Crystallography & Symmetry
patterns derived from their groups
Concept

Fundamental domain — where it appears

The smallest piece of the plane from which a group rebuilds the whole pattern. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Growing the p4 orbit. One motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.

The orbit is the pattern

A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.

operations · What symmetry is A fundamental domain for p4m. One representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

The fundamental domain

The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.

operations · Fundamental domain p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

Domains of a subgroup

A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

operations · Fundamental domain 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 p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group.

Every wall names a generator

The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

operations · Fundamental domain p4m: 10 of 36 wavevectors have to be visited. The Brillouin zone of the square lattice with a grid of 36 wavevectors on it, of which 10 are drawn solid: one per star, which is everything a calculation over a p4m-symmetric operator has to visit. The share is 27.8 per cent against the 12.5 per cent that the order of the point group would give, and it is larger for a reason worth naming — the wavevectors on the boundary of the wedge have short stars, so they are over-counted by any argument that only divides by the group order. The identity that is checked is that the star sizes add to the whole grid.

The domain in reciprocal space

A fundamental domain is the piece of a pattern the group repeats, and this collection has drawn several. The same idea in reciprocal space is what makes a calculation over a crystal affordable — and its share of the zone is larger than one part in the group's order, for a reason worth measuring.

operations · Fundamental domain The domain is a polygon, and its edges are elements. The Dirichlet domain of a point whose stabiliser is trivial: the set of points at least as close to it as to any other point of its orbit. It is a convex polygon, it is a fundamental domain, and each of its edges lies on the bisector of the base point and one image of it — so each edge already carries the element that produced it, with no search. Edges are drawn by kind: paired with another edge, fixed pointwise by a reflection, or folded in half by a half turn.

The relations a polygon dictates

Poincaré's theorem has two halves. The walls of a fundamental domain name the generators, which is the half this collection already computes; walking round its corners names the relations, which needs a domain with corners rather than a domain made of pixels. Building the Dirichlet polygon exactly gives a presentation of each of the seventeen — and coset enumeration says every one of them is right.

operations · Fundamental domain

Named alongside it

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

Asymmetric unitBasis reductionChange of basisCosetModular groupModuli spaceOrbifoldOrbitPlane groupQuadratic formStabiliserTiling by a group

All concepts