Crystallography & Symmetry
patterns derived from their groups
The subject, divided

Fields

The order these are listed in is the order the argument builds — what a symmetry operation is, then the lattices that decide which ones are possible, then the classification that follows, then what a lattice forbids, then the things that live outside the rule, then how any of it is known from an experiment, and last the step from the plane into space.

A field is the coarsest cut this collection makes: it says which part of the subject an essay belongs to, and there are 9 of them holding 393 essays. The order below is authored and is the only thing on this page that is — it is the order the argument builds, so a reader who starts at the top and reads down never meets a term before the field that defines it.

Everything else is read out of the essays themselves. The ladders on each card are that field's anchors — an anchor being one idea, and its ladder the distinct arguments that stand against it, from the essay that introduces it to the one that assumes all the others. A field's ladders are not declared anywhere: they are what its essays turned out to be about, counted at build time, which is why a card can say deepest chain 7 and be right the day after an eighth is written.

A field is not the only cut. Themes run the other way, across the fields, and ladders ignore both and sort by depth alone. The same 393 essays sit under all three.

A rotation. The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies.

Operations

Rotations, mirrors, glides and translations — what a symmetry is, how they compose, and why the composition is a group rather than a list.

41 essays · 7 ladders · deepest chain 9
The hexagonal lattice. Every periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry.

Lattices

The repeating scaffold underneath every crystal. Five in the plane, fourteen in space, and no others — a classification with a short proof.

43 essays · 7 ladders · deepest chain 11
The seventeen wallpaper groups. Every way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.

The classification

Exactly seventeen ways to fill the plane with a repeating pattern. Not a fact to memorise: a theorem, with a finite argument and no eighteenth case.

42 essays · 10 ladders · deepest chain 9
The five rotations a lattice will carry. One motif and every rotation a plane lattice permits: orders 1, 2, 3, 4, 6, and nothing else up to 12. Each panel turns the motif by its own operation as many times as the order allows, on the lattice that operation requires — oblique for the identity and the half turn, hexagonal for the third and the sixth of a turn, square for the quarter. The trace printed under each is the sum of the diagonal of the operation's matrix written in the lattice's own basis, and it is a whole number in every panel, which is the entire content of the crystallographic restriction. The list of orders is produced twice, once from that trace condition and once from the degree of a cyclotomic polynomial, and the figure refuses to draw if the two disagree.

What a lattice forbids

Only two-, three-, four- and six-fold rotations are compatible with periodicity. The proof takes one line and the exception took a Nobel Prize.

39 essays · 5 ladders · deepest chain 8
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 and quasicrystals — patterns with long-range order, five-fold symmetry, and no repeating cell at all.

40 essays · 6 ladders · deepest chain 8
A lattice and its reciprocal. The reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out.

How it is known

A crystal's structure is read from the pattern it scatters. The reciprocal lattice, systematic absences, and what an experiment can and cannot see.

42 essays · 9 ladders · deepest chain 9
P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

Into space

Seventeen becomes two hundred and thirty. Screw axes, glide planes, and the translations that no choice of origin can remove — the operations a flat surface has no room for.

41 essays · 7 ladders · deepest chain 8
The thirty-two crystal classes. Every crystallographic point group, as a stereogram. Each was found by enumerating the subgroups of m3̅m and of 6/mmm, and each diagram is the orbit of one general direction under the group, filled where the pole is in the upper hemisphere and open where it is in the lower — which is the only thing in the picture that tells a rotation from a rotoinversion.

What symmetry decides

Thirty-two classes, enumerated rather than listed — and then the one thing they are for outside crystallography's own bookkeeping: which components of a physical property a crystal is permitted to have before anybody measures it.

42 essays · 8 ladders · deepest chain 9
The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it.

Symmetry at work

Crystal forms, twins, domain walls and grain boundaries. Mineralogy, metallurgy and ferroelectrics each worked these out separately, and every one of them turns out to be an orbit or a coset of a group already built here.

63 essays · 10 ladders · deepest chain 10

Every essay · The themes · All ladders