Ladders
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.
The lattice underneath
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.
The seventeen
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.
What a symmetry actually is
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.
The reciprocal lattice
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.
The crystallographic restriction
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.
The motif must be a comma
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.
Centring, and why cm is not pm
A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.
Seven friezes
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.
The fundamental domain
The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.
The groups ornament actually uses
Seventeen exist and decoration does not use them evenly. Which are common is an empirical question that published surveys answer differently, and part of the reason is a hazard this site can measure exactly.
What Shechtman measured
A diffraction pattern with sharp spots and tenfold symmetry, in April 1982. Sharpness meant order and tenfold meant no lattice, and the two had been believed inseparable — so the interesting question is what the observation had to rule out before it could mean anything.