Phyllotaxis
the spirals are counted, not admired
Concept

Permutation test — where it appears

Scoring a measured arrangement against a null built by shuffling labels among the same cells, so the counts are held and only the arrangement moves. It is the comparison behind two defect results here, and its designed weakness is that it already knows the head it is testing.

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

264 exceptions in a 2400-organ head, and eight circles. A golden, 137.508° head of 2400 organs, tessellated inside 86% of its radius. Every interior cell is drawn, and the 135 five-sided cells and 129 seven-sided ones are marked apart from the 1367 hexagons, and the pale circles are radii computed from the divergence angle through the lattice's third-shortest vector — nothing is fitted. Every defect sits within 0.64 of a cell spacing of one of those eight circles, and between them there is not one exception in hundreds of cells.

The defects lie on rings

The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.

tissue · Topological charge 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 The floor test on one section: whether the shortest spacings follow the spacing around them. The outer whorl of one section cut into 16 runs of equal count; each run drawn at its median spacing, relative to the section's median run, and at its shortest spacing divided by the median of the 41 spacings around it. Filled: a volume clock read through a limit of 0.003 of the outer radius, a floor of 0.315 in length, reading p = 1.87; rank correlation -0.74, one-sided p 0.001. Open: an unlimited shell whose law drifted from 3 to 1.57, reading p = 1.87; rank correlation 0.12, p 0.675. An animal's short days are short in proportion to the spacing around them, so its divided floors do not move with the run's spacing; a limit's floor is one length, so where the spacing is wider the divided floor is lower. Both from an animal whose days vary by a tenth.

A limit is a length, and a short day is a share

A shell section that cannot resolve its finest growth lines names a steep law as a shallower one, and so does a shell whose animal really did change its law. Matched to name the same law from the same whorls, the two agree in their counts, and the test that compares how irregular each whorl is goes blind on both across the same band of limits. What separates them is what a limit is. Every spacing it leaves is at least one length, wherever on the whorl it sits, while an animal's short days are short in proportion to the spacing around them. Asked whether the shortest spacings along the outer whorl follow the local spacing, forty sections of a drifting or stepped law are flagged at most twice, and a limited volume clock in the blind band thirty-three to forty times.

shells · Spiral

Named alongside it

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

Cell spacingDislocationHonest limitsNull modelTopological chargeAnnulusBound pairClaim testingControlDefect ringDelaunayDisorder

All concepts