The symmetry a net was written with

A net has no coordinates. That is the whole reason it is interesting: throw away the distances and the angles from a structure, keep only which atom is bonded to which and how the bonds wrap around the cell, and something survives that a list of coordinates was hiding. Two crystals with quite different cell edges can have the same net, and the net is the thing a chemist means when saying two materials have the same framework.

Having no coordinates leaves a question, though, and the placement nobody chose answered it: to ask what symmetry a net has, put every vertex at the average of its neighbours, and measure the symmetry of the point set that results. That placement is canonical — it is the solution of a linear system with no choices in it — and it is optimal, in the sense that no other placement of a stable net has more symmetry. So a net’s group is well defined after all, and this collection has been reporting it for two phases.

It was reporting the wrong thing, and the way it was wrong is the shape this whole site is built to catch. The barycentric placement is canonical, and its coordinates are not: they are written against whichever pair of translations the description happened to use. The detector that reads them is not.

The same net, typed four ways #

A quotient graph is a few vertices, a few edges, and on each edge a pair of integers saying which cell its far end sits in. The pair is measured against a basis of the translation lattice, and the basis is a choice — the same freedom every cell on this site has. Applying an integer matrix of determinant ±1 to every voltage describes the same net against a different pair of translations, and sameNet searches exactly that freedom, along with relabelling the vertices and moving one of them into a different cell.

So the six rows of the figure above are six descriptions of one object. The barycentric placement of each is the same placement seen in a different coordinate system: the honeycomb’s two vertices come out at a third and two thirds along both axes in the description this file has always used, and at some sheared image of that in the others.

The detector, given those coordinates, returns p6m for the first and p2 for the fourth. Both answers are correct reports about the matrices they were allowed to try, and only one of them is a statement about the net.

The pattern is easiest to see in the two panels drawn side by side, because the left one does not look like a wrong drawing. It looks like a perfectly ordinary oblique net, which is what a sheared honeycomb is.

Why the detector cannot see it #

The detector on this site works the same way for nets as it does for patterns, and the way it works is the reason the bug exists.

For a given lattice type it takes that type’s holohedry — the full symmetry group of the lattice, as a list of integer matrices — and for each matrix, and each pair of points, computes the translation that would carry the first point onto the second. That is a complete search: every symmetry of the set must send some point to some point, so nothing can be missed. It is the half of the round trip that does not know what it is looking for, and it is why this collection can say a pattern has a symmetry rather than nearly does.

The completeness is over translations. It is not over linear parts. HOLOHEDRY.hexagonal is a list of twelve specific matrices — the ones that are the hexagonal operations when the basis is the standard hexagonal one, a unit vector and its rotation by a hundred and twenty degrees. Conjugate the lattice by some other unimodular matrix and the same twelve operations become twelve different matrices, none of which is in the list, and the search dutifully finds none of them.

Five lattice types were tried, and trying five types is not trying five lattices. It is trying five sets of matrices, one per type, in one basis. That distinction is invisible for a pattern, because a pattern arrives with a declared lattice type and a basis that matches it. It is not invisible for a net, because a net has neither, and the description’s basis is whatever the enumeration or the typist produced.

The clearest statement of the failure is that the answer was not a function of the net. Hand the same net in twice, spelled differently, and get two answers. Every gate this site runs passed throughout: the drawings were well formed, the labels fitted, the assertions inside the generators held, and the round trip that checks a pattern against its group had nothing to say, because the object here is a graph and the group was never generated from anything.

What a net does have: the form its own edges make #

A net has no metric. It does have edges, and the edges of the barycentric placement are a set of vectors, and a set of vectors makes a quadratic form: Q, the sum of dₑ dₑᵀ over the edges of one cell. Any symmetry of the net permutes the edges and acts linearly on the plane, so it carries that sum to itself: M Q Mᵀ = Q for every symmetry M. The form is invariant under a group nobody has computed yet, which is what makes it useful — it can be built before the group is known, and it is built from the net rather than chosen.

There is one place to be careful, and getting it wrong produces an answer that looks plausible and is not. Q is a sum over vectors, so a symmetry acts on it as M Q Mᵀ. A metric is a form on coordinates, and a symmetry acts on it the other way round, as Mᵀ g M. The two conditions agree exactly when g is proportional to the inverse of Q. In two dimensions the inverse is the adjugate over the determinant, and the determinant is a positive number no reduction cares about, so the adjugate is used directly and the arithmetic stays in integers throughout. Using Q itself as the metric hands back a basis in which the honeycomb’s own form is (18 9; 9 6) — reduced-looking, hexagonal in neither convention, and giving an answer worse than the one it was meant to repair.

With the metric in hand the rest is the reduction this collection already has: Lagrange and Gauss’s algorithm, which swaps the two basis vectors so the shorter comes first, subtracts the nearest integer multiple of one from the other, and repeats until 2|b| ≤ a ≤ c. It terminates because the first diagonal entry strictly falls at every step but the last, and it is the same procedure that produces the shortest basis of a lattice and underlies the cell that settles the argument.

The point of reducing is a classical fact that does the work here: a reduced binary form’s automorphism group is the holohedry of its own lattice type, in standard position. So in the basis that reduces the metric, the detector’s fixed lists mean what they are supposed to mean, and the five types between them cover every symmetry the net can have.

The one shape reduction does not reach #

A reduced form has an automorphism beyond ±1 in exactly three cases, and they are the three lattice types with a mirror: b = 0, which is rectangular; a = c, which is rhombic; and a = −2b, which is rhombic as well and does not look it.

The third is the case that has to be handled by hand, and a net found it. When a = −2b the vector v₁ + v₂ has |v₁ + v₂|² = a + 2b + c = c, so it has exactly the length of v₂ — the two equal vectors of the centred lattice are v₂ and v₁ + v₂, not v₁ and v₂, and the reduced basis is not the one the rhombic convention wants. The five-coordinated net in this collection’s registry is precisely this: its reduced metric is (6 −3; −3 14), its lattice is centred rectangular with a mirror, and matched against the five standard shapes it matched none. The detector was offered the oblique list and returned a group of order four that had no name.

Rewritten on (v₂, v₁ + v₂) the form is (14 11; 11 14), the swap is a symmetry, and the group is cmm. When a = c as well the lattice is hexagonal and already standard, so the transform is applied only when the two differ — which is a condition rather than a case analysis, and is checked rather than assumed.

What the census was actually measuring #

The one-vertex census is where this mattered most, because a census is the one place on this site where nets arrive unchosen. Every other net here was written down by somebody who knew what it was for, in the basis that made it legible. A census produces its members in whatever basis the enumeration reached them in, which is a fact about the loop bounds and nothing else.

So the census’s headline number was a statistic about the enumeration order. Of the hundred and forty-three one-vertex three-edge nets inside a box of three, exactly one was reported as having more symmetry than the inversion centre every such placement has. Measured on each net’s own basis, seventeen do.

The correction is a factor of seventeen and the qualitative claim survives it. Symmetry is still the exception — a hundred and twenty-six of the hundred and forty-three are p2 and nothing more — and p2 is still a floor rather than a coincidence, for the reason that essay gives: a one-vertex net has its only vertex at the origin, its edges leave in ± pairs, and the inversion is free.

What does not survive is the ratio, and the ratio is the whole content of a census. “One in a hundred and forty-three” and “one in eight” are different statements about how special a symmetric net is, and only the second is about nets.

The two nets in the registry that changed #

Only two of the collection’s named nets moved, and both are instructive about how quiet the failure was.

The skew net was reported as p2 and is pmm. It was written with voltages (1,0), (0,1) and (2,1), which is a sheared way of writing a net whose edge form is rectangular; its reduced metric is (2 0; 0 3) and the two mirrors are there in every basis, unnamed in this one. That net was the census essay’s chosen illustration of a typical member — drawn precisely because it was thought to have nothing beyond the floor — and it is one of the seventeen. A figure captioned “this is what an ordinary member looks like” was showing one of the exceptions.

The five-coordinated net was reported as p2 and is cmm, for the centred-form reason above.

Neither is a large correction to the subject and both are the right size to be alarming, because they are exactly what a systematic instrument error produces: not a wild answer that somebody notices, but a plausible answer that is too small, on the nets nobody was looking at.

Naming the group was wrong for a second reason #

Fixing the basis exposed a second failure underneath it, which is the ordinary way of things: the identification of which of the seventeen an operation set is compared linear parts against a fixed spelling.

That comparison was chosen for a good reason and it is the right test for the pair it was written against: p3m1 and p31m have their mirrors in different directions relative to the lattice, are not conjugate as groups of integer matrices, and no count of operations separates them. The trouble is that it is also too strict. A centred rectangular lattice has a mirror along each diagonal; cm is spelled here with one of them; and a net whose mirror is the other one matched nothing at all. Nine of the ninety-one five-edge two-vertex nets came back unnamed for that reason alone.

The repair says what “the same wallpaper group” means: the operation sets are compared up to conjugation by a change of basis, searched over matrices with entries up to two. A bounded search of that kind can miss a match and cannot invent one, and two of the seventeen matching would mean those two were conjugate, which no two of them are. The lattice type then stops entering the name at all — which is right, since a net whose metric is hexagonal may perfectly well have a group that is not.

How it was found, which was not by looking #

Nothing found this for two phases, and it is worth being precise about why, because the reason is a general one about which checks can catch which errors.

Every gate the fleet runs on a figure asks whether the drawing is well formed: whether the ink is inside the box, whether a label fits, whether it contrasts against its ground, whether the generator threw. Every one of them passed on a honeycomb reported as p2, because a sheared honeycomb drawn on an oblique lattice with an inversion centre marked is a perfectly good drawing of a perfectly real thing. The assertions inside the generators passed too: they asked whether the placement separated the vertices, whether the cycles generated the translations, whether the detected operations were closed — all true, all in the wrong basis.

The round trip that is this site’s characteristic instrument had nothing to say either, and the reason is structural. For a pattern, the round trip generates a point set from a group and then rediscovers the group from the point set, so an answer that is too small fails immediately: the generated symmetries are known and the detector must return all of them. For a net there is no generating group. The graph is the input and the group is the output, so there is nothing on the other side of the trip to compare against, and an answer that is too small is indistinguishable from a net that is not very symmetric.

What found it was writing the two-vertex census and watching the honeycomb — the one net in that family anybody has named — come back as p2 at one box size and p6m at another. A census reports the same object at two bounds and is therefore its own consistency check, which is a property worth having on purpose: the number that moved was not a number anybody had a prior about, and it moved because the enumeration order changed and nothing else did.

It found a second defect on the way, and the second is the one that changed more numbers. sameNet is a bounded search, so it is not transitive: three descriptions of the honeycomb appeared in one box, each matched with the honeycomb and none with the other two, and the census counted them as three nets. That is repaired by the same reduction, used to build a canonical key rather than a comparison — and it moved the published one-vertex count from a hundred and forty-four to a hundred and forty-three.

The general form of it: an instrument whose input has no independent description cannot be checked by asking it the same question twice. It has to be asked the same question in two languages. Writing the net six ways and demanding one answer is that, and it is now a refusal rather than a remark.

Where the exactness stops #

Computed here: the quadratic form of every net’s own barycentric edges, as integers with the denominators cleared once; its inverse as an adjugate; the Lagrange–Gauss reduction of that; the basis change back; and the detection, in that basis, at all five lattice types with the largest surviving operation set kept.

Nothing here is approximate and one thing is bounded. The reduction, the form and the detection are exact rational arithmetic, as everything on this site is. The equivalence between descriptions is not: sameNet searches basis matrices with entries up to four, so two descriptions it separates might in principle be joined by a matrix with an entry of five, and the naming search is bounded at two. A negative answer from either is negative within its bound, which is the same caveat the census already carries and is stated again because this essay adds a second bounded search rather than removing one.

A more symmetric placement than the barycentric one is still ruled out for stable nets only. The claim that nothing beats the equilibrium placement is a claim about nets whose vertices do not collide, and a third of the two-vertex census collides. For those there is no drawing at all and therefore no group, which is a gap in the question rather than in the answer.

And the reduction decides a basis, not a metric. The form the edges make is one invariant form among many — a different weighting of the edges gives a different form with the same symmetries — and it is used here because it exists without a group having been computed. It is not a claim that a net has a preferred metric, which it does not.

Where the ladder goes next #

Back, to the two essays this one corrects: the placement nobody chose, whose registry table has two new entries in it, and every net with one vertex, whose census statistic moved by a factor of seventeen and whose chosen typical example turned out to be exceptional.

Sideways, to the arithmetic the repair runs on: the shortest basis, where the same reduction appears as a question about lattices rather than about nets, and thirteen ways to hold a lattice, which is the classification of the finite groups of integer matrices that the whole notion of a standard-position holohedry rests on.

Onward, to every net with two vertices, the census this correction had to land before, where the floor of p2 disappears entirely and a description can be a supercell in a second way.