Crystallography & Symmetry
patterns derived from their groups
Generator

hexagonal: 0.5 and 0.577

hexagonal: 0.5 and 0.577
hexagonal: 0.5 and 0.577. The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid.

The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid.

4 essays call two-radii. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

hexagonal: 0.5 and 0.577. The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid. Lattices

Covering and packing want different lattices

A lattice has two natural radii — the largest spheres on its points that do not overlap, and the smallest that leave no gap — and both are radii of the same Voronoi cell. In the plane one lattice is best at both. In space the best packer and the best coverer are different lattices, and they are duals of one another.

 incommensurate: "dense on a line". The shortest non-zero vector a subgroup contains, as the search widens, against the square lattice drawn flat behind it as a control. For a lattice the answer is constant: the shortest vector is the shortest vector, and looking further finds nothing nearer. For a subgroup that is not a lattice it falls without limit, because the convergents of a continued fraction give integers making the combination arbitrarily small. This one falls from 0.414 to 1.2e-2 over bounds 1 to 64, which is the verdict "dense on a line" arrived at by measurement rather than by reading a definition. Nothing here is decided by asking whether a ratio is rational; the ratio is a float and the question would be undecidable of one. Lattices

Discrete, or dense, and nothing between

Every count in this collection rests on a hypothesis nobody states, because it is built into the word lattice: the translations of a pattern form a discrete subgroup of the plane. Drop it and the counts do not become larger — they stop existing, because the object stops being a lattice. A subgroup of the plane is one of five things, and only two of them are lattices.

 (17, 5) and (23, 7) reduced in 3 steps. Lagrange's reduction, run on the basis (17, 5), (23, 7). Each step subtracts a whole multiple of the shorter vector from the longer and swaps them; after 3 steps neither can be shortened by the other and the pair is reduced. The faint arrows are the intermediate bases and the solid pair is the answer, of length 1.41. The procedure always terminates and always finds the shortest vector, and in the plane that is a theorem rather than a hope. Lattices

The shortest vector, and where it stops being easy

Two moves find the shortest vector of a plane lattice, and they always terminate. Nothing on this site has ever needed more, because every lattice here has two or three dimensions. In general the same question is NP-hard, the best polynomial procedure returns an answer that may be exponentially too long, and an entire branch of cryptography is built on the gap.

 Perfection is a rank, and most lattices do not reach it. For each lattice, the rank of the matrices vvᵀ built from its shortest vectors, against the dimension of the space of symmetric matrices those live in. Reaching it means the shortest vectors pin the form down completely: no deformation keeps every one of them at its length. Falling short means there is a direction left to move in, and the lattice is not a local maximum of density. Lattices

One perfect form in space

Which lattice packs spheres most densely is a question about a maximum over a continuum, and Voronoi turned it into a rank calculation and a sign check. A lattice is a local maximum exactly when its shortest vectors pin its shape down completely and its inverse can be written over them with positive coefficients. Searching every reduced integer form of minimum two finds one such lattice in the plane and one in space.

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