Crystallography & Symmetry
patterns derived from their groups
Concept

Unimodular — where it appears

An integer matrix of determinant plus or minus one, which is exactly a change of lattice basis that keeps the lattice. Counting the ones with bounded entries is how the cost of the phrase "the cell is a choice" gets a number.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Reducing a basis. An awkward basis and the reduced one Gauss's algorithm returns. Both describe the same lattice — the change of basis has determinant one — and the reduced pair is the shortest vector together with the shortest independent of it, checked against an exhaustive search.

Reduction, and the shortest basis

Every lattice has infinitely many bases and no arithmetic picks a preferred one — until a rule is imposed. Reduction is that rule, it terminates in a handful of steps, and it is what lets a database decide whether two reported crystals are the same crystal.

lattices · Lattice In the plane, the lengths do name the lattice. Every reduced binary form with coefficients up to 20 — 1750 lattices — with its theta series computed to 120 terms. No two of them agree. That is Schiemann's theorem for binary forms, which says the theta series determines the lattice in two dimensions and in three, confirmed here as far as the search reaches rather than proved. The closest pair is worth the space: two lattices whose shortest vectors both have squared length twenty agree for 38 terms — because neither has any vector before then — and part at the next one.

The lengths do not name the lattice

Seventeen hundred plane lattices, every one with a theta series shared with no other — the lengths determine the lattice, and an exhaustive search says so. In sixteen dimensions two different lattices have identical counts at every distance, and the example is sixty years old.

lattices · Lengths Averaging a metric over the group. The 3 pale ellipses are the unit circle carried by each element of a finite group of rational matrices — none of them a rotation, because the group has been skewed out of the orthogonal ones on purpose. Their average is the heavy ellipse, and it is invariant: MᵀAM = A for every element, exactly, in rational arithmetic. So a finite group of matrices is always a group of isometries of some inner product, and every question about how large such a group can be becomes a question about the symmetries of an ellipse. The space of invariant forms here is 1-dimensional, so up to scale the average is the only one.

The average that makes it finite

Two arguments every classification leans on are usually assumed rather than made: that a finite group of motions fixes a point, and that a finite group of integer matrices preserves a metric. They are the same trick — average over the group — and the trick fails exactly where it should.

restriction · Finiteness E₈ has the smallest sum at every width. For four eight-dimensional lattices of unit covolume — D₈, its dual D₈, the cubic lattice Z⁸ and two copies of D₄ — the ratio of their Gaussian sums, less the constant term, to E₈'s, at widths from 0.2 to 5, logarithmic. Every ratio exceeds one everywhere: E₈ wins at every width against every rival. The closest approach is at t = 1, where D₈ and its dual are equal, a factor of 1.31 above E₈. D₈ is the better of the pair at wide widths and D₈ at narrow ones, and neither comes close to E₈ at either end.

In eight dimensions one lattice wins at every width

In space no lattice has the smallest Gaussian sum at every width, because the densest lattice wins when the Gaussian is narrow and its dual wins when it is wide, and fcc is not bcc. The argument fails exactly when the densest lattice is its own dual. In eight dimensions it is: E₈, with 240 shortest vectors, beats D₈, its dual, the cubic lattice and a sum of two four-dimensional lattices at every width tried, by a factor of at least 1.31, and its sum is mirror-symmetric in the logarithm of the width.

lattices · Lengths

Named alongside it

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

Basis reductionLatticeTheta seriesClose packingConjugationDecidabilityDual latticeDualityEnumerationEquivalenceFinite groupFixed point

All concepts