Crystallography & Symmetry
patterns derived from their groups
Concept

The Euler characteristic — where it appears

A number computed from a surface that no bending changes, and which for an orbifold takes fractional contributions from its marks. It is two for a sphere and zero for the plane, and that single difference is why a closed cage needs twelve pentagons and a flat net needs none.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

A fundamental domain for p6m. One representative from every orbit of p6m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

Orbifold notation, the shorter language

Fold a pattern up along its own symmetries and what remains is a small surface with marked points. Its shape is a complete name for the group, and reading the name off costs an arithmetic sum that has to come to two.

classification · Seventeen The seventeen signatures, and the seventeen groups. Every combination of features costing exactly two, beside the plane group each one names. The left column is produced by an accounting identity that has never heard of a lattice; the right by reading seventeen groups' own operations — their rotation centres and orders, which of those lie on mirrors, and how many closed curves the mirror lines make once equivalent lines are identified. The map between the two lists is a bijection, and the figure does not appear unless it is one — in both directions. A signature with no group and a group whose signature is not on the list are both refused, and so is the failure that actually happens: two groups deriving one signature, which costs exactly two and passes every check but injectivity.

Seventeen dollars

Conway's magic theorem prices the features a folded-up pattern can have — a handle costs two, a mirror boundary one, a cone point of order n almost one — and requires the total to come to exactly two. There are seventeen ways to pay, and the classification falls out of an accounting identity that never mentions a lattice.

classification · Seventeen The (2, 3, 7) group, in the Poincaré disk. A triangle with angles π/2, π/3 and π/7, reflected in its own three sides until depth 12: 380 triangles, alternating in handedness because every generator is a reflection. The sum 1/2 + 1/3 + 1/7 is less than one, so the triangle does not fit in the flat plane and the drawing is of the hyperbolic one, with the whole plane squeezed inside a disk. Every triangle has the same hyperbolic area; the ones near the edge look small because the model shrinks distances there, and the tiling stops at the edge of the drawing rather than at the edge of anything.

Past two, the list does not stop

Conway's accounting says a wallpaper group costs exactly two dollars, and there are seventeen ways to spend it. Spend less and the answer is a finite group. Spend more and the list is infinite — but the cheapest thing past two costs two and one eighty-fourth, and nothing at all lies in between.

classification · Seventeen Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear.

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

lattices · Wigner–Seitz cells Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs.

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

restriction · Finite groups 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 general quadrilateral tiles. A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested.

Which shapes tile by themselves

Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.

classification · Tilings 11 nets, and one accounting. Every plane net folds onto a torus when its own translations are divided out, and a torus has Euler characteristic zero — so the quotient's vertices, edges and faces satisfy n − e + f = 0 and the number of faces is not something to be counted off a drawing but e − n. Dividing through gives one over the mean face size plus one over the mean degree equal to a half, which is the same relation that forbids a plane tiling by pentagons, reached here with no geometry in it at all. It holds for every net in the table.

Every net folds onto a torus

Divide a plane net by its own translations and the quotient is a finite graph drawn on a doughnut. A doughnut has Euler characteristic zero, so the number of faces is not something to count — it is forced, and with it a relation between how many edges meet at a vertex and how many bound a face.

classification · Flat space A circuit that closes on the wrong point: (1, 0). A square lattice with one extra half-column, drawn as a graph: the rows above the core have one more site than the rows below, and the core is the site at the end of the extra column. The path is 4 steps east, 4 north, 4 west and 4 south — the same number out as back — and it ends one lattice vector from where it started. Every one of the 12 circuits in the survey that goes round the core fails by that vector, and all 10 that miss it close exactly.

The circuit that does not close

A defect in a crystal is usually introduced as a picture — an extra half-row of atoms, a wedge taken out. What makes a defect a crystallographic object rather than a drawing is a closure failure: walk a closed circuit through the lattice and come back to the wrong point, by an amount the lattice itself decides.

applied · Defects 3 whole-number solutions: (6, 3), (4, 4), (3, 6). Every pair of whole numbers from three to 12, with the mean face size across and the mean degree down. A square in the first colour is a pair satisfying one over p plus one over q equals a half exactly — the flat case, where a periodic net is possible — and there are 3 of them: 6 and 3, 4 and 4, 3 and 6. The lighter squares above and to the left have a sum greater than a half, which is a closed polyhedron rather than a plane tiling; the ones below and to the right have a sum less than a half and belong to a surface of negative curvature. The plane is the boundary between them and it is thin.

Three answers in whole numbers

One over the face size plus one over the degree equals a half. Ask for whole numbers and there are exactly three answers, which are the three nets everybody has drawn since childhood — and the pairs on either side of them are a closed polyhedron and a plane the plane has no room for.

classification · Tilings 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 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 Where the sphere and the projective plane have no net. The number of different closed nets with three bonds at every atom and faces that are pentagons and hexagons only. On the sphere, with twelve pentagons and k hexagons for k up to 12, every count has at least one net except k = 1. On the projective plane, with six pentagons and h hexagons, each count sits under the sphere count it lifts to, since every hexagon of a projective net becomes two on the sphere. The projective counts for h = 0 to 6 are 1, 0, 0, 1, 1, 3, 3, so the projective plane has no net at h = 1 or 2: two gaps where the sphere has one. Every sphere count was found by enumeration and agrees with the published one.

A gap the sphere does not have

A net of pentagons and hexagons on the projective plane must have six pentagons, and the count permits any number of hexagons. Not every number happens. Lifting each net to the sphere turns the question into one about which cages have a centre — and the answer leaves two gaps where the sphere has one.

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

Named alongside it

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

Combinatorial curvatureEnumerationCrystal netPolyhedronCountingCrystallographic restrictionDualityOrbifoldClosed surfaceCone pointExhaustive searchFree action

All concepts