The drawings
Every drawing on this site comes from one of twelve systems in three families: chaotic flows, self-similar figures and neural networks. A seed picks the system and every random choice in it, so the same seed always gives the same drawing. Here is each one, drawn from a single seed. A thirteenth, a spiral that escapes, lives on the page you get at a broken link.
Chaotic
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Lorenz system
dx/dt = σ(y − x)dy/dt = x(ρ − z) − ydz/dt = xy − βzσ = 10, ρ = 28, β = 8/3
Edward Lorenz’s 1963 model of convection in the atmosphere. Start it a hair differently and it soon takes a completely different path: the butterfly effect.
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Rössler system
dx/dt = −y − zdy/dt = x + aydz/dt = b + z(x − c)a = 0.2, b = 0.2, c = 5.7
Otto Rössler’s 1976 system, designed to behave like Lorenz’s but be easier to analyse: a spiral band that keeps folding back into itself.
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Aizawa attractor
dx/dt = (z − b)x − dydy/dt = dx + (z − b)ydz/dt = c + az − z³/3 − (x² + y²)(1 + ez) + fzx³a = 0.95, b = 0.7, c = 0.6, d = 3.5, e = 0.25, f = 0.1
A three-variable chaotic flow whose single trajectory traces a sphere-like shape with a tube running along its axis.
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Halvorsen attractor
dx/dt = −ax − 4y − 4z − y²dy/dt = −ay − 4z − 4x − z²dz/dt = −az − 4x − 4y − x²a = 1.4
A chaotic flow with cyclic symmetry: shift its three variables round (x to y, y to z, z to x) and the equations stay the same.
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Forced Duffing oscillator
dx/dt = vdv/dt = −δv + x − x³ + γ cos ωtδ = 0.3, γ = 0.5, ω = 1.2
A periodically pushed spring with two resting points, a form of the equation Georg Duffing studied in 1918. With the right push, its motion never settles.
Self-similar
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Heighway dragon
start: FF → F+GG → F−GF and G draw a step, + and − turn 90°rewritten 10 to 12 times
Fold a strip of paper in half again and again, then open every fold to a right angle: this is the shape. Found by NASA physicists John Heighway, Bruce Banks and William Harter in the 1960s.
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Sierpiński arrowhead
start: AA → B−A−BB → A+B+AA and B draw a step, + and − turn 60°rewritten 6 or 7 times
One unbroken line that traces out Wacław Sierpiński’s triangle.
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Gosper curve
start: AA → A−B−−B+A++AA+B−B → +A−BB−−B−A++A+BA and B draw a step, + and − turn 60°rewritten 4 or 5 times
Bill Gosper’s “flowsnake”: a single line that fills hexagon-like islands, each made of smaller copies of itself.
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Barnsley fern
each point (x, y) becomes, at random: 1% (0, 0.16y)85% (0.85x + 0.04y, −0.04x + 0.85y + 1.6) 7% (0.2x − 0.26y, 0.23x + 0.22y + 1.6) 7% (−0.15x + 0.28y, 0.26x + 0.24y + 0.44)
Michael Barnsley’s fern, made by applying four simple maps at random, one point at a time.
Neural
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Growing neuron
branch → grow, wander, fork, thin
A neuron grown from a few rules: branches extend, wander slightly, fork and thin out, and the short ticks are spines. Santiago Ramón y Cajal drew neurons like this in India ink, from tissue stained with Camillo Golgi’s method. Each seed grows a pyramidal neuron, a stellate neuron or a Purkinje cell.
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Chaotic network
dx/dt = −x + J·tanh(x), J ~ N(0, g²/N), N = 100, g = 2.2
A hundred model neurons, each connected to every other at random. Sompolinsky, Crisanti and Sommers showed in 1988 that once the connections are strong enough, the activity of such networks becomes chaotic. Drawn here: that activity, projected onto its three main directions of variation.
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Digit reader
64 → 16 → 10, tanh, softmax, Adam
A small network reading handwritten digits it never saw in training. The pen draws each digit, the hidden units it activates and the five that push hardest for the answer, then writes the answer and the network’s score for it; earlier answers fade. The digits come from a dataset Alpaydin and Kaynak published in 1998: handwriting collected on preprinted forms and reduced to 8×8 grids by counting the inked pixels in each block (UCI Machine Learning Repository, CC BY 4.0). On 450 held-out digits it gets 97% right, and the drawing shows its mistakes too.