the crystallographic restriction
Here is a curious prohibition: you can tile a floor with a pattern that has 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry, but you can never make a genuinely repeating pattern with 5-fold symmetry, nor with 7-fold, 8-fold, or anything higher. Pentagonal symmetry, so common in flowers and starfish, simply cannot extend into a wallpaper that repeats forever. This is the crystallographic restriction, and it explains why honeycombs are hexagonal and crystals come in only certain shapes.
The reason ties rotation to repetition. A pattern that repeats by translation has a regular grid (lattice) of equivalent points. If the pattern also has an n-fold rotation, that rotation must map the lattice to itself. Working out the geometry — the rotation must carry one lattice point to another while preserving all the spacings — forces n to be 1, 2, 3, 4, or 6. The value n = 5 fails because a rotation by 360/5 = 72 degrees applied to the lattice would generate points closer together than the smallest lattice spacing, an impossibility; the same clash rules out every n above 6. So only the orders 1, 2, 3, 4, 6 survive — note the conspicuous absence of 5 and the gap after 6.
This single fact is the hidden reason there are exactly seventeen wallpaper groups and not more, and it governs real crystals: the rotational symmetries of any periodic crystal lattice are limited to these same orders. The honest and fascinating twist is that the restriction applies only to PERIODIC patterns. In 1982 Dan Shechtman discovered quasicrystals — real materials with genuine 5-fold symmetry in their diffraction — which is possible precisely because they are aperiodic, never exactly repeating. Penrose tilings are the geometric model of this: they display 5-fold symmetry while sidestepping the restriction by refusing to repeat.
See why 5-fold fails on a lattice. Suppose a periodic pattern had a 5-fold rotation. Take the shortest translation vector in the lattice and rotate it by 72 degrees about a lattice point; the rotated copy must also be a lattice vector. But combining the original and rotated vectors produces a new lattice vector shorter than the one you started with — contradicting 'shortest'. The contradiction proves no periodic pattern can have 5-fold symmetry.
Only 2-, 3-, 4-, and 6-fold rotations are compatible with a repeating lattice — 5-fold is forbidden.
The restriction bans 5-fold symmetry only in PERIODIC patterns. Aperiodic tilings (Penrose) and real quasicrystals genuinely show 5-fold symmetry — they evade the theorem by never exactly repeating, not by breaking it.