Crystallography & Symmetry
patterns derived from their groups
Concept

Barycentric placement — where it appears

The drawing of a net in which every vertex sits at the average of its neighbours, unique up to an affine map once the translations are fixed. It is the placement nobody chose, and its symmetry group is the net's own.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

the honeycomb net: cmm against p6m. the honeycomb net drawn twice. On the left a placement chosen by hand, whose symmetry group is cmm of order 4; on the right the placement in which every vertex sits at the average of its neighbours, whose group is p6m of order 12. The graph is identical in the two — the same vertices joined the same way — so every symmetry of the left-hand drawing is a symmetry of the net and the right-hand drawing has them all. Each detected operation is then required to carry every edge of the quotient graph to an edge, which is what makes it a symmetry of the net rather than of the point set.

The placement nobody chose

A net has no coordinates, so drawing one means inventing them. There is exactly one way to invent them that involves no choice: put every vertex at the average of its neighbours. The drawing that results has the largest symmetry group the net admits, and this site's own detector finds it.

applied · Nets One vertex, two edges: one net. Three edges: no answer at all. Every net with one vertex and the stated number of edges, counted inside boxes of voltages of three sizes, up to change of basis and the sign of an edge. Two edges give one net whatever the box, and the reason is a sentence: two voltages that generate the translations are a basis of ℤ², and every basis is carried to every other. Three edges give more nets in every larger box, and that is not a failure of the search — normalise two of the voltages to a basis and the third is a free pair of integers, so the family is infinite. An enumeration inside a bound reports which of those two situations it is in rather than reporting the count it happened to reach.

Every net with one vertex, counted

A net is a few vertices, a few edges and a pair of integers on each, so a census is available: fix the numbers, bound the integers, enumerate. Two edges give exactly one net at every bound. Three give three, then nineteen, then a hundred and forty-three — and the question changes.

applied · Nets One net, six descriptions, four different answers about its symmetry. The honeycomb written against six bases of ℤ², all of them the same net. The detector tests each lattice type's holohedry in standard position, so a symmetry written against another basis is a matrix that is not in the list and is never tried — and the answer comes back as p6m, or an unnamed group of order four, or p2, or cmm, depending on how the voltages were typed. The metric column is the form the net's own edges make, inverted; the reduced column is that form after Lagrange–Gauss reduction, and it is the same in every row, which is what makes the last column a property of the net.

The symmetry a net was written with

A net has no coordinates, so its symmetry is whatever its best drawing has. This collection measured that by handing the drawing to a detector — and the detector tests a fixed list of matrices, so the answer depended on which pair of translations the voltages had been written against. The honeycomb came back as p6m, or p2, or cmm, or nothing, one net and four answers.

applied · Nets Two vertices and three edges: two nets, at every box size tried. Every net with two quotient vertices and the stated number of edges, counted inside boxes of voltages of several sizes. One cross voltage is set to zero by the gauge — the freedom that moving one vertex into another cell gives — and the rest are drawn from the box. Each entry is the count of nets whose placement separates their vertices, plus the count of those whose does not: the first has a canonical description and stops growing, and the second does not have one and therefore keeps rising with the box. The reducible column is the descriptions thrown away for a reason the one-vertex census never had — cycles generating the whole of ℤ² and a net whose own cell holds one vertex rather than two — and it is empty at every odd edge count, because the swap that would reduce a description pairs its edges and an odd number cannot pair.

Every net with two vertices, counted

The one-vertex census could not contain the honeycomb, because the honeycomb has two vertices in its cell. Adding the second one closes a family at two nets, removes the floor of p2 entirely, makes a third of the members undrawable, and forces the census to refuse a kind of description the first one never met: an honest quotient graph written on twice the cell it needs.

applied · Nets

Named alongside it

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

Crystal netChange of basisQuotient graphVoltageCensusGraph isomorphismAccidental symmetryAutomorphismBasis reductionEquilibrium placementFree actionGraph laplacian

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