Crystallography & Symmetry
patterns derived from their groups
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists the generators, each rendered at its defaults.

No figure on this site is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the p4 plate and the p6m plate without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 17 of them so far.

absences

What pg scattersThe diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen.115 present6 absentthe glide's signaturecomputed from the atom positions, not from the grouppg

compose

Doing one after anotherTwo symmetries of a pattern, and the one you land on by doing both. The third picture is not a new operation drawn to fit — it is the composition, and it was already in the group.first: mirrorthen: mirrorlands on: rotation of order 4closure is what makes it a group, not a listp4m

frieze

The seven frieze groupsEvery way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.p1hopp11gstepp1m1sidlep2spinning hopp2mgspinning sidlep11mjumpp2mmspinning jumpsolid line: a mirror · dashed: a glide · lens: a half-turn centreseven, and no eighth

inflation

InflationOne tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell.one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tilesthe substitution rule applied to a single tile

lattice

The five plane latticesEvery 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.obliqueno constraint on lengths or anglerectangularangle 90°, lengths freecentred rectangularequal lengths, angle freesquareequal lengths, angle 90°hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell

motif-matters

Why p4 cannot be drawn with dotsThe same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.a dot at (1/12, 1/12)8 symmetries per cella motif with no symmetry4 symmetries per cell57 of 121 dot positions give more symmetry than p4both generated with the 4 operations of p4the dot gains 4

no-five-fold

Assuming a 5-fold rotationThe 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.rotate both ways and add|shortest| × 0.618shorter than the shortestso no such lattice existsthe descent argument, drawn5-fold

operation

A rotationThe 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.turn about a point, and nothing moves at that pointrotation

orbit-build

Growing the p4 orbitOne motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.1 of 43 of 44 of 4the pattern is grown from the group, never drawnp4

order-not-repetition

Periodic and aperiodic orderA periodic pattern repeats: there is a translation that maps it exactly onto itself. An aperiodic one does not, and yet it is completely determined and has sharp diffraction — order and repetition are different properties, which is what quasicrystals forced the subject to separate.periodic — a translation maps it to itselfrotation orders limited to 1, 2, 3, 4, 6aperiodic — no translation doesfive-fold symmetry, and sharp diffractionthe restriction assumes periodicity on its first line

penrose

A Penrose tiling, 5 inflationsTwo 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.890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5

reciprocal

A lattice and its reciprocalThe 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.the crystal latticewhere it scatterslong in one is short in the otheraxis ratio 1.7

restriction

Which rotations a lattice will carryThe trace of an n-fold rotation is 2cos(2π/n), and in a lattice basis the matrix is integer so the trace must be a whole number. Only five values in the available range are whole, and each corresponds to exactly one rotation order.-2-10121-fold2.000integer — allowed2-fold-2.000integer — allowed3-fold-1.000integer — allowed4-fold0.000integer — allowed5-fold0.618not an integer6-fold1.000integer — allowed7-fold1.247not an integer8-fold1.414not an integer9-fold1.532not an integer10-fold1.618not an integer11-fold1.683not an integer12-fold1.732not an integerrotationtrace2 cos(2π/n) — a whole number only five timescomputed, not tabulated1, 2, 3, 4, 6

seventeen

The seventeen wallpaper groupsEvery 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.p1p2pmpgcmpmmpmgpggcmmp4p4mp4gp3p3m1p31mp6p6m17 groups, each generated and verified

tenfold

A diffraction pattern with tenfold symmetrySharp spots, arranged with a symmetry that no periodic crystal can have. When this was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered.181 reflections10-fold symmetryforbidden to any latticeinteger combinations of ten star vectorsaperiodic

wallpaper

The wallpaper group p4mA pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.p4msquare lattice · 8 operations per cellelements marked

why-seven

Why sevenThe ingredients available on a strip are a translation, two kinds of mirror, a half-turn and a glide. Combining them produces seven distinct groups, because several combinations generate operations they were not given and collapse onto each other.what is addedgivesnothing but translationp1a glidep11ga vertical mirrorp1m1a half-turnp2a horizontal mirrorp11mbrings a glide with itvertical mirror + half-turnp2mgbrings a glide with itboth mirrorsp2mmbrings a half-turn with itthe collapses are why the count is seven rather than larger7