Crystallography & Symmetry
patterns derived from their groups
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Assuming a 5-fold rotation

Assuming a 5-fold rotation
Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist.

The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist.

5 essays call no-five-fold. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself. Order without repetition

Penrose tilings

Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.

 Assuming a 5-fold rotation. The shortest lattice vector, its rotated copies, and the combination of them that is itself a lattice vector. Where that combination comes out shorter than the vector assumed shortest, the assumed rotation cannot exist. What a lattice forbids

Why five-fold is impossible

A second proof, geometric rather than algebraic: assume a five-fold centre, and out of it construct a lattice vector shorter than the shortest one there is.

 A 5-fold cluster in a crystal that has no 5-fold axis. A cluster of 10 points with an exact 5-fold axis at the centre of each cell, repeated by the lattice. Two measurements, on the same points. The cluster is carried onto itself by a turn of 72° to within 2e-16 of a cell — exact, as far as the arithmetic goes. The pattern is not: applying the same turn about a lattice point sends some atoms 1.19 of a cell from the nearest atom, which is most of the way across it. Both are true at once. The axis is a symmetry of the contents of one cell and not of the crystal, which is what non-crystallographic symmetry means and why a virus with a sixty-fold capsid can crystallise in an ordinary space group. What a lattice forbids

A fivefold axis in an ordinary crystal

A virus with sixty-fold symmetry crystallises in a space group that has none of it. The restriction forbids a fivefold axis to the lattice and says nothing about what sits inside one cell — so the axis is exact, the crystal genuinely lacks it, and both statements are measurable on the same set of atoms.

 Thirteen ways to hold a lattice. Every finite group of integer matrices in two dimensions, up to a change of integer basis: 13 of them. Ten different abstract groups appear, and three of the ten hold a lattice in two inequivalent ways — a mirror along an axis or along a diagonal, and the same for 2mm and for 3m. The enumeration is a search: every subgroup of the two maximal holohedries, merged by conjugacy under integer matrices of determinant ±1, with the answer checked for not depending on how wide the search was. What a lattice forbids

Thirteen ways to hold a lattice

The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.

 The invariant degrees exist for every n; the lattice permits five of them. The reflection group with an n-fold rotation has an invariant ring generated in degrees 2 and n, for every n whatever — the dimensions on the right are counted by pairing monomials in complex coordinates, which needs no matrix and therefore no lattice. Five of these groups can be written in integer matrices, and those five are named in the middle column; the rest cannot, because a lattice has no five-fold or seven-fold rotation. The crystallographic restriction is usually a statement about traces of matrices. Here it is the statement that only five of these invariant rings belong to a crystal, and the two arguments have nothing in common but their answer. What a lattice forbids

The degrees that name the restriction

The reflection group with an n-fold rotation has invariants of degrees 2 and n — for every n, with no lattice anywhere in the argument. Which of those groups a crystal may have is then the only question left, and its answer is the crystallographic restriction arriving from a direction nobody points it from.

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