Chaos & Fractal Deep Explorer
v0.1 · prototype
A laboratory for emergent complexity

Simple equations.
Infinite universes.

Every system here is governed by rules you could write on a napkin. What unfolds when you let them run is a different matter entirely. Pick a universe and find out what hides inside the math.

Modules online
08 / 08
Render engine
WebGL · Canvas2D
Precision
FP64 / FP32-GPU
Backend
None · client only
A note on scope
Eight working modules. Mandelbrot uses perturbation theory at deep zoom, reaction-diffusion auto-falls back to half-float on systems without full-float render targets, and the Lyapunov map probes any (a, b) on hover. True arbitrary precision past 10¹⁴ and fractal music remain aspirational.
Marty Math by MartinGoetzinger.com
Free tools and simulations to help you explore the ideas behind the math, from chaos and fractals to the calculators that run the numbers on the decisions that matter.
center−0.5 + 0i
zoom×1.0
iter256
precisionfp32DEEP
drag pan · wheel zoom · click set Julia c
Module 01 · Complex dynamics

The Mandelbrot set

Every point c on the screen is tested by iterating z → z² + c starting from zero. Black means the orbit stays bounded forever. Color means it escaped — and how fast.

View mode
Palette
Max iterations256
Curiosities
What you're seeing The boundary of this shape has a fractal dimension of 2: it is so wrinkled it fills the plane. Around every magnification you will find ghost copies of the whole set, floating in their own galaxies of detail.
time seriesx(n+1) = r · x(1 − x)
bifurcation diagramr ∈ [2.4, 4.0]
drag to zoom · double-click reset
Module 02 · Discrete dynamics

The logistic map

One equation. One parameter. As you turn the dial, a peaceful population suddenly splits in two. Then four. Then eight. Then it shatters into chaos. This is the simplest gateway into the deep.

Growth rate r3.7000
Initial x₀0.4000
Quick stops
Lyapunov λ+0.000
Regime—
Period—
Feigenbaum's secret The ratio of successive bifurcation gaps converges to δ ≈ 4.6692. It shows up in dripping faucets, heart arrhythmias, and convecting fluids. Different physics, same number. That shouldn't happen, and yet.
σ10.00
ρ28.00
β2.67
t0.00
drag orbit · wheel zoom
Module 03 · Continuous dynamics

The Lorenz attractor

Edward Lorenz's 1963 weather model. Three coupled equations. The trajectory never repeats, never crosses itself, and never leaves a bounded region. The geometry of weather, simplified to its bones.

σ Prandtl10.0
ρ Rayleigh28.0
β geometry2.67
Particles1
Multiple particles start a hair's breadth apart. Watch them diverge.
Butterfly wings The two lobes are not periodic orbits. The trajectory wanders unpredictably between them, but remains forever trapped on a structure of fractal dimension ≈ 2.06 — neither a surface nor a volume.
|Δθ|0.000
energy0.00
Module 04 · Sensitivity demo

The double pendulum

Two rods. Two masses. Newton's laws, nothing more. For a while the twin pendulums look identical — then they don't. This is the most honest picture of chaos you can build at home.

Mass m₁1.0
Mass m₂1.0
Length L₁1.0
Length L₂1.0
Gravity g9.81
Initial separation1e−6
The Lyapunov story For tiny separations Δθ₀, the gap grows roughly as Δθ(t) ≈ Δθ₀ · e^(λt). A positive λ is the mathematical fingerprint of chaos: any initial-condition error eventually dominates everything you thought you knew.
gen0
alive0
click toggle cell · drag paint
Module 05 · Emergence

Conway's Game of Life

Four rules, applied to a grid of cells. From them: gliders that travel, oscillators that beat, guns that fire, and patterns we still cannot predict. Computation, born from arithmetic.

Speed10/s
Seed pattern
A small theorem The Game of Life is Turing-complete. Inside this grid you can build adders, memory, clocks, and ultimately a computer running this very page. From four rules. From nothing.
Module 06 · Comparative chaos

Strange attractor gallery

Different equations. Same answer: structure without repetition, bounded without periodicity. Click any one to drive it. Each is computing live; each trajectory is genuinely chaotic.

Focused attractorLorenz
Trail length3000
Equation
A taxonomy of chaos Each of these is a flow in 3D phase space that satisfies the same conditions: bounded, non-periodic, and exponentially sensitive. The geometry differs wildly. Aizawa looks spherical, Halvorsen tetrahedral, Rössler a folded band. They are all attractors in the same family.
F feed0.0367
k kill0.0649
step0
click / drag to seed · grid 512²
Module 07 · Continuous PDE

Reaction-diffusion

Two chemicals on a lattice. One feeds the other. Both diffuse. From this alone you get the spots on a leopard, the stripes on a tiger, and patterns Turing predicted in 1952 before they were ever measured in a beaker.

Preset
F · feed rate0.0367
k · kill rate0.0649
Palette
The Turing instability Diffusion usually smooths things out. The miracle of this system is that two diffusing chemicals, coupled by a reaction, can be unstable to homogeneity. Identical starting conditions plus a hair of noise grow into permanent, structured pattern.
center(2.5, 3.5)
zoom×1.0
seqBBABA
λ here—
drag pan · wheel zoom · hover to probe λ
Module 08 · Parameter space

Lyapunov fractal

At each (a, b), we run the logistic map cycling between r = a and r = b through a fixed sequence, then measure how fast nearby trajectories diverge. Warm means stable. Cool means chaotic. The boundary is where order surrenders.

Sequence
Iterations500
Vantage points
λ at cursor+0.000
regime—
Markus & Hudson, 1989 The structures here are sometimes called swallows, shrimps, or the Zoo. They appear in pure mathematics, but the same patterns turn up in population biology, chemical oscillators, and laser dynamics. Same math, different physics.