Concept

Similar sublattice — where it appears

A sublattice that is the whole lattice turned and stretched, so that a drawing of it alone would be a drawing of the parent. Which indices allow one is a question about the integers a quadratic form represents, and its turn is a fraction of a full turn only at 45° and 30° in the plane.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Sums of two squares, arriving as superstructures. Which indices admit a sublattice of the same shape as the square lattice, drawn as a bar per index whose height is how many there are. The pattern is not a pattern about lattices at all: an index works exactly when it is a sum of two squares, because a similar sublattice of the square lattice is multiplication by a Gaussian integer and its index is that integer's norm. The indices that work up to 30 are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, and the same list is produced here a second time by factorising rather than by searching, with the two required to agree.

The sublattices that are the same shape

Thinning a lattice usually changes its shape. Sometimes it does not: the sublattice is the parent rotated and scaled, and a drawing of it alone would be a drawing of the parent. Which indices allow it turns out to be a question Fermat answered in 1640.

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

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.

restriction · Similarity

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.

restriction · Similarity

Named alongside it

The objects these essays reach for when they reach for this one.

Companion matrix Crystallographic restriction Eigenvalue Eisenstein integer Gaussian integer Similarity Cyclotomic polynomials Index Multiplicative function Quadratic form Sublattice Superstructure

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