Concept

Golden angle — where it appears

The divergence of 137.508 degrees, one full turn divided by the square of the golden ratio.

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

Also named here as lattice, whorled — the same set of essays touches all of them, so they are one junction rather than several.

mean sides per cell(forced to six)mean squared departure from six(not forced)whorled, 144°5.9860.023golden, 137.508°5.9900.253rational, 137.5°5.9900.255Lucas, 99.502°6.0360.255137.0°5.9900.291Poisson5.9691.830six433–637 interior cells each, inside 86% of the radiussame cells, same cut, two statistics

The second moment is the measurement

The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. The mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

tissue · sixsides
a window on the head, 60% of its widthshare of all cell contactsby difference in placement index3431%5527%2117%8915%136%82%counted:34 and 55665 nodes · 1903 contacts · 5.72 per nodecoordinates in placement order, nothing else

The six are the spirals

Label every contact between two cells in a seed head with the difference between the two nodes' placement indices. The labels are the parastichy numbers — 34, 55, 21, 89 — and the six sides Euler forces turn out to be about two from one family, one and a half from the next, and one each from two more.

tissue · contact network
whorled, 144°from 2 and 3+2235golden, 137.508°from 8 and 13+8+13+21+3481321345589Lucas, 99.502°from 11 and 18+11+18+291118294776rational, 137.5°from 8 and 13+8+13+21+3481321345589137.0°from 8 and 13+8+8+21+21+21+218132129507192113contact families above 2% of all cell contactsfilled dots are the two that are not sums

Every family but two is a sum

A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.

lattices · counting

Named alongside it

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

LatticeMeasurementRational angleWhorledCell areaDelaunayDisorderEuler's formulaLattice offsetPackingParastichyRim effect

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