Phyllotaxis
the spirals are counted, not admired
Concept

Parastichy transition — where it appears

The place on a stem or head where the counted pair changes, as one family of spirals gives way to the next. It falls at a stated rise for each pair, a factor of the golden ratio squared apart on a noble angle, and the rise is what fixes where it is seen.

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

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.

The counts change with radius

The same head gives 13 and 21 near the centre, 21 and 34 further out, 34 and 55 beyond that, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.

lattices · Transitions A golden head of 1200 organs whose radius grows as the organ's index to the power 0.5, with the radii where its counted pair changes. The same divergence at every setting: only the exponent of the radius changes. With organ i at i to the power 0.5, the pair 21/34 arrives at organ 65, radius 8.1; the pair 34/55 arrives at organ 187, radius 13.7; the pair 55/89 arrives at organ 517, radius 22.7. A head that crowds its rim has its transitions earlier in radius than one that spreads it.

A growth law moves the counts

A head's counted pair changes at radii a factor of φ apart, and that factor has only ever been read on heads whose radius is the square root of an organ's index. Free the exponent and the arithmetic says two things: the same pair arrives at an organ in proportion to the exponent, and consecutive changes of pair are a factor of φ to the power twice the exponent apart in radius. Measured on six exponents from 0.4 to 0.7, both hold, the second to about a hundredth of the exponent, and it needs no organ counted from the centre. It also survives displacement of a fifth of a spacing, where a reading from the cells has long since failed.

lattices · Transitions

Named alongside it

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

FibonacciRiseTransitionsAnnulusBranchContinued fractionContinued fraction convergentDivergence angleHonest limitsLattice offsetMeasurementParastichy pair

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