The surfaces a count by genus skips

As many heptagons as pentagons ends with a ladder of surfaces and a charge for each: the sphere demands twelve pentagons, the torus demands nothing, and every extra hole demands twelve more heptagons. It is indexed by genus, so it reads 12, 0, −12, −24, and it steps in twelves.

That essay records what the indexing cost it: “A Klein bottle also has Euler characteristic zero and also admits a hexagonal net, and it is not on the ladder because the drawing of the ladder is in genus and a Klein bottle has none.”

The gaps between those twelves are not gaps. A closed surface may have any integer Euler characteristic at or below two, and the odd values are perfectly ordinary surfaces — they are simply the ones that cannot be oriented, and genus cannot name them. The first is the projective plane, at χ = 1, and its charge is six: six pentagons on a closed surface, which is half of the sphere’s twelve and is a number no orientable surface ever asks for.

Half of a sphere, and what that means #

The projective plane can be described in a way that makes its charge obvious before any accounting is done: it is the sphere with every point identified with the point opposite it.

Do that to a polyhedron rather than to a bare sphere and the result is a net. Every vertex is glued to the vertex opposite, every edge to the edge opposite, every face to the face opposite — so the vertex, edge and face counts all halve, the characteristic halves from two to one, and the charge halves from twelve to six.

It works only when the identification fixes nothing. A map with a fixed point does not have a surface for a quotient: the point where a cell is glued to itself is not a place where the surface looks like a plane. So the polyhedron must be centrosymmetric — every vertex must have a vertex opposite it, and no cell may be its own image — and that is checked here rather than assumed.

The tetrahedron has no centre of inversion, and it is the only regular solid that does not. So it has no antipodal map to quotient by, contributes no net to the projective plane, and appears in the table as a refusal rather than as a row. That is a fact about the solid the earlier rungs never had occasion to use — the tetrahedron is the self-dual one, it is the one whose rotation group is T rather than O or I, and this is the same absence showing up in a third place.

Six pentagons, and three squares #

The dodecahedron’s quotient is the useful case, because it puts a number on the surface that the sphere cannot supply.

Twelve pentagons become six. Twenty vertices become ten, thirty edges become fifteen, and 10 − 15 + 6 = 1. The charge is 6 × 1 = 6, and six pentagons at one apiece is exactly six. There is a closed net of six pentagons, and no orientable surface admits one.

The cube’s quotient is worth setting beside it. Three squares, four vertices, six edges — and 4 − 6 + 3 = 1 again. A square carries a charge of two at three edges a vertex, so three of them come to six, and the same bill has been paid in a different denomination.

That the two quotients pay the same bill differently is the projective plane’s version of an observation the sphere’s rung makes about twelve, and it carries over unchanged because the accounting never asked which surface it was on. What changed is one integer.

The graph this quotient turns out to be #

The hemi-dodecahedron has ten vertices, fifteen edges and three edges at every vertex, and a reader who has met graph theory will already have recognised it. It is the Petersen graph — the standard counterexample, the smallest graph that is neither planar nor three-edge-colourable, and the object that appears in a textbook whenever a plausible-sounding claim about graphs needs refuting.

That is checked here rather than recognised. The quotient’s adjacency is built from the dodecahedron’s own faces with antipodal vertices identified; the degrees come out three throughout; and the girth — the length of the shortest cycle — comes out five, found by breadth-first search from every vertex rather than by looking.

The girth is what proves the graph non-planar, and it does it with Euler’s relation and nothing else. A simple graph drawn in the plane satisfies F = 2 − V + E, which here is seven faces. Every face of such a drawing has at least as many sides as the girth, so 2E ≥ 5F would have to hold — thirty against thirty-five. It does not, so no planar drawing exists. That is the same three lines this whole anchor is built on, used to refuse a drawing rather than to count pentagons.

So the object in the figure above is a graph that cannot be drawn in the plane without crossings and can be drawn on the projective plane with none. Its faces there are the six pentagons, and the six pentagons are why: seven faces of at least five sides is too many for the plane and six is exactly right for a surface charging six.

That is worth carrying past this essay. The question which surface does this net need is not a curiosity about topology; for a graph it is the question of how much room is required to draw it without crossings, and the answer is bounded below by the same accounting. A net that will not lie flat is one whose charge the plane cannot pay.

What the identification does to the symmetry #

The quotient halves the counts, and it halves something else with them.

The dodecahedron’s full symmetry group has order one hundred and twenty and contains the inversion; its rotations alone number sixty. The antipodal map is the inversion, so dividing by it divides the full group by a normal subgroup of order two and leaves a group of order sixty acting on the quotient. The rotations survive intact and the improper operations become the same operations as the proper ones they differ from by a centre.

That is the same arithmetic the most of an icosahedron a crystal can keep uses from the other side, where the question is which subgroup of the sixty rotations a lattice can hold. Here the sixty are what is left after a quotient rather than what is available before a restriction, and the number is the same because the group is.

The consequence for the net is that the quotient is as symmetric as the solid was, in the sense that matters: every one of the six pentagons is equivalent to every other, and every one of the ten vertices to every other. A quotient by a free action does not break symmetry, it removes the operations that were acting trivially on the quotient anyway — which is the same statement a free action makes when a plane group is divided out of a plane net.

Where the odd numbers come from #

The reason the earlier ladder could not reach χ = 1 is worth stating as arithmetic rather than as a limitation of notation.

An orientable closed surface is a sphere with g handles and has χ = 2 − 2g. That expression is even for every g, and there is no handle count that gives an odd answer — so the whole family occupies the even rows and nothing else.

A non-orientable closed surface is a sphere with k crosscaps and has χ = 2 − k. That runs through every integer at or below one: one crosscap gives the projective plane at χ = 1, two give the Klein bottle at χ = 0, three give χ = −1, and so on downwards without limit.

So every odd characteristic belongs to a non-orientable surface and to no other kind, and every even one at or below zero belongs to one of each. The classification of closed surfaces is that short, and reading the charge off it is a substitution.

The consequence for nets is direct. A charge of six, or of minus six, or of any other value not a multiple of twelve, is a charge that can only be presented by a surface with a crosscap in it. A carbon cage of six pentagons is not a strange fullerene; it is not a fullerene at all, because a fullerene lives on a sphere and this does not.

Two surfaces at zero, and the count that cannot separate them #

The row at χ = 0 is where this gets sharp, because it holds two surfaces and the previous rung’s figure asserts that it holds one.

The assertion there is that exactly one surface admits a trivalent net of hexagons alone, and it is the torus. That is true of the ladder it is asserted over, which contains only orientable surfaces. The Klein bottle also has characteristic zero, also charges nothing, and also admits a net of hexagons alone — so on the full ladder the count is two.

Rather than argue it, both are built. One brick-wall honeycomb, wrapped two ways: plainly, which joins each column to itself, and with the column index negated on crossing the seam, which is the flip that makes the surface non-orientable.

The result is the reason this rung exists.

Every quantity the charge is computed from is the same for both. The vertices, the edges, the faces, the face sizes, the characteristic, the total. A reader handed the two censuses and asked which is which has nothing to work with, because the accounting is a function of the face vector and the face vectors are equal.

That is the honest limit of this whole family of results, and it is worth saying in the strongest form available. The three lines settle what a surface charges; they do not settle which surface it is. Two different closed surfaces present the same bill and admit the same net, and no amount of counting faces distinguishes them — the distinguishing property is orientability, which is about how the faces are glued rather than about how many there are.

Deciding orientability, rather than declaring it #

A net presented as a graph with a cyclic order at each vertex does not announce its surface, and the machinery here had to be taught to find out.

A plain rotation system can only describe an orientable embedding. The cyclic orders are read in one global sense of rotation — clockwise, say — and a surface without a consistent notion of clockwise has no such reading. Tracing the flipped net with a plain rotation system produced nine hexagons, one eighteen-sided face and a characteristic of minus two: a perfectly well-formed embedding of the right graph in a surface with two handles.

The repair is standard and its shape is worth carrying. An embedding scheme is a rotation system together with a signature on each edge, and it describes an embedding in any closed surface rather than in an orientable one only. The trace multiplies its running sign by each edge’s signature on crossing it, and takes the neighbour after the one it arrived on when the sign is positive and the one before when it is negative.

Orientability is then decided rather than asserted. Give each vertex a sign by propagating outwards along a spanning tree; the net is orientable exactly when every remaining edge agrees with the signs its two ends already carry. An edge that disagrees is a closed walk that comes back reversed, which is what a crosscap is, and the walk is reported.

Nothing in that test looks at a picture, and nothing in it consults a name. That matters here more than in the earlier rungs, because the earlier rungs could take the surface as given: a cage is on a sphere, a wrapped honeycomb is on a torus. Here the surface is the thing in question, and a method that assumed it would be assuming the answer.

What this does not settle #

A net on a surface, not a substance. Nothing above says a carbon cage of six pentagons could be made. The projective plane does not embed in three-dimensional space without passing through itself, so a hemi-dodecahedron is a combinatorial object rather than a shape a chemist could hold — and that is a much stronger obstruction than the ones the toroidal case runs into, which are about stability rather than about existence.

The quotient route reaches one characteristic. Every net above with χ = 1 came from halving a centrosymmetric solid, which is a construction that gives χ = 1 and nothing else. Nets on the surfaces at χ = −1 and below are permitted by the arithmetic and are not built here; whether the ones the accounting allows exist is the same open question the sphere’s rung records, where a necessary condition is known to be satisfied by one face vector that no net realises.

And the degrees are not mixed with the surfaces. The degree ladder varies the coordination number at a fixed sphere; this varies the surface at a fixed set of solids. The four-valent and five-valent quotients above are the only points where the two directions meet, and they meet by accident rather than by design — the octahedron and icosahedron happen to be centrosymmetric, so their quotients came along with the others.

The same distinction, in the plane groups #

The division this rung turns on is one the collection already has under another name, and connecting the two is the point of having both.

The two that fold into a surface is about which plane groups act on the plane without fixed points, and there are two: p1, whose quotient is a torus, and pg, whose quotient is a Klein bottle. That is exactly the pair above — the two closed surfaces of characteristic zero — arriving from the theory of groups rather than from a wrapping rule.

And the reason pg gives the non-orientable one is the reason the flip does here. A glide reflection reverses orientation, so a path that follows one comes back mirrored; the quotient has a closed walk that reverses, which is a crosscap. The signature on an edge in the trace above is doing the same bookkeeping the glide does in the group: it records that crossing here reverses which way round is which.

So the two objects at χ = 0 are not merely two surfaces that happen to charge the same. They are the two quotients of the plane by a group acting freely, and the difference between them is whether that group contains an orientation-reversing element — which is the same criterion that decides whether a crystal may be built from a single hand.

The order the pieces arrived in #

The arithmetic here is older than any of the objects it is applied to, and the order is worth knowing because it explains why the genus indexing became standard.

Euler’s relation is 1750 and was stated for convex polyhedra, where the surface is a sphere and there is nothing to index. The one-sided surfaces are 1858, found in the same year by Möbius and by Listing working separately, and they were curiosities rather than a family until the surfaces were classified — which is Dehn and Heegaard in 1907, and which is where χ = 2 − k for k crosscaps comes from.

The polyhedra that live on them came later still. Coxeter’s hemi-polyhedra — the quotients used above — are a twentieth-century construction, and they were studied as regular maps rather than as solids, because a solid is a thing in space and these are not. That is the reason a chemist’s ladder of surfaces is indexed by genus: the surfaces a molecule can sit on are the orientable ones, and a language built for cages has no need of the rest.

The graph came in from a third direction entirely. Petersen published his graph in 1898 as a counterexample in the theory of factorisations, with no reference to a surface at all; that it is the dodecahedron halved was noticed afterwards. Three subjects arrived at one object for three reasons, and the accounting above is what makes them the same object rather than three that resemble one another.

The moral this collection keeps returning to applies here without modification. A classification is only as complete as the thing indexing it, and an index chosen for the cases in hand will silently omit the cases that were never in hand.

Why a crystallographer meets the odd rows, and what the ladder owes after them #

The projective plane is not a surface a crystal sits on, and the reason to have it is not that a structure is shaped like one.

It is where a quotient lands. The whole method of reading a net as a group acting on a cover divides a periodic structure by its own symmetry and studies what is left, and what is left is a quotient whose surface depends on which operations were divided out. Divide by translations alone and the quotient is orientable; divide by anything that reverses orientation and it is not. A crystallographer working with quotients meets non-orientable surfaces the moment glides are included, which is most of the time.

And the odd charge is a signature of that. A quotient object whose accounting comes to six rather than twelve or nothing has had an orientation-reversing operation divided out of it, and the charge says so before anything else does. That is a diagnostic rather than a theorem, and it is the practical use of knowing which rows of the ladder are which.

The next question this leaves is the one the earlier rungs also leave and which nothing here answers: an arithmetic that says what a net must charge does not say that a net exists. The projective plane’s six is realised, twice over, by construction. Whether every face vector the odd rows permit is realised is not known here, and the sphere’s single exception is the standing reminder that the answer is not automatically yes.