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Mathematics

Fractals, cellular automata and complex-plane sets — mathematical objects of infinite complexity, generated by a few lines of code. This category turns abstract mathematics into something you can see, touch and steer. Each interactive Mathematics model runs directly in your browser, so you can pan and zoom into the Mandelbrot set, rebuild a waveform from rotating Fourier circles, watch a Pythagoras tree grow, or trace prime numbers spiralling outward. You will learn how calculus, geometry, number theory and linear algebra connect, building genuine intuition that static textbook diagrams rarely give. Whether you are a curious beginner, a GCSE or A-Level student, or a university learner, these visualisations make the ideas behind the formulae tangible — and reveal why mathematics matters far beyond the classroom, from computer graphics to data science.

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.

ζ
New ★★★★ Advanced
Riemann Zeta Function
Visualise the Riemann zeta function on the complex plane: domain-colour ζ(s), walk the critical line Re(s)=½ and watch |ζ| dip to zero at the non-t…
Riemann zeta complex analysis critical line
🌹
★☆☆ Easy
Rose Curves (Rhodonea)
Polar roses r = a·cos(kθ): odd k gives k petals, even k gives 2k. Rational k = n/d makes intricate blooms.
Polar Curves Trigonometry Canvas 2D
➿
★★☆ Moderate
3D Lissajous Curves
Three perpendicular oscillators woven into a rotating 3D curve. Small coprime ratios close into harmonic knots and lattices.
Lissajous 3D Projection Canvas 2D
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★★☆ Moderate
Hypocycloids & Epicycloids
A circle rolls inside or outside a fixed circle, tracing roulette curves — deltoid, astroid, cardioid, nephroid. Cusps = R/gcd(R,r).
Parametric Curves Roulette Canvas 2D
🪻
★★★ Advanced
Newton's Method Fractal
Every pixel iterates Newton's method in ℂ toward a root, colored by which basin it reaches. Switch polynomials and zoom the fractal boundary.
Fractal Basins of Attraction Canvas 2D
🌌
★★★ Advanced
Mandelbulb — 3D Fractal
Real-time 3D Mandelbulb raymarched on the GPU with an honest distance estimator de = 0.5·log(r)·r/dr. Slide the power n to morph the spiky shell, set iterations, swap palettes.
Three.js GLSL Raymarching Fractal
🌀
★★★ Advanced
Fourier Epicycles
Draw any closed shape and watch a chain of rotating circles redraw it. The radii, frequencies and phases come from the complex DFT of your path — Fourier series made visual.
Canvas 2D Fourier DFT Epicycles
🌀
★★★ Advanced
Space-Filling Curves
Hilbert, Peano and Morton (Z-order) curves filling the plane. Colour by path position to see how Hilbert preserves locality far better than Z-order; inspect the 1D↔2D mapping.
Canvas 2D Hilbert Fractal Locality
🔢
★★★ Advanced
Ulam Spiral — Patterns in Primes
Integers on a square spiral with primes marked — striking diagonals emerge. Highlight prime-rich polynomials like n²+n+41, try the Sacks spiral, inspect any cell's factorization.
Canvas 2D Prime Numbers Number Theory Sieve
🔬
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
🔺
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
🟦
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
🔢
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
🌿
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
🎲
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
🗺️
★★☆ 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
🌿
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
🌊
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
🧩
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
🎨
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
📈
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
🔢
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
⬜
★☆☆ 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
🎯
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
🎵
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
📐
★★☆ Moderate
Eigenvalues & Eigenvectors — Linear Transformation Visualiser
Drag 2×2 matrix entries to see eigenvectors remain on their span while all other vectors rotate. Displays eigenvalues, characteristic polynomial and determinant live.
Eigenvalues Linear Algebra Matrix Determinant
🫧
★★★ Advanced
Soap Bubble
Simulate soap films finding minimal surfaces. Plateau relaxation minimizes surface area. Explore catenoid, helicoid, saddle surfaces.
Minimal Surfaces Plateau Canvas 2D
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★★☆ Moderate
3D Cellular Automaton
Explore 3D cellular automata with Moore neighborhood. Famous rules: Cloud, Crystal Growth, Amoeba, Pyroclastic. WebGL voxel rendering.
Cellular Automata WebGL Voxel
🔮
★★★ Advanced
Hyperbolic Geometry Tiling
Tessellate the Poincaré disk model of hyperbolic plane with {p,q} regular tilings. Möbius transforma...
Hyperbolic Geometry Poincaré Disk Tessellation
🔢
★★★ Advanced
p-adic Number Ultrametric Space
Visualize p-adic numbers as a fractal tree. The p-adic distance |x-y|_p = p^{-v_p(x-y)} where v_p is...
p-adic Numbers Ultrametric Number Theory
🔷
★★☆ Moderate
Prime Distribution & Gaps
Visualize prime number distribution: spiral (Ulam), prime gaps histogram, and the Prime Number Theor...
Prime Numbers Ulam Spiral Number Theory
📉
★★★ Advanced
Weierstrass Nowhere-Differentiable Function
The Weierstrass function W(x) = Σ aⁿ·cos(bⁿπx) is continuous everywhere but differentiable nowhere w...
Weierstrass Fractal Analysis
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.

Other Categories

Every Mathematics simulation in this collection runs free in your browser, with no installs or sign-ups. Use each interactive Mathematics model to learn Mathematics online at your own pace — zoom fractal boundaries, decompose signals with Fourier analysis, or explore prime patterns and topology. These same techniques power real-world applications such as JPEG image compression, MRI medical imaging, computer graphics and machine-learning algorithms, showing how the mathematics you visualise here underpins the technology you use every day. Bookmark the page and keep experimenting with the parameters to deepen your mathematical intuition.