Hexagonal numbers

Hexagonal numbers sit in the Greek figurate tradition, counting dots packed into a hexagon grown outward from one fixed corner. They have an immediate relationship to the triangular numbers that is easy to state and slightly surprising: every hexagonal number is triangular. Specifically H(n) = T(2n−1), so the hexagonal numbers are exactly the triangular numbers at odd index — 1, 6, 15, 28, 45 are the 1st, 3rd, 5th, 7th and 9th triangular numbers. The converse fails, since most triangular numbers are not hexagonal.

The centred variant is the one that escapes mathematics. Centred hexagonal numbers — 1, 7, 19, 37, 61 — count a central dot surrounded by complete rings, and that is precisely how identical circles pack most densely in a plane. Each interior circle touches six others, which is why the arrangement appears wherever efficient planar packing matters: in honeycomb, in graphene, in the cross-section of a bundle of optical fibres, in the way cannonballs were once stacked.

The hexagon's efficiency has a theorem behind it. The honeycomb conjecture holds that a hexagonal grid is the least-perimeter way to divide a plane into regions of equal area, which makes it the cheapest arrangement in wax for a given storage volume. Pappus of Alexandria discussed the idea in the fourth century and credited the bees with geometric sense; a complete proof arrived only in 1999, from Thomas Hales.

Figurate numbers in general received their most quoted result from Fermat, who asserted in 1638 that every positive integer is the sum of at most n n-gonal numbers — three triangular, four square, five pentagonal, six hexagonal. Cauchy proved it in 1813.

Key properties

• H(n) = n(2n−1), giving 1, 6, 15, 28, 45, 66, 91, 120, 153, 190.
• Every hexagonal number is a triangular number: H(n) = T(2n−1), the triangular numbers at odd index.
• The converse is false — most triangular numbers, such as 3 and 10, are not hexagonal.
• Centred hexagonal numbers follow 3n(n−1) + 1, giving 1, 7, 19, 37, 61 — the counts in hexagonal close packing.
• The difference between consecutive centred hexagonal numbers is always a multiple of 6, since each new ring adds 6(n−1) dots.
• The sum of the first n centred hexagonal numbers is n³, so the cubes are their running totals.
• Cauchy’s polygonal number theorem: every positive integer is the sum of at most six hexagonal numbers.

Where they turn up

• Honeycomb, where the hexagonal grid is the least-perimeter division of a plane into equal areas — proved by Thomas Hales in 1999.
• Graphene and benzene, whose carbon atoms sit on a hexagonal lattice.
• Hexagonal close packing, one of the two densest ways to pack identical spheres, and the densest way to pack circles in a plane.
• Board games and strategy maps, which use hex grids because every neighbour is equidistant — unlike a square grid, where diagonals are further away.
• The basalt columns of the Giant’s Causeway, where contraction on cooling produced an approximately hexagonal fracture pattern.