Crystallography & Symmetry
patterns derived from their groups
Series

Similarity — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two lattices carried onto part of themselves by a turn and a stretch. Left: the square lattice, with the points of its similar sublattice of index two enlarged and one cell of it outlined; the sublattice is the parent turned by 45° and stretched by √2. Right: the hexagonal lattice and its similar sublattice of index three, the parent turned by 30° and stretched by √3. The shaded cell in each is the parent's own. These two turns, an eighth and a twelfth of a full turn, are the only fractions of a turn a plane similarity adds to the five a plane lattice already has.

    A stretch buys an eighth and a twelfth of a turn

    A lattice can be turned onto itself only by a half, a third, a quarter or a sixth of a turn. Allow the turn to come with a stretch, so that the lattice lands on part of itself, and the plane gains exactly two more: an eighth, with a stretch of √2, and a twelfth, with √3. Every lattice with a rational metric gains a quarter turn as well. Space gives all of it back: no similarity of a lattice in three dimensions turns by an eighth or a twelfth.

    part 1 · restriction
  2. Two similarities in four dimensions that turn by a sixteenth and a twenty-fourth. The eigenvalues of the companion matrices of x⁴ − 2x² + 2 and x⁴ − 3x² + 3. Each has all four on one circle, of radius ⁴√2 and ⁴√3, so each matrix is a similarity of a lattice in four dimensions; the small ticks mark sixteenths and twenty-fourths of a turn, and the eigenvalues sit at 22.5° and 15° and their mirror images. Neither order is an order any rotation of a lattice can have in four dimensions.

    Four dimensions adds a sixteenth and a twenty-fourth

    A similarity of a lattice stretches it by r, so its determinant is rⁿ and the n-th power of its turn is a rational rotation. That gives a ceiling on the orders a stretch can buy in any dimension. In four dimensions the ceiling admits fifteen orders and eleven occur: the nine a rotation already has, plus sixteen and twenty-four, from x⁴ − 2x² + 2 and x⁴ − 3x² + 3. The other four are kept out by the field the turn's fourth power generates. And the plane's eighth and twelfth have relatives in every even dimension, so a stretch lets the order of a turn grow in step with the dimension.

    part 2 · restriction

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