The drawings

Every drawing on this site comes from one of twelve systems: six chaotic flows and six self-similar figures. A seed picks the system, its starting point and the view, 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.

• 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.

  Seed 100043 · Draw this on the home page

• 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.

  Seed 100026 · Draw this on the home page

• Thomas attractor

  dx/dt = sin y − bxdy/dt = sin z − bydz/dt = sin x − bzb = 0.208186

  René Thomas’s cyclically symmetric system: something like a particle wandering through a lattice of forces, slowed by a little friction.

  Seed 100007 · Draw this on the home page

• 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.

  Seed 100010 · Draw this on the home page

• 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.

  Seed 100003 · Draw this on the home page

• 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.

  Seed 100012 · Draw this on the home page

• 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.

  Seed 100025 · Draw this on the home page

• Koch snowflake

  start: F−−F−−FF → F+F−−F+FF draws a step, + and − turn 60°rewritten 4 or 5 times

  Helge von Koch’s 1904 curve, repeated on the three sides of a triangle: an infinitely long border around a finite area.

  Seed 100027 · Draw this on the home page

• 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.

  Seed 100053 · Draw this on the home page

• 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.

  Seed 100013 · Draw this on the home page

• Branching plant

  start: XX → F+[[X]−X]−F[−FX]+X or F−[[X]+X]+F[+FX]−X or F[+X][−X]FX, picked at randomF → FFF draws a step, + and − turn 25° ± 6°[ starts a branch, ] returns from itrewritten 5 times

  Aristid Lindenmayer introduced these rewriting rules in 1968 to describe how simple multicellular organisms grow; they were later used to model whole plants. Here each branch picks its rule and angle at random, so no two plants are alike.

  Seed 100019 · Draw this on the home page

• 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.

  Seed 100006 · Draw this on the home page