Crystallography & Symmetry
patterns derived from their groups
Concept

Periodic graph — where it appears

An infinite graph carrying a free action of a lattice of translations whose quotient is finite. It is the formal object a crystal net is, and its definition mentions no distances at all, which is what makes questions about it decidable by integer arithmetic.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Rotation orders 1, 2, 3, 4, 6 and no others. Every net in this collection, with the orders of the rotations its own symmetry group has, and the degrees of its vertices beside them. The orders are 1, 2, 3, 4, 6 — the crystallographic restriction, arrived at with no length anywhere in the argument: the translations of a net are ℤ² by construction, an automorphism carries translations to translations, so it acts on ℤ² by an integer matrix, and an integer trace in the interval from minus two to two is one of five numbers. The degree column is there because the two are constantly confused: a net may perfectly well have vertices of degree five, and one here does.

The restriction, with no lattice assumed

The proof that only two-, three-, four- and six-fold rotations are possible is usually stated about a lattice, and every step of it turns out to need no lengths at all. A periodic graph has the same theorem, proved the same way — and a graph may have a five-fold symmetry the plane cannot receive.

restriction · Restriction the honeycomb net, unfolded over 3×3 cells. The infinite graph the quotient graph names, drawn over 3 by 3 cells with the home cell outlined. Each edge of the quotient becomes one edge per cell, running to the cell its voltage names; the drawing adds coordinates the net does not have, and they are the placement in which every vertex sits at the average of its neighbours. two vertices, three edges, degree three — the graph of graphene and of every hexagonal mesh.

A structure with the distances thrown away

Keep which atoms are joined and throw away where they are, and what is left is an infinite graph that can be written on a postcard: a few vertices, a few edges, and a pair of integers on each. Two things about that writing-down are free, and neither of them changes the net.

applied · Nets Nine graphs against two conditions. Every candidate graph with the two quantities Steinitz's theorem asks for: the largest number of vertices that can be removed while it stays connected, capped at three because three is all the theorem needs, and the number of edges against the most a planar graph on that many vertices can have. The connectivity is decided by removing every pair and testing what is left, which is the definition rather than a proxy for it. Five of the nine pass both and are the graphs of convex polyhedra; the other four fail exactly one condition each, which is why they are here.

A polyhedron is two properties of a graph

Steinitz's theorem says a graph is the corner-and-edge graph of a convex polyhedron exactly when it can be drawn in the plane without crossings and stays connected after any two vertices are removed. No lengths, no angles, no convexity — the conditions are about the graph alone, and each one is needed, which four small counterexamples show.

applied · Nets

Named alongside it

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

Crystal netTranslation subgroupAutomorphismChange of basisCrystallographic restrictionDegreeThe Euler characteristicGauge freedomGraph isomorphismInteger matrixPolyhedronQuotient graph

All concepts