Crystallography & Symmetry
patterns derived from their groups
Generator

the honeycomb net at p6m, order 12

the honeycomb net at p6m, order 12
the honeycomb net at p6m, order 12. the honeycomb net drawn at the placement in which every vertex sits at the average of its neighbours in the cells its voltages name. The placement is the solution of one linear system per coordinate and is exact in the lattice basis, so its symmetry is detected by the same round trip every pattern here goes through: 12 operations, group p6m. 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 honeycomb net drawn at the placement in which every vertex sits at the average of its neighbours in the cells its voltages name. The placement is the solution of one linear system per coordinate and is exact in the lattice basis, so its symmetry is detected by the same round trip every pattern here goes through: 12 operations, group p6m. 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.

5 essays call net-placement. 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.

 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: 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.

 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.

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