Crystallography & Symmetry
patterns derived from their groups
Concept

Similarity — where it appears

A map that changes size and orientation and nothing else, so that two objects related by one have the same shape. Two lattices are similar when a rotation and a scale factor carry one to the other, which is the equivalence under which a lattice has a shape rather than a size.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

The space every lattice lives in

Five lattices in the plane is the number of *kinds*. The number of lattices is a continuum — and it has a shape: one two-dimensional region with two corners, three edges and an interior, where the five kinds turn out to be a region, three arcs and two points rather than five things of one sort.

lattices · Moduli One change of basis turns a Gram into its own adjugate. For each Gram matrix: the matrix after the basis change by a right-angle rotation, and the adjugate. They are equal, always — and the adjugate is the determinant times the inverse, which is the dual lattice's Gram. So the dual is the same lattice on a rotated basis, scaled by one over the determinant. Five rows are the named plane lattice types and the rest have entries picked at random, because the claim is an identity in integers and not a property of the five.

Every plane lattice is its own dual

The dual of a lattice has the inverse Gram matrix, and in two dimensions the inverse is the adjugate over the determinant — which is what one particular change of basis does to a Gram. So a plane lattice's dual is the lattice itself, turned through a right angle and scaled, for every lattice with no exception. In three dimensions it is a condition, and the face-centred and body-centred cubic lattices are duals of each other rather than of themselves.

lattices · Lattice How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have.

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

lattices · Sublattices

Named alongside it

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

Bravais latticeChange of basisGram matrixLatticeQuadratic formUnimodular matrixBasis reductionDeterminantDual latticeFundamental domainHolohedryIndex

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