Illustrated essays on symmetry in the plane and in crystals: the seventeen wallpaper groups, the rotations a lattice forbids, and the quasicrystals that appeared to break the rule. https://www.crystal-symmetry.com/ 2026-08-04T15:28:25.768Z https://www.crystal-symmetry.com/essays/what-a-symmetry-is/ 2026-08-04T15:28:25.768Z

Not a property of a shape but a motion that leaves it alone. Once symmetry is a verb rather than an adjective, everything else in the subject follows — including why there can only ever be seventeen wallpapers.

https://www.crystal-symmetry.com/essays/the-lattice-underneath/ 2026-08-04T15:28:25.768Z

Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.

https://www.crystal-symmetry.com/essays/the-crystallographic-restriction/ 2026-08-04T15:28:25.768Z

A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.

https://www.crystal-symmetry.com/essays/the-seventeen/ 2026-08-04T15:28:25.768Z

Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.

https://www.crystal-symmetry.com/essays/penrose-tilings/ 2026-08-04T15:28:25.768Z

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.

https://www.crystal-symmetry.com/essays/the-reciprocal-lattice/ 2026-08-04T15:28:25.768Z

Nobody has seen a space group. Crystals are read from where they scatter, and where they scatter is a second lattice in which long has become short and short has become long.

https://www.crystal-symmetry.com/essays/the-four-motions/ 2026-08-04T15:28:25.768Z

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

https://www.crystal-symmetry.com/essays/five-lattices-and-no-others/ 2026-08-04T15:28:25.768Z

A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long.

https://www.crystal-symmetry.com/essays/why-five-fold-is-impossible/ 2026-08-04T15:28:25.768Z

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.

https://www.crystal-symmetry.com/essays/reading-hermann-mauguin/ 2026-08-04T15:28:25.768Z

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.

https://www.crystal-symmetry.com/essays/systematic-absences/ 2026-08-04T15:28:25.768Z

The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.

https://www.crystal-symmetry.com/essays/inflation-and-the-golden-ratio/ 2026-08-04T15:28:25.768Z

A Penrose tiling is not laid out tile by tile. It is grown by cutting every tile into smaller ones and rescaling, and the growth matrix's dominant eigenvalue is the golden ratio.

https://www.crystal-symmetry.com/essays/why-it-is-a-group/ 2026-08-04T15:28:25.768Z

The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.

https://www.crystal-symmetry.com/essays/the-cell-is-a-choice/ 2026-08-04T15:28:25.768Z

Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.

https://www.crystal-symmetry.com/essays/p3m1-and-p31m/ 2026-08-04T15:28:25.768Z

Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence.

https://www.crystal-symmetry.com/essays/order-is-not-periodicity/ 2026-08-04T15:28:25.768Z

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

https://www.crystal-symmetry.com/essays/the-orbit-is-the-pattern/ 2026-08-04T15:28:25.768Z

A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.

https://www.crystal-symmetry.com/essays/the-motif-must-be-a-comma/ 2026-08-04T15:28:25.768Z

A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption.

https://www.crystal-symmetry.com/essays/seven-friezes/ 2026-08-04T15:28:25.768Z

The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.