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Mathematics

Fractals, cellular automata and complex-plane sets — mathematical objects of infinite complexity, generated by a few lines of code.

10+ simulations WebGL · GLSL · Canvas 2D Fractals · CA · Geometry

Category Simulations

Simulations in development — stay tuned

Fractal geometry — objects of self-similar structure that repeat at every level of scale. The Mandelbrot set, Barnsley fern and Sierpinski triangle are generated by extremely simple iterative rules.

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Ready★★★ Advanced
Fractal Explorer
Mandelbrot & Julia sets rendered with smooth colouring GLSL shaders. Pan, zoom, switch modes and morph Julia parameters in real time.
WebGL2 GLSL Fractal GPU
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New★☆☆ Easy
Sierpiński Triangle
Two methods: Chaos Game (random vertex jumps) and recursive subdivision. Hausdorff dimension ≈ 1.585. Pan, zoom, choose color scheme.
Canvas 2D Recursion Chaos Game
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New★★☆ Moderate
1D Cellular Automata — Wolfram
All 256 Wolfram rules — Rule 30 (chaos), Rule 90 (Sierpinski), Rule 110 (Turing-complete). Toggle bits to build custom rules live.
Canvas 2D Wolfram Cellular Automata
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New★★☆ Moderate
Number Spirals
Ulam spiral of prime numbers, Sacks spiral and Fibonacci sunflower. Discover hidden diagonal lines of primes with pan & zoom.
Canvas 2D Number Theory Primes Fibonacci
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New ★☆☆ Easy
Kaleidoscope
Draw in one sector and watch it mirror in N-fold symmetry (3/4/6/8/12). Rainbow mode, spin, colour picker, save as PNG.
Canvas 2D Symmetry Kids Art
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New ★★☆ Moderate
Pythagoras Tree
Recursive fractal tree built from the Pythagorean theorem. Adjust branch angle, lean, depth up to 14 levels. Animated growth.
Canvas 2D Fractal Recursion Kids
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New ★☆☆ Easy
Barnsley Fern
Infinite fern generated by four affine transformations via the Chaos Game. Switch between Classic Fern, Black Spleenwort, Modified Fern and Maple Leaf presets. Color by transform index.
IFS Fractal Chaos Game Nature
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New ★★★ Advanced
Voronoi Diagram
Interactive Voronoi tessellation with Lloyd's algorithm for centroidal relaxation. Euclidean, Manhattan or Chebyshev distance metrics.
Canvas 2D Voronoi Tessellation Geometry
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New ★★☆ Moderate
Random Walk
Brownian motion and random walk simulation. Gaussian, Lévy flight and lattice walk types. Watch the √t displacement law emerge.
Canvas 2D Stochastics Brownian Motion Diffusion
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★★☆ Moderate
Geodesic Domes
Subdivide icosahedra, octahedra and tetrahedra onto a sphere at frequencies 1v–5v. Explore Buckminster Fuller's structural geometry with Euler's formula.
Three.js Geodesic Buckminster Euler
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★★☆ Moderate
Spherical Projections
Compare Mercator, Mollweide, Lambert azimuthal, sinusoidal and equirectangular projections. Tissot's indicatrices reveal area and shape distortion everywhere.
Canvas 2D Cartography Tissot Mollweide
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New ★★☆ Moderate
Bifurcation Diagram
Logistic map bifurcation — explore the route to chaos. Zoom into the Feigenbaum constant and self-similar structure at every scale.
Chaos Logistic Map Canvas 2D
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New ★★☆ Moderate
Flow Fields
Thousands of particles follow a Perlin-noise or curl-noise vector field. Adjust field scale, particle count and decay for generative art compositions.
Perlin Noise Generative Art Canvas 2D
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New ★★★ Advanced
Wave Function Collapse
Procedural tile placement inspired by quantum superposition. Cells collapse from maximum entropy to a single tile based on neighbour constraints.
Procedural Constraint Canvas 2D
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New ★★☆ Moderate
Stippling
Weighted Voronoi stippling via Lloyd relaxation — convert any image into a dot-art representation with controllable density and point count.
Voronoi Generative Art Canvas 2D
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New ★★☆ Moderate
Taylor Series Visualizer
Watch how polynomial partial sums of a Taylor/Maclaurin series converge to sin, cos, exp, ln and more. Animate term-by-term addition and explore the radius of convergence.
Taylor Series Calculus Convergence
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New ★☆☆ Easy
Collatz Conjecture
Explore the 3n+1 hailstone sequences, plot stopping-time heatmaps and navigate the reverse Collatz tree — all around an unsolved problem in number theory.
3n+1 Number Theory Stopping Time
∫
New ★☆☆ Easy
Riemann Integral
Visualize left, right, midpoint, trapezoid and Simpson approximations converging to the exact definite integral. Adjust n up to 200 subdivisions.
Calculus Numerical Integration Simpson Rule
∞
New ★☆☆ Easy
Möbius Strip
Drag to rotate the 3D Möbius strip. Trace a path to prove it has only one side, adjust half-turns, and explore non-orientable topology.
Topology One-Sided Surface Non-Orientable
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★☆☆ Easy New
Matrix Transformations
Edit the 2×2 matrix to see it stretch, rotate, shear and reflect the grid. Live eigenvectors, unit circle ellipse, determinant, and 8 presets.
Linear Algebra Eigenvalue Determinant
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Ready★★☆ Moderate New
Bézier Curves
Drag control points to sculpt Bézier curves (degree 1–5) built with the De Casteljau algorithm. Animate construction and compare against a B-spline.
Canvas 2D De Casteljau Spline
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New ★★☆ Moderate
DFT & STFT Visualiser
Generate signals and watch their Fourier transforms live. Switch between DFT magnitude spectrum and STFT spectrogram to see how frequency content evolves over time.
Fourier Transform Spectrogram Signal Processing
Fractal Geometry & IFS Iterated functional systems. Hausdorff dimension. Smooth coloring in GLSL. → Article Wolfram Cellular Automata Complexity classes I–IV. Rule 110 and Turing completeness. Totalistic CA in 2D. → Article Numbers & Spirals: Number Theory in Pixels Sieve of Eratosthenes. Ulam spiral. Fibonacci sequence and the golden angle. →
All articles →

About Mathematics Simulations

Fourier transforms, fractals, prime spirals, and calculus — animated

Mathematics simulations make abstract concepts tangible by animating equations in real time. The Fourier series visualiser constructs any periodic wave from a sum of rotating circles, directly showing how frequency components add. The Mandelbrot and Julia set explorers reveal the infinite complexity that emerges from a single quadratic iteration. Number spirals expose hidden prime-distribution patterns by arranging integers on an Archimedean spiral.

Each simulation is built on exact mathematical definitions — no approximations or artistic license. Exploring the parameter space of a fractal, watching a Pythagoras tree grow with angle controls, or decomposing a square wave into harmonics builds genuine mathematical intuition. These visualisations are used in undergraduate courses on complex analysis, signal processing, and discrete mathematics worldwide.

Mathematics simulations reveal the beauty hidden in abstract structures. The Mandelbrot set is computed from a one-line recurrence relation yet contains infinitely complex geometry at every scale. L-Systems produce realistic trees and ferns from symbol-rewriting rules that fit on a single line. Fourier analysis underpins every digital audio codec, image compression algorithm, and radio system on the planet. These visualisations make the abstract tangible.

Key Concepts

Topics and algorithms you'll explore in this category

FractalsSelf-similar structures with non-integer Hausdorff dimension
Fourier AnalysisDecomposing signals into sine/cosine components
L-SystemsLindenmayer grammar for botanical and fractal geometry
Cellular AutomataRule-110 and 1D automata — Wolfram classes
Number TheoryPrime spirals, Ulam spiral, number sequences
Iterative MapsMandelbrot set, Julia sets, Newton fractals

📐 Test Your Maths Knowledge

5 questions — fractals, Fourier, primes, and more

Frequently Asked Questions

Common questions about this simulation category

What is the Mandelbrot set?
The Mandelbrot set is the set of complex numbers c for which the iteration z_{n+1} = z_n² + c remains bounded. Points inside the set are coloured black; points outside are coloured by escape-time to reveal the fractal boundary with infinite self-similar detail at every zoom level.
How does Fourier transform decompose a signal?
The Fourier transform expresses any periodic function as a sum of sine and cosine waves at different frequencies and amplitudes. The simulation lets you add harmonics interactively and watch the time-domain waveform reconstruct in real time — making the synthesis/analysis duality visually clear.
What are L-Systems?
Lindenmayer Systems are string-rewriting grammars that generate complex fractal plant shapes and geometric patterns from a few simple rules applied iteratively. A single rule like F→F[+F]F[-F]F can produce a realistic-looking tree after five iterations.

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