No figure on this site is a drawing that was made once and saved. Each one is a function:
it takes parameters and returns SVG, so the same generator produces the p4 plate and the
p6m plate without either being redrawn.
That is the reason the collection can keep growing without the illustrations drifting apart.
A generator is written once, checked once, and every essay that calls it inherits the same
line weights, the same colour roles, and the same behaviour in dark mode. There are
28 of them so far.
bifurcation100 125 150 175 0.500 1 1.50 growth parameter G angle the model settles on (°) golden two-whorl filled: converged · hollow: still wandering one rule, one knob
cells-of-a-head171 bounded cells mean 5.87 sides
centre-sensitivity0 20 40 60 0 5 10 15 20 25 assumed centre, as a percentage of the spiral's own radius, away from the truth error in the fitted growth factor (%) the one free choice in the measurement and it is not free enough to matter
continued-fractionsgolden ratio [1; 1, 1, 1, 1, 1, 1, 1, 1, …] best approximations 2/1 3/2 5/3 √2 [1; 2, 2, 2, 2, 2, 2, 2, 2, …] best approximations 3/2 7/5 17/12 π [3; 7, 15, 1, 292, 1, 1, 1, 2, …] best approximations 22/7 333/106 355/113 partial quotients all ones is the extreme case
counted-spirals21 and 34 spirals counted, not assumed
counts-by-anglegolden 137.51° 55 and 89 Fibonacci Lucas 99.50° 47 and 76 Lucas 151.14° 50 and 81 neither 77.96° 37 and 60 neither counted from the points one sequence per branch
counts-change-with-radius0 25 50 75 40 60 80 fraction of the head's radius parastichy numbers found in a band there 1000 primordia at the golden angle 4 different pairs
fitted-exponent0 2 4 6 8 2 3 4 exponent k how badly r₀ᵏ = Σ rᵢᵏ fails at that k k = 3 127 junctions fitted k = 3.000
gaps-as-the-head-fills0 2.50 5 7.50 10 500 1e+3 2e+3 2e+3 number of primordia largest empty gap, in units of the mean spacing 137.508° (golden) 137.3° 135° = 3/8 of a turn gaps measured from the triangulation an asymptotic claim, not a contest
golden-against-nautilusgolden — 6.85× per turn nautilus — 3.2× per turn same construction, two factors a factor of 2.14 apart
growth-factors3× 4× 5× 6× 7× measured nautilus, low measured nautilus, typical measured nautilus, high golden spiral, φ⁴ shaded: the measured range the claim sits outside it
head-at-angledivergence 137.508° closest pair 1.60 × mean spacing
how-many-sides5 sides 134 20% 6 sides 458 69% 7 sides 73 11% 665 bounded cells mean 5.908 sides six is forced, not chosen
how-rational-is-it0 0.200 0.400 20 40 60 denominator q how nearly q × angle is a whole turn (0 = exactly) golden 137.508° 137.3° 135° = 3/8 distance to the nearest whole turn no dips means no rows
l-system491 symbols · 130 segments a description, not a mechanism
morphospace0 0.250 0.500 0.750 2 3 4 5 6 W — whorl expansion per turn D — distance of the opening from the axis darker: whorls in contact the curve is W·D = 1
murray-treebranch angle 30° · daughter ratio 1.00 63 junctions checked
nautilus-overlaygolden — 6.85× per turn nautilus — 3.2× per turn same construction, same start 2.14× apart in growth
packing-four-ways0 0.250 0.500 0.750 1 120 130 140 150 divergence angle (°) closest pair, as a fraction of the mean spacing (higher is better) 137.508° 400 points per angle three criteria, three winners
raup-shellW = 2.40 · D = 0.42 W·D = 1.01 — evolute
recover-the-anglethe first of the four used to build counts recovered 137.508° 55 · 89 137.520° 99.502° 47 · 76 99.500° 151.100° 31 · 81 151.105° 77.960° 37 · 60 77.960° worst error 0.012° counts in, angle out the recovery never sees the angle
settling120 130 140 150 0 25 50 75 100 step divergence angle produced at that step (°) golden angle growth 0.40 settled spread 0.00°
spiral-at-growthgrowth 3.20× per turn recovered 3.200×
the-rule0 5e+5 1e+6 2e+6 0 100 200 300 angle around the boundary (°) repulsion from what is there growth 0.40 · 14 elements in play the minimum is where the next one goes
three-regimesG = 0.30 139.2° — golden G = 0.62 143.0° — other G = 1.10 180.0° — whorled (half) one rule, three growth rates the angle is an output
where-the-claim-holds0 0.200 0.400 100 120 140 160 divergence angle (°) resistance to rational approximation (higher is more irrational) 1/√5 — the bound 137.508° — 0.438 1500 angles on a 0.05° grid, plus the golden angle exactly this claim is sharp
where-the-model-fails0 50 100 0.250 0.500 0.750 1 growth parameter G spread of the last 30 steps (°) — 0 means settled filled dark: converged the usable range is stated, not implied
why-three0 10 20 0.500 1 1.50 2 vessel radius cost per unit length least total cost pumping falls, upkeep rises the sum has one minimum