Phyllotaxis
the spirals are counted, not admired
Concept

Grain boundary — where it appears

A line in a crystal where two pieces of one lattice meet at different orientations, made of dislocations spaced to account for the angle. The word is often borrowed for chains of five- and seven-sided cells on a seed head, though no lattice rotates there and no circuit fails to close.

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

Every five is a cell away from a seven, and the loneliest is 0.927 spacings out. How far a five-sided cell is from the nearest seven-sided one, in cell spacings, on a golden, 137.508° head of 2400 organs. The measured bar runs from the closest five to the loneliest — 0.833 to 0.927, with a median of 0.919. It stops a single cell out, so there is no unpaired tail at all rather than a small one. The nulls are seeded permutations over 200 draws: relabelling which defects are fives puts the average five 1.254 ± 0.064 spacings away, and scattering the whole multiset over the interior cells puts it 1.786 ± 0.092. 100.0% of the fives share a wall with a seven against 68.2% for the strong null, z = 7.3.

Every five is bound to a seven

A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.

tissue · Topological charge A Burgers circuit round a dislocation in a crystal and round a five–seven pair on a golden head's ring. Left: a triangular lattice moved by the elastic field of one edge dislocation, its five- and seven-sided cells marked, with the tightest circuit round them, the 8 organs touching either. Rounded to whole lattice vectors the steps sum to -1, 0: the circuit fails by one lattice vector. Right: a five–seven pair on the ring at radius 29.51 of a 4,000-organ golden head and the 8 organs touching either of the two. Its steps sum to 0, 0, and the circuit closes.

A dislocation that closes

A defect ring on a seed head looks exactly like a low-angle grain boundary: a closed chain of bound five–seven pairs one cell thick. The defining test of a dislocation is a loop that fails to close. On an elastic crystal it fails by one lattice vector at every size. On a golden head's rings the tightest loop round a five–seven pair closes every time, 272 pairs out of 272, and one set of lattice labels holds every patch across the outer rings without a single broken edge. No lattice rotates at the ring, where a real boundary of that density would need thirty-two degrees. What a ring actually is sits in one number: its walls are within two over its own count of being replaced by the other diagonal of their cell, on one curve for all four rings.

tissue · Topological charge

Named alongside it

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

Bound pairDelaunayDislocationHonest limitsCell spacingControlDefect ringDisorderLattice vectorsNegative resultNull modelPermutation test

All concepts