Fractal Fern

Four affine maps and a weighted coin, thrown a hundred thousand times at an eight-thousand-pixel canvas.

The chaos game

Start at the origin. Pick one of four affine maps with fixed probabilities, apply it, plot where you land, repeat. What the walk converges on is the set the four maps leave unchanged — the attractor — and nothing else about the fern is stored anywhere.

var rand = random.NextDouble();
if (rand <= 0.01)      { newX = 0;                        newY = 0.16f * y; }
else if (rand <= 0.7)  { newX = 0.76f * x + 0.04f * y;    newY = -0.04f * x + 0.85f * y + 1.6f; }
else if (rand <= 0.93) { newX = 0.2f  * x - 0.26f * y;    newY = 0.23f  * x + 0.22f * y + 1.6f; }
else                   { …right frond… }

Step 01

A hundred points, and some of the first few are not on the fern at all. The walk has to fall onto the attractor before it can stay there, which it does within a dozen steps because every one of the four maps contracts distances.

Not quite Barnsley’s numbers

This is the classic Barnsley fern with one coefficient changed. Barnsley’s main frond scales by 0.85 in x; this one uses 0.76:

newX = 0.76f * x + 0.04f * y;
newY = -0.04f * x + 0.85f * y + 1.6f;

Contracting horizontally more than vertically makes the fern narrower than the canonical one — taller in proportion, with the leaflets held closer to the spine. That is why the canvas is 5000 × 8000 rather than square: the aspect ratio of the attractor changed, and the frame followed it.

The probabilities are unchanged at 1%, 69%, 23% and 7%. They are not aesthetic choices — they are roughly proportional to the area of the attractor each map is responsible for covering, which is what makes the point density even across the whole shape.

Step 02

Two thousand points and the shape is settled.

Step 03

Twenty thousand. Each leaflet is a copy of the entire fern — the main frond map applied once more, at 0.76 by 0.85 and a few degrees of rotation.

The stem has no area

The first map is degenerate:

newX = 0;   newY = 0.16f * y;

Its linear part has determinant zero: it collapses the whole plane onto a vertical segment. There is no area for it to cover, and one throw in a hundred is simply how often the fern needs a stem drawn at all.

Step 04

Half the points. From here the picture stops changing shape and starts changing tone.

Step 05

A hundred thousand. The last regions to fill are the leaflet tips, which are reached only through long chains of the two side maps and are therefore the rarest destinations in the walk.

Points are drawn as filled marks with no blending, so a pixel hit twice looks like a pixel hit once — tone comes from coverage, not accumulation. Raising the count brightens the sparse regions and does nothing to the dense ones.

One fern, seven renderings

The maps are identical in all seven renderings, so the plant is always the same fern. What changes is what each rendering does with it — how close it stands, how much ink it throws, what marks it throws, and how many passes:

Rendering	Points	Mark	Composition	Ink
Matrix Green	100,000	12 px squares	full frond, apex clear of the top edge, stem exiting the bottom	neon green on black
Crown	140,000	10 px squares	1.95× crop of the upper frond, bleeding every edge	teal-to-mint ramp
Halo	200,000	two passes: 26 px, then 10 px	full frond	dim green aura, bright frond
Drift	30,000	8 px discs	centred at 0.6×	lavender dotwork on violet-black
Sumi	85,000	14 px squares	full frond	ink on paper
Nocturne	150,000	8 px squares	full frond	ivory on ink-navy
Ember	70,000	15 px squares	full frond	amber on warm charcoal

Crown stands close enough that the leaflets stop reading as a plant and become a field of abstract forms; Halo draws the walk twice, a coarse dim under-pass whose big marks bloom past the silhouette and a fine bright pass over it; Drift sets the whole plant small and sparse, as round dots, with room to breathe around it. The other four keep the full-frond composition and differ in mark size, density and palette — deliberate, curated choices rather than recolors of one drawing.

The gallery closes with the piece as it would hang: the canonical fern on a warm wall, behind an off-white mat and a near-black frame that casts a soft shadow. That plate is a presentation of the lead, not an eighth rendering — the frame is a layer drawn around the artwork’s own pixels, and there is no framed copy of any other variant.

Determinism

Every walk is seeded from its preset’s Seed — or, for layered renderings, from each layer’s own — so the same points land in the same places every run. The generator is System.Random, which the project’s determinism policy flags as not guaranteed stable across .NET versions: reproducible on a given runtime, not pinnable by hash across all of them.

Progression 05

Variant 07

Gallery 01