Crystallography & Symmetry
patterns derived from their groups
Concept

Modular group — where it appears

The group of integer matrices of determinant one acting on the shapes of plane lattices, generated by a shear and a swap. Its two relations say that one generator has order two and another order three, and its fundamental region is where reduction sends every lattice.

Named by 2 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 The region, and its copies. Words in S and T up to length 4, each carrying the region somewhere else. The copies do not overlap and they do not leave gaps: the upper half-plane is tiled by them, one copy per change of basis. That is the whole content of the claim that reduction picks a canonical basis — every basis of every lattice is in exactly one copy, and reduction is the walk back to the shaded one.

Two moves reach every basis

A lattice has infinitely many bases and reduction picks one. Why it can is a fact about a group with two generators and two relations — and the fundamental region tiles the plane with its own copies, one per basis, which is what makes the walk home finite.

lattices · Moduli

Named alongside it

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

Basis reductionChange of basisFundamental domainModuli spaceQuadratic formUnimodular matrixContinued fractionGeneratorsGroupHolohedryLattice typeMeasure

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