Crystallography & Symmetry
patterns derived from their groups
Ladder

Restriction — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. -2-10121-fold2.000integer — allowed2-fold-2.000integer — allowed3-fold-1.000integer — allowed4-fold0.000integer — allowed5-fold0.618not an integer6-fold1.000integer — allowed7-fold1.247not an integer8-fold1.414not an integer9-fold1.532not an integer10-fold1.618not an integer11-fold1.683not an integer12-fold1.732not an integerrotationtrace2 cos(2π/n) — a whole number only five timescomputed, not tabulated1, 2, 3, 4, 6

    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.

    rung 1 · restriction
  2. rotate both ways and add|shortest| × 0.618shorter than the shortestso no such lattice existsthe descent argument, drawn5-fold

    Why five-fold is impossible

    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.

    rung 2 · restriction
  3. 48 operations24 rotations, 24 improperorders 1, 2, 3, 43 four-fold axes4 three-fold axes6 two-fold axesno six-fold anywheresigned permutations of three axes, enumerated48

    The restriction in three dimensions

    Space is roomier than the plane in every other respect, so the natural expectation is that it permits more rotation orders. It permits exactly the same five, and seeing why is more interesting than the result.

    rung 3 · restriction
  4. 4 dimensionsorder 5integer entries625 shadow points 0 0 0 -1 1 0 0 -1 0 1 0 -1 0 0 1 -1the lattice's shadow on the plane it rotates5-fold

    Where five-fold becomes legal

    A five-fold rotation with whole-number entries exists — in four dimensions, as a four-by-four matrix that can be written down. The plane forbids it because the plane is too small, and knowing which dimension is large enough changes what a quasicrystal is.

    rung 4 · restriction

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