Crystallography & Symmetry
patterns derived from their groups
Generator

The Wigner–Seitz cell of the hexagonal lattice

The Wigner–Seitz cell of the hexagonal lattice
The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell.

Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell.

6 essays call voronoi. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell. Lattices

The cell nobody chose

Every unit cell on this site is a convention, and one construction escapes the warning entirely: the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no rule — and its combinatorics are decided in integers, with the square roots confined to drawing it.

 The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell. Lattices

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

 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. Lattices

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.

 hexagonal: 0.5 and 0.577. The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid. Lattices

Covering and packing want different lattices

A lattice has two natural radii — the largest spheres on its points that do not overlap, and the smallest that leave no gap — and both are radii of the same Voronoi cell. In the plane one lattice is best at both. In space the best packer and the best coverer are different lattices, and they are duals of one another.

 Every plane lattice, shaded by Σ|v|^(−4). The region every plane lattice is one point of, with each point shaded by the sum of the inverse powers of the lengths of that lattice's own vectors, at equal cell area — dark where the sum is small. The square lattice is the ringed point on the vertical axis and the hexagonal one is at the corners, which are the same lattice on two bases. The minimum is at the corner, and it is at the corner at every exponent tried. That is not the same statement as the densest packing, which is decided by the shortest vector alone: this sum counts every shell, and there was no reason in advance for the two questions to have the same answer. Lattices

The lattice that minimises a sum

Packing discs asks about the shortest vector alone. Summing an inverse power over every vector of a lattice asks about all of them at once, and there was no reason in advance for the two questions to have the same answer. They do — at every exponent, and the measurement says by how much and where it cannot say.

 Three places to put the boundary. A large site and a small one, with three candidate boundaries between them. Halfway is the ordinary Voronoi cell and it cuts through the large sphere. Splitting in the ratio of the radii is the natural repair and its surfaces are not planes, so the cells do not fit together. The power plane sits where the tangent lengths agree, which is further from the large site than halfway and is still a plane — and being a plane is the whole reason the construction works. Lattices

Where the boundary goes when the atoms differ

Assigning each point of space to the nearest atom is the right rule only when every atom is the same size. Splitting the distance in the ratio of the radii is the obvious repair and it produces curved faces that do not fit together. The repair that works measures to a sphere rather than to a point: the boundary stays a plane, the cells still tile exactly, and a small enough atom loses its cell altogether — at a radius ratio of exactly one in eight.

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