Barnsley Fern

Four affine maps, a weighted coin, and a million points that land on a fern.

Four rules, a million throws

Michael Barnsley’s fern is defined by four affine maps of the plane and nothing else. Each takes a point and moves it:

(x, y) → (a·x + b·y + e,  c·x + d·y + f)

Map	a	b	c	d	f	p
Stem	0.00	0.00	0.00	0.16	0.00	0.01
Main frond	0.85	0.04	−0.04	0.85	1.60	0.85
Left frond	0.20	−0.26	0.23	0.22	1.60	0.07
Right frond	−0.15	0.28	0.26	0.24	0.44	0.07

Start anywhere, pick a map at random with those probabilities, apply it, plot where you land, repeat. That procedure is the chaos game, and what it converges on is the set that the four maps together leave unchanged — the attractor. The fern is not drawn; it is the fixed point of the four rules.

Step 01

A hundred points. Read on its own this is noise, and the first few points are not even on the fern — the walk has to fall onto the attractor before it can stay there. It does so quickly, because every one of the four maps is a contraction.

Why it converges

Each map shrinks distances. The main frond scales by about 0.85, the side fronds by about a third, the stem flattens everything onto a line. Two points started anywhere at all are pulled together by every step, so the walk forgets where it began: within a dozen throws it is on the attractor to within a pixel, and it never leaves.

That is also why the picture does not shift between the stages below. Each stage is the same random walk, stopped earlier. Nothing moves; density accumulates.

Step 02

Two thousand points, and the shape is settled. The stem, the spine, the sweep of the frond — all of it decided by four rows of six numbers.

The probabilities are areas, not importance

The weights are not aesthetic choices. They are proportional to how much of the attractor each map is responsible for covering, which is roughly the determinant of its linear part:

Map	Determinant	Probability
Stem	0.00	0.01
Main frond	0.7241	0.85
Left frond	0.0642	0.07
Right frond	−0.1088	0.07

Weighted this way, points arrive at an even density across the whole shape. Weight them evenly instead and the fern still appears — the attractor does not depend on the probabilities at all — but the stem, which needs one throw in a hundred to be drawn at all, would be rendered a hundred times too heavily and the tips would be starved.

The stem’s determinant is exactly zero: a, b and c are all zero, so that map collapses the whole plane onto a vertical segment. It has no area to cover, and one per cent is simply how often the fern needs a stem.

Step 03

Thirty thousand. The leaflets separate, and each one is a copy of the entire fern — the main frond map applied once more, at 0.85 scale and four degrees of rotation.

Step 04

Two hundred thousand. From here on, extra points are tone rather than shape.

Where it stops

Step 05

A million throws. The limit is arbitrary: the attractor is infinite and the walk would keep landing on it forever. What a million buys is the tips of the leaflets, which are the parts reached only through long chains of the side maps and therefore the last to fill in.

Points are drawn as unantialiased PointSize squares with no blending, so a pixel struck twice looks exactly like a pixel struck once. Tone comes from coverage rather than from accumulation — which is why raising the iteration count brightens the sparse regions and does nothing at all to the dense ones.

Seven renderings

The variants differ along three deliberate axes. Four of them converge to structurally different attractors; the rest change how the canonical attractor is framed, stopped, and inked rather than recolouring the same picture.

Rendering	Structure	Iterations	Point size	Ink
Emerald	canonical — slight bilateral curve	1,000,000	1.5px	`#2BD57D` on `#04120A`
Golden	swept — trunk arcs strongly rightward	1,000,000	1.5px	`#FFD700` on `#1A1200`
Rose	leaning — the whole plant bends left	1,000,000	1.5px	`#F2A0B5` on `#160A10`
Violet	wide-leaflet — broader frond fans	1,500,000	1.0px	`#CC88FF` on `#0A0010`
Crown	canonical — the whole upper canopy	1,500,000	1.0px	`#5ADBC8` on `#03151A`
Ember	canonical — landscape crop of the widest leaflets	120,000	2.4px	`#FF8C42` on `#0F0A05`
Paper	canonical, ink on a light ground	1,000,000	1.4px	`#1D2B38` on `#F4EFE4`

The golden variant doubles the main-frond rotation coefficients (b, c: 0.04 → 0.08), bending the trunk into a visible arc. The rose variant adds a horizontal drift to the frond maps (e: 0 → −0.10), sweeping the entire plant left. The violet variant scales the side-frond linear parts up ~50%, fanning the leaflets further from the axis. All four remain recognisably Barnsley ferns — the same four-map IFS structure, different geometry.

Crown crops the canonical attractor to its upper canopy on a square canvas — the whole canopy, frond tip included, since the tip sits far right of the trunk. Ember crops the landscape canvas to the heart of the fern, where the leaflets are widest, and stops the walk after 120,000 throws and draws larger points, so the grain of the chaos game stays visible as individual marks. Paper inverts the ground, printing slate ink on warm paper. Every variant carries its own seed, so no two share the same walk.

Crops end in a clean line rather than a band of stray marks: a preset whose window cuts into the attractor skips the points outside it, so nothing leaks into the canvas edge where the crop stops.

The museum treatment

One plate is not a rendering but a presentation: the emerald fern hung as a finished piece — on a gallery wall, behind a mat, in a dark frame, with an engraved plaque beneath it. The wall, the mat, the frame, and the plaque are drawn by a presentation layer that wraps the lead preset without touching it; the fern inside is byte for byte the emerald render. The plaque’s engraving is cut in the piece’s own mark — unantialiased squares from a five-by-seven pixel font — so it borrows no font from whichever machine renders it.

It is one plate, not a family: the museum treatment frames the lead preset only, because a framed copy of every variant would be a wall of duplicates rather than a presentation.

Determinism

Each preset’s walk is seeded from its own Seed, so the same points land in the same places on every run. The generator is System.Random, which the project’s determinism policy flags as not guaranteed stable across .NET versions; the piece is reproducible on a given runtime but is not one of the pieces that can be pinned by hash across all of them.

Progression 05

Variant 08