Crystallography & Symmetry
patterns derived from their groups
Generator

the honeycomb net: 2 vertices, 3 edges, quotient graph

the honeycomb net: 2 vertices, 3 edges, quotient graph
the honeycomb net: 2 vertices, 3 edges, quotient graph. The quotient graph of the honeycomb net: 2 vertices, 3 edges, and on each edge the pair of integers saying which cell its far end sits in. That is the whole net — an infinite graph written as a finite one. two vertices, three edges, degree three — the graph of graphene and of every hexagonal mesh. Its cycles generate the translations at index 1, which is what makes this description honest rather than one written on too large a cell.

The quotient graph of the honeycomb net: 2 vertices, 3 edges, and on each edge the pair of integers saying which cell its far end sits in. That is the whole net — an infinite graph written as a finite one. two vertices, three edges, degree three — the graph of graphene and of every hexagonal mesh. Its cycles generate the translations at index 1, which is what makes this description honest rather than one written on too large a cell.

13 essays call net-quotient. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. What a lattice forbids

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.

 11 nets, and one accounting. Every plane net folds onto a torus when its own translations are divided out, and a torus has Euler characteristic zero — so the quotient's vertices, edges and faces satisfy n − e + f = 0 and the number of faces is not something to be counted off a drawing but e − n. Dividing through gives one over the mean face size plus one over the mean degree equal to a half, which is the same relation that forbids a plane tiling by pentagons, reached here with no geometry in it at all. It holds for every net in the table. The classification

Every net folds onto a torus

Divide a plane net by its own translations and the quotient is a finite graph drawn on a doughnut. A doughnut has Euler characteristic zero, so the number of faces is not something to count — it is forced, and with it a relation between how many edges meet at a vertex and how many bound a face.

 3 whole-number solutions: (6, 3), (4, 4), (3, 6). Every pair of whole numbers from three to 12, with the mean face size across and the mean degree down. A square in the first colour is a pair satisfying one over p plus one over q equals a half exactly — the flat case, where a periodic net is possible — and there are 3 of them: 6 and 3, 4 and 4, 3 and 6. The lighter squares above and to the left have a sum greater than a half, which is a closed polyhedron rather than a plane tiling; the ones below and to the right have a sum less than a half and belong to a surface of negative curvature. The plane is the boundary between them and it is thin. The classification

Three answers in whole numbers

One over the face size plus one over the degree equals a half. Ask for whole numbers and there are exactly three answers, which are the three nets everybody has drawn since childhood — and the pairs on either side of them are a closed polyhedron and a plane the plane has no room for.

 The honeycomb's two levels meet at K, exactly. The two levels of the honeycomb net along a line from the centre of the zone to its corner. The off-diagonal entry of its two-by-two matrix is the sum of the phases of three bonds, and at the corner those phases are the three cube roots of unity, whose sum is zero — exactly, as an identity in the ring the phases live in rather than as a number that came out small. So the matrix there is the zero matrix and both levels are zero. It is the shortest exact statement of a crossing in this collection. Into space

The crossing at the corner

The honeycomb's two levels meet at the corner of its zone, and the meeting is not approximate. Three phases sum to zero there — an identity between cube roots of unity — so the matrix is the zero matrix, and making the two sites differ opens a gap of exactly that difference.

 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. Symmetry at work

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.

 the kagome net: 4, 8, 14, 18 at the first four shells. The vertices of the kagome net at graph distance one, two, three and four from a chosen vertex, each marked with its distance. Distance here is a number of edges and nothing else — no length enters, and the shells are drawn on the barycentric placement only so that they can be seen. The counts are 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, which is the net's coordination sequence. Symmetry at work

Counting outwards

How many vertices lie one step from a vertex, two steps, three? The counts settle into a straight line — but for some nets only along the even distances, with a different line along the odd ones, alternating for ever. The period is measured, and it is not always one.

 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. Symmetry at work

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.

 two sites in a hexagonal cell: 3 cutoffs, degree 3 to 12. Two atoms per hexagonal cell, at the positions graphite's carbons occupy, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so. Symmetry at work

A net is a choice of what counts as a bond

A list of atomic positions does not contain a net. It contains distances, and somebody has to decide which of them are bonds — so the net is a fact about the cutoff as much as about the crystal, and moving the cutoff past a shell of neighbours changes the answer.

 The kagome net's level that does not move. Three levels of the kagome net across the zone, one of them flat. The reason is drawn beside it: a state that alternates in sign round one hexagon and vanishes everywhere else is an exact eigenvector of the adjacency operator at −2, because every site outside the hexagon that touches it touches exactly two of its vertices and those two carry opposite signs. The check is integer arithmetic in a supercell of 27 sites, with a residual of exactly zero. A state confined to one hexagon has no wavevector, and a level made of such states cannot depend on one — which is what a flat line across a zone means. Symmetry at work

The level that does not move

Three levels cross the kagome net's zone and one of them is a horizontal line. The reason is a state that alternates in sign round a single hexagon and is exactly zero everywhere else — a solution with no wavevector in it at all, which is why no wavevector can move it.

 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. Symmetry at work

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.

 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. Symmetry at work

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.

 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. Symmetry at work

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.

 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. Symmetry at work

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.

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