Phyllotaxis
the spirals are counted, not admired
Concept

Bound pair — where it appears

A five-sided cell and a seven-sided cell of a tessellated head that share a wall, read as one defect rather than two. On a golden head every five outside the centre is bound to a seven across a wall of the larger spiral family it keeps, and the pairs run round each defect ring in the order of the Fibonacci word.

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 The five kinds of cell across the ring where the 89 spirals begin, with every contact labelled. One cell of each kind from the ring of a 2,400-organ golden head where the family of 89 enters and the family of 21 leaves, drawn with a spoke to each neighbour labelled by the difference of their placement indices. Inside the ring, a hexagon has two contacts each of 21, 34 and 55. A seven has those six and one 89. A traded hexagon has one 21, two 34s, two 55s and one 89. A five has two 34s, two 55s and one 89. Outside, a hexagon has two each of 34, 55 and 89. Read outward, the cells give up the 21s and take on the 89s, and the seven and the five are the two ways of doing it unevenly.

A defect ring is where a family of spirals begins

Label every wall between two cells of a seed head with the difference of their placement indices and a hexagon is two contacts from each of three spiral families. The cells that are not hexagons sit on rings, and read by their labels each ring is a handover: one family of spirals ends there and the next begins. On a golden head the ring where 89 enters holds 34 sevens, 21 hexagons that have traded one contact and 34 fives — 34 + 21 + 34 = 89 — because every chain of the entering family begins at one of its cells and every chain of the leaving family ends at a traded hexagon. Joined by their walls, the fives and sevens make 21 clusters, and the clusters run round the ring in the order of the Fibonacci word.

tissue · Contact network

Named alongside it

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

DelaunayHonest limitsTopological chargeCell spacingClaim testingControlDefect ringDislocationDisorderEuler's formulaGrain boundaryLucas numbers

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