Crystallography & Symmetry
patterns derived from their groups
Concept

Trivalent net — where it appears

A net in which exactly three edges meet at every vertex, so that three faces meet there too. It is the shape a carbon cage and a honeycomb both take, and the one for which a face of six sides is flat — which makes six the size every count of curvature is measured against.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

60 vertices, 12 pentagons. A closed net with three edges at every vertex: 60 vertices, 90 edges and 32 faces, of which 12 are pentagons and 20 are hexagons. The pentagons are picked out in the second colour. Their number is not a property of this cage — it is twelve for every closed trivalent net of pentagons and hexagons, at any size, and the hexagon count is free.

Twelve pentagons, and no way round them

The crystallographic restriction forbids a five-fold face in a flat repeating net. Curve the net into a closed cage and the same three lines of arithmetic require exactly twelve of them — at any size, with the hexagon count free. What a lattice forbids, closing up compels.

restriction · Curvature A centre at every other ring, and never between. Two families of closed cage, each a tube of hexagons closed at both ends by a cap of six pentagons, taken from no rings of hexagons to 8. The top row has five faces to a ring and a pentagon at each pole; the bottom row has six and a hexagon. Each box holds the cage's number of atoms with its number of hexagons beneath, and a box is drawn solid with a dot under it when the cage has a symmetry that reverses orientation and fixes nothing — a centre, which is what lets the cage halve onto the projective plane. The five-family has one at even numbers of rings and the six-family at odd ones, so their hexagon counts are 0, 10, 20, 30 … and 8, 20, 32, 44 … — two arithmetic progressions rather than two rows.

A centre at every other ring

A census cannot settle an infinite row, and the construction proposed to settle it was a tube capped at both ends, lengthened a ring at a time. Carried out, it alternates: a centre appears at every other ring and never between, the two families it permits reach two arithmetic progressions rather than a row, and the first of them opens with exactly the cage the census found could not halve.

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 The fewest pentagon contacts, from twenty atoms to sixty. The least number of pairs of pentagons sharing a bond, over every cage of pentagons and hexagons at each size from 20 to 60 atoms, with two lower bounds: 30 − 3h from counting edges, and 20 − h from counting the pentagons round each hexagon, which is also the prediction of one fewer contact for each hexagon added. The measured values past 44 atoms are 8, 8, 7, 5, 5, 4, 4, 3, 0; the bound is met at 20, 24, 26, 28, 36, 38, 40, 42, 44, 50, 60 atoms and nowhere else.

The last three contacts go at once

The fewest pentagon contacts a cage can have fell by one for every hexagon added over eight sizes up to forty-four atoms, which pointed straight at the truncated icosahedron at sixty. Carried to sixty by a faster search, the fall does not continue evenly. It pauses, jumps, pauses again, and loses its last three contacts in a single step. The steady fall was not a trend. It was a lower bound, twenty minus the number of hexagons, which a short count round each hexagon proves, and the census now shows where cages meet it and where they cannot.

restriction · Curvature Pentagon contacts as pieces of the icosahedron. The icosahedron drawn with one vertex at the centre, two rings of five, and the twelfth vertex as the outer circle; each vertex is a pentagon and each edge a possible contact. Left: the dodecahedron, whose twelve pentagons all touch five others, uses every edge. Centre: the commonest contact graph among the 5,770 cages to sixty atoms, 6 contacts in clusters of 3 + 3 + 2 + 2 + 1 + 1, carried by 179 cages, placed on the icosahedron with every contact on an edge. Right: the one cage with no contacts at all. All but two of the 340 contact graphs can be placed this way.

Three hundred and forty ways for twelve pentagons to touch

Every cage of pentagons and hexagons has twelve pentagons, and which of them share a bond is a graph on twelve points. Over all 5,770 cages from twenty atoms to sixty there are only 340 such graphs, and the graph almost never tells one cage from another: the commonest is shared by 179 cages and one recurs at thirteen different sizes. And every one of them except two, both from small cages, fits inside a single graph, the icosahedron's, which is what the dodecahedron's twelve pentagons make when every one touches five others.

restriction · Curvature

Named alongside it

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

Combinatorial curvatureEnumerationFullereneCountingIcosahedral symmetryCensusClosed surfaceThe Euler characteristicGraph isomorphismPolyhedronExhaustive searchFace vector

All concepts