The Golden Angle
Each dot is a seed, placed one at a time: turn 137.508 degrees, step a little further out, place the next. That one angle, the golden angle, packs the head of a sunflower evenly at every size, and the eye reads the result as spirals, 55 sweeping one way and 89 the other in the default picture. The counts are always Fibonacci numbers, the same series on which the years of Venus and the Earth sit, close to 8 to 13.
| Golden angle | 137.508° |
|---|---|
| Golden ratio, phi | 1.618 |
| Rim spirals at 1,200 seeds | 55 and 89 |
| Earth year in Venus years | 1.6255, near 13/8 |
What you are looking at
Every dot is a seed, and the seeds arrive in order from the middle outwards. Each one is turned 137.508 degrees past the one before and set a little further from the centre, so that every seed gets the same amount of room.
The coloured lines join each seed to its two nearest neighbours further out. Gold lines wind one way around the centre and violet lines the other. Count the spirals near the rim and you get 55 of one colour and 89 of the other. Look further in and the counts step down through 34, 21, 13 and 8. They are always two neighbouring Fibonacci numbers, which is why the pattern seems to change character as your eye moves from the edge to the middle.
Try the dial. Set 137.50 degrees exactly with 3,000 seeds and the outer seeds stop spiralling and line up in 144 straight spokes. At 137.3 degrees spirals of 42 and 97 appear, and neither is a Fibonacci number. Only the golden angle keeps the head evenly packed all the way out.
The math
The golden ratio is phi = (1 + square root of 5) / 2 = 1.6180. Cut a circle into two arcs whose lengths are in that ratio and the smaller arc is 360 / phi squared = 360 x (2 - phi) = 137.508 degrees, the golden angle.
Why that angle packs so well: phi is the number that fractions approximate worst, so no number of turns ever brings a seed back onto the ray of an earlier one. The best near misses are fractions made of Fibonacci numbers. The golden angle is 0.38197 of a turn, and 21/55, 34/89 and 55/144 of a turn are each a little closer to it. After 55 seeds the turning has almost come back to a whole number of turns, so seed 55 sits almost on the ray of seed 0, seed 56 near seed 1, and so on: that near miss is what draws a family of 55 spirals. The family you see at a given radius is the near miss that leaves the smallest gap there.
The same Fibonacci ladder turns up in the solar system. The Earth takes 365.256 days to go round the Sun and Venus 224.701, so the Earth's year lasts 1.6255 Venus years. That is very close to 13/8 = 1.625, a ratio of two neighbouring Fibonacci numbers, which is why the pattern Venus draws repeats after 8 Earth years and 13 Venus years. It is near phi too, but not on it: 1.6255 is 0.46 percent above 1.6180.
How it is computed
Seed k is placed at a distance from the centre proportional to the square root of k, at an angle of k times the turn. The square root gives every seed equal area. This is the model Helmut Vogel published in 1979 for the head of a sunflower.
The spiral lines are measured, not assumed. For every seed the piece checks the distances to the next 240 seeds, keeps the two nearest, and colours each link by which way it winds around the centre. The spiral counts in the caption are the most common link lengths, in steps, among seeds in a band near the rim. With the golden angle every link found is a Fibonacci number of steps, and the tests check that. Everything is computed in your browser when you change a setting.
Questions
Why is the golden angle 137.5 degrees?
It is the smaller of the two arcs you get by cutting a full circle in the golden ratio: 360 divided by phi squared, which is 137.508 degrees. The larger arc is 222.492 degrees, and 222.492 / 137.508 is phi again.
Why do the spiral counts come out as Fibonacci numbers?
Seeds that are a Fibonacci number of steps apart have turned through almost a whole number of turns, so they sit almost on the same ray. Those near misses become each seed's nearest neighbours, and joining every seed to the one a Fibonacci number of steps later draws that many spirals.
Why does 137.5 degrees make straight spokes?
Because 137.5 degrees is exactly 55/144 of a turn. After 144 seeds the turning has come round to exactly 55 full turns, so every seed lands on the same ray as the seed 144 places before it, and with enough seeds (1,500 or more) the outer ones line up in 144 spokes. The golden angle never repeats like that.
Is the link between Venus and the golden ratio real?
The numbers are real: the Earth's year is 1.6255 Venus years, very close to 13/8, a ratio of two neighbouring Fibonacci numbers. It is a near resonance between two orbits. It sits close to phi but not on it, and the numbers on their own say nothing about meaning. Astrologers have long read the Venus cycle in their own traditions.