Golden Phyllotaxis

Two thousand circles placed at the golden angle, which is the only angle that never lines up.

Two lines of arithmetic

double radius = Radius * 2 * Math.Sqrt(i + 0.5);
double angle = GoldenAngle * (i + 0.5);

That is the whole placement. The ith circle sits at radius proportional to √i and at angle i × θ, where

double GoldenAngle = Math.PI * (3 - Math.Sqrt(5));   // ≈ 137.507764°

This is Vogel’s model of phyllotaxis, published in 1979 as a description of how seeds sit in a sunflower head. It has exactly two ingredients, and both of them are doing something specific.

Step 01

The first hundred circles. There is no row structure and no radial structure, and it is worth noticing how unusual that is — almost any other constant angle would produce visible spokes within a dozen placements.

Why the square root

Radius grows as √i because area grows as radius squared. A disc of radius r has area πr², so to place circles at constant density the ith one must sit at radius proportional to √i. Any other exponent crowds the middle or the rim.

That is why the finished piece has no visible gradient of density: the packing is uniform by construction, and the only thing that changes from centre to edge is the alpha, which ramps from 150 to 255 across the sequence so the outer circles read slightly brighter than the inner ones.

Step 02

A sixth of the way out. Spiral arms have appeared — and nothing in the code draws an arm. They are an artefact of the eye connecting circles whose indices differ by a fixed amount.

Why the golden angle

π(3 − √5) radians is 360°/φ², about 137.5°. It is the angle whose ratio to a full turn is the most irrational number there is.

That matters because if the angle were a rational fraction of a turn — say exactly 1/7 — then every seventh circle would land on the same ray and the pattern would be seven spokes with gaps between them. Any rational angle gives spokes; any angle close to a simple rational gives near-spokes, which is worse, because the gaps appear at large radius where there is most room to notice them.

The golden angle is as far from every simple rational as an angle can be. Its continued fraction is all ones, which is the mathematical statement of “hardest to approximate”. So the circles never line up, at any radius, for any period.

Step 03

A third. The arm count has changed. Look at the visible arms near the centre and count them, then count the ones at the rim: both counts are Fibonacci numbers, and they are consecutive. As the radius grows the eye abandons one family of arms for the next Fibonacci number up, because that family is the one whose members are currently closest together.

That is the same reason sunflower heads have 34 and 55 spirals, and pinecones 8 and 13 — the arms are the convergents of the continued fraction of 1/φ², and nothing chose them.

Step 04

Two thirds. The transition between arm families is happening continuously across the radius; there is no point at which it occurs.

Step 05

Two thousand circles. Every one of them at the same radius step, the same angle increment, and the same drawn size.

One radius, two jobs

One detail worth stating plainly: Radius does two jobs. It is the drawn size of the mark, and it is also the step of the spiral — the lattice is set by Radius × 2, so at the placement radius the neighbouring circles are about a diameter apart.

The two jobs can be separated. The drawn size is scaled per variant (RadiusScaleInner to RadiusScaleOuter); the lattice is not. Sunflower tapers its discs from 102% to 72% of the lattice step toward the rim, so the seeds sit apart; Fibonacci shrinks them to 30% and links them, so each bead marks a lattice point.

Near the very centre the packing breaks down in every variant — the first few circles overlap heavily, because √i grows fastest there and the angle has not had enough turns to separate them. It is the only place in the piece where the packing fails, and it is what gives the middle its dense knot.

Six renderings

Rendering	Circles	Radius	Scale	Mark	Ink
Gilded	2000	6	1.45	stroked outline, 1.25px	`#E6C05F` on `#0d0b08`
Sunflower	2800	7	1.05	filled disc, tapering to 72%	`#8A5220` → `#F2C14E` on `#171008`
Currents	2600	6	1.25	110° arc along the flow, fading outward	`#A5DDE8` on `#0a1420`
Fibonacci	2200	6	1.40	30% beads, linked 34 indices apart	`#5CB881` on `#0b0e0a`
Blueprint	3600	4	1.60	fine outline, 0.9px	`#2C4A66` on `#F3EEE2`
Ember	2100	6	1.40	jittered outline, fading outward	`#F07A3A` on `#0c0503`

Gilded is the piece itself: one ink, one size, uniform density. The other five each change one structural decision and let the palette follow.

Sunflower fills the circles instead of outlining them, and ramps the ink from seed-brown at the heart to gold at the rim, like the real head the geometry is borrowed from. Currents draws only part of each circle — a 110° arc centred on the direction of travel — so the disc becomes a field of commas streaming along the spiral, bright at the eye and dissolving outward. Fibonacci draws the arms themselves: a small bead at every lattice point, and a segment from each point to the one 34 indices ahead — exactly the arm family the eye assembles near the middle of the disc. Blueprint is the same disc on paper instead of on black, finer and denser than the others. Ember perturbs every point by up to a few pixels from a seeded stream and lets the rim fade, so the exact lattice reads as a drift of sparks.

The disc’s natural radius is 2 × Radius × √CircleCount — about 537, 741, 612, 563, 480 and 550 units for the six renderings above — so the lattices differ in scale as well as in structure, and left alone they would float in the frame to very different degrees. Each rendering therefore carries a Scale: a uniform zoom about the centre, chosen so the outermost mark ends the same ~112–132 pixels from the frame edge in all six. The renderings differ in structure, not in how deliberately they fill the canvas.

The framed piece

The seventh gallery plate is the gilded disc hung as a finished work: mounted inside an ivory mat behind a thin dark frame on a dark gallery wall, a soft drop shadow lifting the frame off the wall. It is drawn by MuseumFrame, a presentation layer that knows nothing about phyllotaxis — it draws any piece directly into the mat opening, contain-fitted and clipped to it. The framed plate is a way of showing the piece, not a family of new renderings: there is exactly one, and the six variants are not copied into frames.

Determinism

Nothing is seeded because nothing is random — with one exception that is seeded precisely so that it can be random. Every circle’s position is a closed-form function of its index, computed in double and drawn in index order. Ember’s jitter is drawn from SeededRandom(7), a PCG32 stream that produces the same offsets on every run and every machine; the seed is a literal in the preset, not a clock reading. Two runs produce identical output.

Progression 05

Variant 06

Gallery 01