the crystallographic restriction theorem

Try to tile a bathroom floor with regular pentagons and you cannot, gaps or overlaps always appear. The same stubborn fact governs crystals: a pattern that repeats perfectly in every direction (a lattice) can only carry rotation axes of order 1, 2, 3, 4, or 6. Five-fold, seven-fold, eight-fold and up are simply impossible for a periodic crystal. This is the crystallographic restriction theorem.

Here is the short proof, in plain arithmetic. Take a row of lattice points spaced a apart. If the lattice has an n-fold axis, rotating a neighbouring point by the angle alpha = 360/n, and also by -alpha, must land on other lattice points, and those must lie on the same row at some whole-number multiple m of the spacing. The geometry forces 2 cos(alpha) = m, an integer. Since cosine lies between -1 and +1, m can only be -2, -1, 0, 1, 2. Solving 2 cos(alpha) = m gives alpha = 180, 120, 90, 60, or 360 degrees, that is n = 2, 3, 4, 6, 1. For 5-fold, alpha = 72 degrees and 2 cos(72) = 0.618, not a whole number, so it is ruled out.

This theorem is exactly why the count of Bravais lattices (14), point groups (32), and space groups (230) is finite and closed. It also explains the shock of 1984: Dan Shechtman found sharp 10-fold diffraction from an aluminium-manganese alloy, forbidden for a periodic crystal. The resolution was quasicrystals, which are perfectly ordered but not periodic, so the theorem simply does not apply to them. The theorem constrains periodic crystals, not all ordered matter.