Crystallography & Symmetry
patterns derived from their groups
Concept

Polyhedron — where it appears

A body bounded by flat faces meeting along edges. Whether it is rigid, whether it tiles space and whether it can be cut into another of the same volume are three different questions about it, and each is settled by a different invariant.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The five, as generator counts. Each of the five convex bodies that tile space by translation, built as the set of combinations of a handful of vectors with coefficients between zero and one. Three generators give a cube, four give either a hexagonal prism or a rhombic dodecahedron depending on whether three of them are coplanar, five give the elongated dodecahedron and six the truncated octahedron. The last column is what the same number of generators would give in general position, and the shortfall is the number of faces lost to coplanarity.

Every parallelohedron is a shadow of a cube

Take a few vectors and form every combination of them with coefficients between zero and one. All five of the convex bodies that tile space by translation come out of that recipe, from three vectors, four, four, five and six — and since the recipe is exactly the image of a cube of that many dimensions, the truncated octahedron is a three-dimensional shadow of a six-dimensional cube. The five are not the generic answers: they are the degenerate ones, and the degeneracy is what the tiling demands.

classification · Wigner–Seitz cells Five solids, twice each. Each Platonic solid as a framework of rods hinged at the corners, and again with its faces made rigid by adding their diagonals. The rank of the rigidity matrix reaches 3V − 6 exactly when the framework cannot move; the shortfall counts the ways it can. Three of the five are rigid as rods and all five are rigid as plates, which is Cauchy's theorem in the form a rank computation can see.

The polyhedra that can flex

A cube of rods folds and a cube of cardboard does not, and the difference is a rank. Cauchy proved in 1813 that a convex polyhedron with rigid faces is rigid; the rank of a rigidity matrix sees it directly, and it also sees where the hypothesis is doing the work. Drop convexity and an octahedron flexes — followed here for forty steps with every edge length held to five parts in a thousand million million.

applied · Rigidity 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.

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.

applied · Nets One determinant, three dimensions. The Cayley–Menger determinant of a set of squared distances, at three sizes. Its value is the squared content of the simplex those distances describe, times a factor that alternates in sign with the dimension. At three points it is Heron's formula rewritten; at four it gives a tetrahedron's volume from its six edge lengths with no coordinates anywhere. The alternating sign is not a convention — a value of the wrong sign means the distances belong to no set of points at all.

What six lengths decide and nine do not

A tetrahedron's volume is a determinant in its six edge lengths, with no coordinates anywhere. Add a fifth vertex and the lengths stop deciding: two shapes with identical edges and identical faces have volumes in the ratio 2.6. What survives is that the possibilities are finite — which is the whole reason a flexing polyhedron cannot change its volume.

applied · Rigidity The same accounting, at every coordination number. One row per number of edges at a vertex. The bill a sphere charges is 2dχ; the face worth nothing is 2d/(d − 2), which is a whole number at three, four and six and is 10/3 at five; the faces that can pay are those with fewer sides than that; and the last column is every way of paying the whole bill with faces of a single size. At three edges a vertex there are three such ways and twelve pentagons is one of them. At six there are none, which is the statement that six-fold coordination belongs to the plane and to no closed surface at all.

The twelve belongs to the vertex

Twelve pentagons is read as a fact about closing a surface. It is not: it is a fact about three edges meeting at a point. Let four edges meet instead and the sphere charges eight triangles; let five meet and it charges twenty; let six meet and it cannot be paid at all.

restriction · Curvature Every closed surface, and the two that charge nothing. The same accounting indexed by Euler characteristic rather than by genus. An orientable surface has χ = 2 − 2g, so it only ever occupies an even row; a non-orientable one has χ = 2 − k and occupies every row from one downwards. The odd rows therefore belong to surfaces that cannot be oriented and to nothing else — and the first of them, the projective plane, charges six. Six pentagons is a bill no orientable surface presents.

The surfaces a count by genus skips

A count indexed by genus steps in twelves and lands only on even numbers. A closed surface can have any characteristic at or below two, and the odd ones belong to the surfaces that cannot be oriented — where the projective plane charges six pentagons, a bill no orientable surface ever presents.

restriction · Curvature Every vector realised, and not at the same hexagon count. Each row is a set of faces other than hexagons whose charge — the sum of 6 − k over them — comes to twelve, which is what a closed trivalent net on the sphere must pay. Each column is a number of hexagons added to that set, and the entry is how many different solids exist with exactly those faces, found by winding up every arrangement of them into a spiral. A dash means the search found none; a question mark means the planar reader declined the row and it is not evidence either way. Every row has an entry somewhere, which is Eberhard's theorem, and the first one is at 0, 2, 3, 4 hexagons depending on the row — so the charge decides everything except the number of hexagons, and the number of hexagons is not a function of the charge.

Everything except the hexagons

Three counts of what a closed net must carry end on the same admission: an arithmetic saying what a net must charge does not say that a net exists. Eberhard's theorem says how close the charge comes to being enough, and the answer has a shape nobody would guess — it fixes every face count except the hexagons, and the hexagons are exactly the entry it cannot see.

restriction · Curvature The fewest contacts twelve pentagons can have, by size. For every cage of pentagons and hexagons up to forty-four atoms, the number of pairs of pentagons sharing a bond. The lower line is the fewest any cage of that size achieves — 30, 24, 21, 18, 17, 15, 14, 12, 11, 10, 9, 8 — the upper line the most, and the dashed line the bound that counting edges gives: the twelve pentagons carry sixty edges between them, a contact uses two and an edge to a hexagon uses one, so the contacts cannot fall below 30 − 3h with h hexagons. The bound is attained while the hexagons are few and goes loose at five, after which each extra hexagon removes about one contact rather than three. The number of cages at each size is printed beneath, and it is the least rather than the average that the bound is about.

How close the twelve must be

The charge fixes twelve pentagons and says nothing about where they go, because it is a sum over faces and cannot see which face touches which. What it cannot see is a graph on twelve points, and the fewest edges that graph can have falls from thirty to eight over the cages a census reaches — then keeps falling at a rate that puts its first zero exactly where the truncated icosahedron is.

restriction · Curvature

Named alongside it

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

Combinatorial curvatureEnumerationThe Euler characteristicConvexityCrystal netCensusCoordination numberCountingDegrees of freedomFace vectorRigidityTrivalent net

All concepts