The figure library
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 — and both of them make the round trip, in which the pattern is grown from the group, the group is forgotten, and the detector has to find its way back to exactly the operations it started from.
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 163 of them, and between them 393 essays call them 799 times.
The count beside each name is how many essays instantiate it, because that number is the one
worth watching. A generator written for a single essay and never called again is a design
failure rather than a neutral cost — reusecheck fails the build on one, and on a
library-wide ratio below four — and the whole economy here is that a picture drawn for one
argument turns out to be the picture for six. It is also the blast radius: changing a
generator changes every figure on its page at once, which is what has to be rebuilt and
looked at before the change is believed.
Operations
A single motion, its element located from its own matrix, and the orbit that grows out of applying it.
-
compose1 -
operation4 -
orbit-build5 -
orbit-count3 -
product4
The seventeen
Pattern plates — one cell of a group, generated from a motif and handed back to the detector.
-
motif-matters10 -
seventeen11 -
wallpaper24 -
window-size1
Lattices in the plane
The five plane lattices at their own metrics, and the two ways of refusing a five-fold rotation.
-
lattice10 -
no-five-fold5
Friezes
The same classification on a strip, where sixteen candidates collapse to seven.
Order without repetition
Penrose tilings, inflation, and a window in which a periodic and an aperiodic patch look alike.
Diffraction
The reciprocal lattice, systematic absences, and the symmetry a diffraction pattern adds.
-
absences9 -
bijvoet2 -
disorder-scatter3 -
friedel6 -
reciprocal14 -
tenfold6
Fundamental domains
The piece of the plane the group repeats, the subgroups that subdivide it, and the special positions inside.
Centring and reduction
A centred cell against its primitive description, and the shortest basis of a lattice.
-
centring4 -
niggli2 -
reduction8 -
sublattice1
Other dimensions
The restriction as arithmetic, so that it can be asked in four dimensions and in three.
Cut and project
A five-fold plane pattern as a slice of a periodic lattice, and the matching rules that force it.
-
cut-project7 -
matching-rules2 -
phason2
What an experiment loses
Powder patterns and the phase problem — the two places information leaves the measurement.
-
phase-problem5 -
powder8 -
resolution6 -
wilson8
Colour symmetry
Index-two subgroups, drawn as a swap of two colours, and the count they come to.
The Fibonacci chain
One dimension, where substitution, projection and diffraction are all short enough to check by hand.
Space groups
Plans and axonometrics of a group in space, with every element re-located rather than translated.
-
origins2 -
plan-readback1 -
roundtrip-3d4 -
setting-map3 -
settings-six1 -
space-axo5 -
space-plan19 -
symmorphic-split2
Screws and glides
The operations that exist only in space, and the split of a translation into intrinsic and locative parts.
The fourteen
The Bravais lattices, enumerated — candidates, the ones destroyed, and the duplicates with their change of cell.
Extensions and absences
Assignments of translations to generators, and the extinction conditions they produce.
The thirty-two classes
Stereograms of every crystal class, the enumeration that produced them, and the Laue classes diffraction collapses them onto.
-
abstract-type1 -
class-map3 -
enumeration4 -
laue6 -
stereogram11 -
symbol-read2
What symmetry decides
Neumann's principle as a character sum — which components of a physical property a class permits, and how many are independent.
-
indicatrix3 -
neumann-sum4 -
permits4 -
property-table10 -
tensor-shape5
Crystal forms
The faces a class requires to appear together, as orbits — and whether they close.
-
form-census3 -
form-plate6 -
goniometer3 -
habit4
Twins
Two orientations of one structure sharing a lattice, and the cosets that list them.
Transitions and domains
What a crystal does when it loses symmetry: how many domains, in what orientations, and where the walls are.
Interfaces
Coincidence site lattices, the indices they come in, and the misfit where the exactness stops.
-
csl6 -
misfit4 -
sigma-series5
Sublattices and subgroups
The sublattices of a given index, how many there are, and the two ways a plane group has a subgroup.
Before the lattice
The finite groups of the plane and of the sphere, the cut the restriction makes in them, the symmetry a crystal has locally and not at all — and the bound that makes every count on this site finite.
Layers
Two-dimensional periodicity in three-dimensional space, and the classes that keep a direction.
-
layer-groups2 -
section2
Modulated structures
Satellites, the second integer they need, and the extra dimension in which the structure is periodic again.
The Patterson function
The map an experiment can always compute, and the sections symmetry stacks its vectors on.
-
harker5 -
patterson-map10 -
phase-circles1
Magnetic groups
Time reversal as an operation, the hundred and twenty-two groups it makes, and what each of them permits.
Close packing
Two stackings of one density, the groups that tell them apart, and the polytypes between them.
Normalisers
The operations that move a description of a pattern without moving the pattern, and the origins that change nothing.
The cell nobody chose
The Wigner–Seitz cell, which needs no basis and no convention, and its reciprocal twin the Brillouin zone.
-
parallelohedron3 -
voronoi6 -
zone-stack2
Orbifold arithmetic
Conway's magic theorem: what each feature of a folded-up pattern costs, and the ways of spending exactly two.
Two notations
The thirty-two classes named twice — directions in Hermann–Mauguin, a construction in Schoenflies — both derived.
Letters and relations
A group with the plane taken away: generators, relators, the cosets they count, and the invariants that survive.
-
abelian3 -
coset-table2 -
generating-set1 -
presentation4
Tilings by regular polygons
The angle equation, the parity argument that kills ten of its twenty-one answers, the eleven that survive, and their duals.
Subgroups and words
Subgroups of small index counted from permutation actions, and the normal form that decides when two words name the same element.
Lengths
How many lattice vectors have each length, as divisor sums, as powder multiplicities, and as the question of whether the lengths name the lattice.
Homometric structures
Different arrangements of atoms with identical interatomic vectors, found by exhaustive search and then built on purpose.
The space of lattices
Every plane lattice as one point of one region, the five kinds as its interior, edges and corners, and reduction as a walk into it.
Folding a pattern up
What is left when a pattern is folded along its own symmetries, and the groups whose fold has no marked points at all — two of the seventeen, thirteen of the two hundred and thirty.
Curvature and the twelve
The accounting that forbids a five-sided face in the plane and requires twelve of them on any closed cage, with the cages grown rather than tabulated.
-
closed-net11
The pinwheel
One triangle subdividing into five copies of itself, turned each time through an angle that never closes — and the near-identical subdivision whose angles do.
One shape, alone
Which single convex polygons tile the plane, grown by their own constructions and checked by sampling — with the two that must fail fed to the same machinery.
-
one-tile2
Packing a lumpy shape
Which symmetry operations put a bump against a hollow rather than against a bump, measured by pushing rows together, and the four plane groups that survive.
Limiting groups
The seven groups a uniform field can have, enumerated by closure, and what is left of a crystal's class once one is applied.
Rod groups
The symmetry of an object periodic in one direction: screws drawn as the orbits they are, and the seventy-five groups enumerated over translation parts.
Eight-fold
The same cut-and-project machinery run on √2 instead of the golden ratio: an octagonal window, an integer inflation matrix, and a tiling of squares and rhombs.
Mirrors, and closure
Every motion taken apart into the mirrors it is a product of — three in the plane and four in space — and the arrangements whose operations fail to close at all.
The seventeen as algebra
The classification reached with no picture in it: translations attached to a point group, the relabellings quotiented away, and a group written on strange axes put back.
What a local rule permits
Whether a set of tiles can cover the plane at all, how many ways a covering rule can be obeyed, and what an average over all of them looks like.
-
local-rule10
One hand
The groups a structure of a single enantiomer may sit in, and the changes of cell that turn a screw into its own mirror image.
-
one-hand8
Absences and phases
Which systematic absences a second scattering event can fill in, and which reflections have a phase symmetry restricts to two values.
-
detour2
Two radii
A lattice measured at both extremes — the largest spheres that do not overlap and the smallest that leave no gap — and the question of whether a set of points is a lattice at all.
One tile
What a shape's boundary decides about the whole plane: the six-arc cut that forces a tiling, the tilings that need two orbits of congruent tiles, and the rings a shape that tiles nothing will still accept.
Gaps, and two periods
The three lengths a rotation of the circle leaves between its own points, and two chains in one crystal whose periods share no multiple.
Averages and phases
What a shell of reflections knows about the content and not the arrangement, and what atomicity forces on the phases it cannot measure.
What a lattice permits
The distinct dislocations a lattice has, as orbits of its short vectors, and the orientations it will accept as twins with their index and their obliquity.
From outside the crystal
Zones and indices read off a crystal's own faces, and a cell recovered from a bag of unlabelled reflections.
The distances thrown away
A structure as a graph and nothing else — the finite description an infinite net is written as, the counting outwards that identifies it, and the placement it has when every vertex sits at the average of its neighbours.
-
net-placement5 -
net-quotient13 -
net-shells4
Bars and joints
The same nets read as frameworks of rigid bars: what a count of freedoms promises, what the rank of the rigidity matrix says instead, and the motions that are left.
-
frame-flex4 -
frame-rank6
A group as a graph
How fast a group grows when the plane is taken away, and the fundamental domain whose walls hand the generators back.
-
cayley3
A group acting on functions
Character tables constructed rather than quoted — the dimensions a class permits, the orthogonality that proves the list complete, and the experiment that separates a forced degeneracy from a coincidence.
What a group leaves alone
The polynomials a group does not move — how many of each degree, the six rings that are free and the four that carry a relation, and the degrees in which the crystallographic restriction is legible.
One tile
The eight-kite shape a search over polykites returns, the patches an exact cover builds from it, and the reflected copies it cannot do without.
One thing described twice
A basis reduced, a structure written down every way its normaliser permits, and the site symmetry a molecule has to match.