Crystallography & Symmetry
patterns derived from their groups
Concept

Voronoi cell — where it appears

The same construction for an arbitrary set of points: the territory of one point, bounded by the bisectors of its neighbours. For a lattice it is the Wigner–Seitz cell, and its combinatorics are decided in integers with the square roots confined to drawing it.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

lattices · Wigner–Seitz cells 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 3.4.6.4 and its dual. The tiling in pale outline with its dual drawn over it: one dual vertex at the centre of every tile, one dual edge across every shared edge, and one dual tile round every vertex. 3.4.6.4 has 3 kinds of tile and one kind of vertex; its dual has one kind of tile and 3 kinds of vertex, and the congruence of those tiles is checked rather than eyeballed — every dual face presents the same cyclic sequence of squared edge lengths, compared exactly. That swap is what the eleven duals are for: read one way the list classifies tilings with all vertices alike, read the other it classifies tilings with all tiles alike.

Eleven duals, one tile each

Swap the vertices of a uniform tiling for its tiles and the eleven come back as eleven tilings by a single repeated shape. Three of those shapes are pentagons — which is worth pausing over on a site whose other essays prove that five-fold symmetry cannot exist.

classification · Tilings 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.

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.

lattices · Wigner–Seitz cells 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.

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.

lattices · Wigner–Seitz cells

Named alongside it

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

Wigner seitz cellRelevant vectorBrillouin zoneDualityUnit cellArchimedean tilingBravais latticeClose packingConvexityCovering radiusCrystallographic restrictionEdge to edge

All concepts