Phyllotaxis
the spirals are counted, not admired
Generator

angles-give-the-pair

five rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.01 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.
The angles against the positions, rise by risefive rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.01 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.risefive stems, read from the angles alonethe position counter0.032refusedrefusedrefusedrefusedrefused3 and 50.0135/85/85/85/85/85 and 80.015/85/85/85/85/85 and 80.0058/138/138/138/138/138 and 130.008refusedrefused5/8refused5/85 and 8seeded at 137.3°, 900 nodes per stemfilled where the two instruments agree

Drawn at its defaults, in the pair, out of the angles. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 6 essays

the blast radius of changing it

The pattern itself

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

 The pattern itself

The angles name the branch

Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.

 The claims, measured

What the pair costs

The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.

 Stems and cones

The order carries the count

Take the divergence angles off a stem, throw away every coordinate, and autocorrelate what is left. The result is periodic at the smaller parastichy number — peaks at it and at every multiple of it. A list of angles, with no picture and no position in it, carries the spiral count.

 Stems and cones

The second comb

The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one.

 Stems and cones

A harmonic is a step taken twice

The spectrum contains the larger parastichy number, their sum, and echoes of the smaller one, and no ranking of peak heights separates them. What separates them is arithmetic: a harmonic is a multiple of the spacing and a family is not, and the two kinds sit in different residue classes.

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