triclinic lattice — no symmetry beyond inversion — and a body-centred cubic one both give truncated octahedra. That is worth pausing on: the shape is decided by how the neighbours sit, not by how symmetric the arrangement is, and the generic triclinic case has fourteen faces for the same reason that the most symmetric case does. The cube, with its six faces, is the special one.

The word doing the work in that comparison is combinatorial. Two cells are of the same type when their faces meet in the same pattern — this many four-sided faces, that many six-sided ones, joined up the same way — and not when they are congruent. The triclinic cell above has fourteen faces no two of which need be congruent and the body-centred cubic one has fourteen faces of exactly two shapes, and they are the same parallelohedron. Fedorov’s theorem is a statement at that level, which is why it can be a list of five rather than a description of a continuum: the shapes vary continuously with the metric and the patterns do not vary at all beyond the five.

That is also the reason the generic case has the most faces rather than the fewest. A face is lost when a bisector is cut off by a nearer one, which needs two shells of neighbours to line up — and lining up is what symmetry does. A lattice with no symmetry has no such coincidences and uses every class it is entitled to, which is seven pairs and fourteen faces. So the fourteen-faced solid is the default, and the cube is what a lattice looks like when four of its seven classes have no relevant vector left in them at all — the four body-diagonal classes of a cubic lattice, whose shortest representatives are cut off by the axial neighbours. Three classes survive, six faces, a cube.

Fourteen is the most a lattice permits, and it is Minkowski’s bound again: a Voronoi cell in n dimensions has at most 2(2ⁿ − 1) faces, which is six in the plane and fourteen in space. The essay on why the classification is finite uses his other bound, on the order of a symmetry group; this is the same mathematician bounding a different thing, and both come out of the arithmetic of lattices rather than out of geometry.

The fourteen do not decide the shape #

The table above suggests a correspondence between Bravais types and shapes. It is not one, and finding out how it fails is the most interesting computation here.

One lattice type, two shapes, and the crossing is at √2. Below the ratio the eight face-diagonal neighbours are near enough to contribute faces; above it, they are cut off by the six axial neighbours and two faces vanish. The crossing is found here by measurement — the cell is recomputed at seventeen ratios and the shape read off the face counts — and it lands where the algebra says it should.

So the Bravais classification and the Voronoi classification are different classifications. The fourteen sort lattices by which symmetry group they have; sorting them by the shape of the cell that needs no convention gives a different and finer answer, which is Delaunay’s: twenty-four lattice characters, distinguished by exactly this kind of transition. Nothing here computes the twenty-four; what is computed is one crossing, which is enough to show that the fourteen cannot be the whole story.

Why this cell rather than a conventional one #

The argument for the Wigner–Seitz cell is the same argument this site makes about every convention, and it is worth restating with the three-dimensional case in hand.

A conventional cell is a choice. The cell is a choice and the lattice is not: a face-centred cubic lattice can be described with a cubic cell containing four lattice points, or with a primitive rhombohedral cell containing one, and the two are the same lattice. Neither cell has the symmetry of the lattice — the primitive one does not look cubic at all.

The Wigner–Seitz cell is not a choice, and it carries the full symmetry of the lattice by construction: every operation preserving the lattice preserves the set of distances to lattice points, so it preserves the cell.

And it is the shape a physicist actually needs. The first Brillouin zone is this construction applied to the reciprocal lattice, and the zones above the first are its higher analogues — which means the truncated octahedron above is not a curiosity but the standard picture of the Brillouin zone of a face-centred cubic crystal, since the reciprocal of a face-centred lattice is a body-centred one.

What a face is, and why fourteen is the ceiling #

The faces have an arithmetic description, and it turns out to be the proof of the bound.

A face is a lattice vector, or rather a pair: the plane bisecting the segment from the origin to v is a face exactly when no other lattice point is nearer to the midpoint than the two ends are. Such a vector is called relevant, and the cell’s faces are its relevant vectors, ±v giving the two faces on opposite sides.

Two relevant vectors cannot be congruent modulo twice the lattice. If v and w differ by twice a lattice vector, then (v − w)/2 is a lattice point sitting where it cuts one of their bisectors off. So the faces fall into distinct classes of the lattice modulo twice itself — and there are 2³ − 1 = 7 non-zero such classes in three dimensions, each contributing one pair of faces.

That is 2(2³ − 1) = 14, and it is where the bound comes from. In the plane it gives 2(2² − 1) = 6, which is the hexagon of the first figure; in n dimensions it gives 2(2ⁿ − 1), which grows exponentially and is attained by a generic lattice. The classes are counted in the table above rather than trusted: every cell’s faces come to exactly half as many classes as faces, and never more than seven.

The count also explains which lattices have few faces. A cubic lattice’s relevant vectors are the six axial neighbours, and the classes of the four space diagonals are represented by vectors too long to be relevant — their bisectors are cut off by the axial ones. Three classes, six faces, a cube. A generic triclinic lattice has no such cancellation and uses all seven.

Where the shape is actually used #

Three uses, and they are in different subjects.

The Brillouin zone. The first zone is this construction in the reciprocal lattice, and since the reciprocal of a face-centred cubic lattice is body-centred cubic, the standard picture of an fcc metal’s Brillouin zone is the truncated octahedron above. Band structures are plotted along paths between its corners, and the corner labels — Γ, X, L, W — are positions on this solid. The zones above the first are the same construction’s higher shells.

Coordination. The cell of a face-centred cubic lattice is a rhombic dodecahedron with twelve faces, one per nearest neighbour, which is the coordination number of a close-packed structure. The cell’s faces are the coordination polyhedron’s directions, so a shape and a count that are usually introduced separately are the same fact.

The approximation Wigner and Seitz actually made. Computing an electron’s wavefunction inside a truncated octahedron is hard; inside a sphere of the same volume it is a textbook exercise. That approximation is what the cell was invented for in physics, and it works because the cell is nearly spherical — which is another way of saying that a lattice’s territory is as compact as it can be.

Where the exactness stops #

The construction is floating-point and the checks are ratios. Metrics are real numbers, the intersection is a solve of three-by-three systems, and the volume test compares a ratio to one within a part in a million. Nothing here is integer arithmetic in the way the group machinery is, and the classification is by counting faces and their edges, which is integral and is where the exactness sits.

Each lattice type is measured at one metric, chosen to be generic. An orthorhombic cell with two equal axes is a tetragonal lattice wearing the wrong label, and a census that used one would be measuring a coincidence. The metrics are stated in the module; the sweep is what shows what happens when a parameter is varied instead.

Nothing here is a proof that the five are five. The count of face-classes bounds the faces at fourteen, which is a proof and is checked; that only five combinatorial types are possible is a much finer statement, and the computation confirms it on fourteen lattices rather than establishing it.

Five is Fedorov’s theorem and is not proved here. What is computed is that the fourteen lattices, at these metrics, give five shapes and that all five are on his list. That no sixth exists for any lattice at any metric is his result, from 1885, and it is quoted.

The face count is a bound and not a formula. Fourteen is attained by the generic case and six by the most symmetric one; there is no rule of the form “more symmetry, more faces”, and the table is the evidence for saying so.

Who found it, and when #

Evgraf Fedorov classified the parallelohedra in 1885, several years before he enumerated the two hundred and thirty space groups. The question he was answering is older than crystallography’s interest in it: which convex solids tile space by translation alone? He found five, and the answer is one of the first classification theorems in the subject.

Georgy Voronoi gave the general construction in 1908, in a memoir on quadratic forms, and proved that Fedorov’s five are exactly the Voronoi cells of three-dimensional lattices. The name attached to the construction is his, and it is used far outside crystallography — in computational geometry, in ecology, in any subject that asks which of several sites is nearest.

Boris Delaunay refined the classification in 1932 into the twenty-four lattice characters, by tracking exactly the kind of transition the sweep above finds. He is also the Delaunay of the triangulation, which is the Voronoi construction’s dual and was found by looking at the same picture the other way up.

Eugene Wigner and Frederick Seitz used the cell in 1933 to compute the electronic structure of sodium, approximating the cell by a sphere of the same volume. The name that stuck in physics is theirs, and the construction they were using was fifty years old. The two names are used interchangeably now, with Voronoi preferred when the points are arbitrary and Wigner–Seitz when they are a lattice.

The one place the plane is misleading #

A reader coming to this from the plane case has an intuition to give up, and it is worth naming because it is the reason the three-dimensional answer is interesting rather than routine.

In the plane the two shapes track symmetry neatly: the two right-angled lattices give rectangles and the other three give hexagons, so the generic case is the hexagon and the special cases are rectangles. Four faces or six, and a reader can hold the whole classification in their head as “squares are special”.

In space that intuition breaks in both directions. The generic case has fourteen faces and so does body-centred cubic, which is as symmetric as a lattice gets; the cube, which sounds like the most symmetric possible cell, belongs to primitive lattices of three different systems including the least symmetric right-angled one. The shape and the symmetry are nearly independent, and the table is where that becomes obvious rather than arguable.

The reason is the counting above. What decides a face is whether a neighbour is near enough for its bisector to survive, which is a statement about lengths; what decides symmetry is which operations preserve the metric, which is a statement about equalities among lengths. A lattice can have many near neighbours without any of them being equal, and it can have equal ones that are too far away to matter.

The question Fedorov’s five open #

The five are the Voronoi cells of three-dimensional lattices. There is a wider question they invite, it has a name, and it is open in general — which is worth knowing because the three-dimensional answer is so clean that the general one looks as though it should be too.

Ask instead: which convex polyhedra tile space by translations alone? That is a question about tilings rather than about lattices, and nothing in it mentions a Voronoi construction. Fedorov’s answer in three dimensions is the same five — every convex translational tile is one of them — but the two questions are not obviously the same question.

Voronoi’s conjecture is that they are the same in every dimension: every convex polytope tiling space by translations is the Voronoi cell of some lattice. It is proved in dimensions up to five and open beyond, and it has been open since 1908.

That is an unusual place for the subject to run out. The classification of lattices continues upward — the Bravais types are known in four dimensions and beyond — and the classification of their Voronoi cells continues with it. What is not known is whether the geometric question and the arithmetic one keep the same answer, and the plane and space give no hint either way, since both are among the settled cases.

Where the sweep’s crossings sit in a classification #

The √2 transition in the tetragonal sweep is presented as a curiosity of one family, and it is an instance of a classification that already exists.

Delaunay sorted the three-dimensional lattices by the combinatorial type of their Voronoi cell — how many faces, of what shapes, meeting how — rather than by their symmetry, and the classification has twenty-four entries. Two lattices are in the same sort when their cells are combinatorially the same polyhedron, whatever their metrics.

The crossings the sweep finds are the boundaries between sorts. Below c/a = √2 the body-centred tetragonal lattice’s cell is one combinatorial type and above it another, and at exactly √2 a set of faces shrinks to nothing — which is what a boundary in that classification is.

So the sweep is not exploring an undescribed continuum: it is walking a path through Delaunay’s classification and crossing a wall. That is worth knowing because it says what to expect from a sweep of any other family — finitely many types, walls where faces appear or vanish, and a combinatorial type that is constant between them.

It also says why the Bravais classification and this one cross rather than nest. One sorts lattices by their symmetry and the other by the shape of a cell, and neither refines the other — which is the essay’s central observation, arriving as a relation between two named classifications rather than as a surprise.

Where this ladder goes #

Three rungs of this anchor have now built one thing three ways: the cell in the plane, the zones above the first in the reciprocal lattice, and the five shapes it takes in space.

What they have in common is the argument for having it at all. Every other cell on this site is a description with a convention inside it, and every one of those conventions has generated an essay — the choice of axes, the choice of origin, the choice of a shortest basis, the reduced cell that settles the argument. This construction has none, and its answer in three dimensions is a list of five.