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Generative Art & Algorithmic Patterns

Mathematics is inherently beautiful. This category covers the phenomena of algorithmic image-making — recursive trees, infinite fractals such as the Mandelbrot and Julia sets, noise-driven flow fields, Voronoi tessellations, L-system grammars, cellular automata and symmetry groups that produce breathtaking visuals from surprisingly simple rules. By adjusting parameters in real time you can learn how iteration, recursion, gradient noise and reaction-diffusion give rise to organic complexity, and see exactly how a single equation becomes a living pattern. It matters because the same techniques power procedural game worlds, film VFX, data visualisation, generative design and creative coding. Every model runs live in your browser, with no installation, making this an ideal playground for students, artists, designers and curious minds exploring where mathematics meets art.

9 simulations Canvas 2D · WebGL · Three.js Fractals · L-Systems · Noise

Category Simulations

Algorithmic beauty from mathematical rules

Generative art bridges mathematics and aesthetics. Every fractal, snowflake and spiral is the visual fingerprint of an underlying equation or rule system. From the self-similar coastlines of Mandelbrot to the branching grammar of L-systems — complexity emerges from iteration.

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New ★★☆ Moderate
Worley Noise
Worley (cellular) noise colours each pixel by distance to feature points, producing cells, cracks and stone textures.
Worley noise cellular noise Voronoi
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New ★★★ Advanced
Domain Warping
Domain warping feeds noise back into its own coordinates — f(p + f(p + f(p))) — turning plain fBm into swirling, marbled, cloud-like fields.
domain warping fBm Perlin noise
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★★☆ Moderate
Phyllotaxis
Vogel's sunflower model: seed n at angle n·137.5°, radius c·√n. Only the golden angle packs Fibonacci spirals without gaps.
Golden Angle Fibonacci Canvas 2D
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★★★ Advanced
Apollonian Gasket
An infinite fractal of mutually tangent circles built with the complex Descartes Circle Theorem, recursively filling every curvilinear gap.
Fractal Tangent Circles Canvas 2D
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★★☆ Moderate
Rule 30 — Elementary CA
Wolfram's elementary cellular automata: pick any rule 0–255 (chaotic 30, Sierpiński 90, universal 110) and watch the space-time triangle grow.
Cellular Automata Wolfram Canvas 2D
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★★☆ Moderate
Neon Tunnel
Fly endlessly through a glowing neon tunnel of pulsing rings — polar tunnel-mapping shader.
Three.js GLSL Raymarching
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★★☆ Moderate
Morphing Sphere
An organic sphere morphing via 3D simplex-noise vertex displacement — iridescent and glossy.
Three.js GLSL Noise
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★★☆ Moderate New
Voronoi Growth
Crystalline growth from Johnson-Mehl-Avrami nucleation: seeds grow into a Voronoi-like mosaic (straight or curved boundaries) with three distance metrics and Lloyd relaxation.
Voronoi Johnson-Mehl Crystallisation Canvas 2D
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★★★ Advanced
Superformula 3D
Gielis's superformula r(φ) modulates a sphere along two axes to produce stars, flowers, twisted donuts and crystals. Live BufferGeometry rebuilds as you slide eight parameters.
Three.js Superformula Parametric Procedural
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★★★ Advanced
Lloyd's Relaxation — Centroidal Voronoi
Move each point to the centroid of its Voronoi cell and repeat — points drift into a centroidal Voronoi tessellation, beautifully uniform and hexagonal-tending. Energy E falls monotonically.
Canvas 2D Voronoi Centroidal Relaxation
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★★★ Advanced
Poisson-Disk Sampling — Blue Noise
Bridson's blue-noise sampler: an active list grows by accepting candidates at distance r..2r — refusing anything closer than r. Compare to uniform random and inspect the radial spectrum.
Canvas 2D Blue Noise Bridson Sampling
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★★★ Advanced
Generative City
Infinite top-down procedural city. Recursive block subdivision lays out roads, zoned parcels fill with extruded buildings, and downtown intensity drives density. Reseed, restyle and export as PNG.
Canvas 2D Procedural Subdivision City
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★★★ Advanced
Computational Photo Art
Turn any image into algorithmic art: ASCII text, photomosaic, stipple dots or Floyd–Steinberg dithering. Upload your own or use built-in sources, tune resolution and palette, export as PNG.
Canvas 2D ASCII Art Dithering Image Processing
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★★☆ Moderate
Fractal Explorer
Mandelbrot and Julia set renderer with GPU-accelerated iteration. Zoom into infinite detail, switch colour palettes, animate the Julia parameter and export high-resolution images.
WebGL Complex Numbers Mandelbrot Julia Set
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★☆☆ Beginner
Kaleidoscope
Real-time kaleidoscope built from reflection symmetry groups. Adjust fold count (k-fold symmetry), draw freely or use the built-in noise driver to generate autonomous evolving patterns.
Canvas 2D Symmetry Reflection k-fold
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★★☆ Moderate
L-Systems
Lindenmayer systems: string-rewriting grammars that grow plants, snowflakes and space-filling curves. Edit axiom, production rules and angle; watch the geometry evolve generation by generation.
Canvas 2D Turtle Graphics Grammar Recursion
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★☆☆ Beginner
Pythagoras Tree
A binary tree where each node sprouts two right-triangle branches. Tune the split angle and size ratio to morph from an elegant oak to a symmetric cross to a spiral of squares.
Canvas 2D Recursion Self-Similarity Pythagorean
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★☆☆ Beginner
Sierpiński Triangle
Chaos game and recursive subdivision — two very different algorithms that converge to the same fractal. Watch the triangle emerge from thousands of random points bouncing between vertices.
Canvas 2D Chaos Game IFS Fractal
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★★☆ Moderate
Snowflake Growth
Cellular automaton that replicates Reiter's snowflake model. Adjust vapour density, anisotropy and diffusion to produce dendrites, plates, needles and hollow columns — the full morphology diagram.
Canvas 2D Cellular Automaton Diffusion 6-fold Symmetry
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★☆☆ Beginner
Number Spirals
Ulam spiral, Sacks spiral and prime rose — integers arranged in spirals reveal hidden patterns in prime distribution. Zoom out far enough and diagonal lines emerge from seemingly random noise.
Canvas 2D Prime Numbers Ulam Spiral Number Theory
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★★☆ Moderate
Flow Fields
3 000 particles trace a continuously evolving Perlin noise vector field. Switch between Ocean, Van Gogh, Fire and Mono palettes; tune noise scale, speed and trail fade.
Canvas 2D Perlin Noise Curl Noise Particles
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★★☆ Moderate
Stippling & Pointillism
Points self-organise into stipple art via weighted Voronoi relaxation. Choose from Waves, Spiral, Mandelbrot and Star cluster density patterns; watch Lloyd's algorithm converge.
Canvas 2D Voronoi Lloyd's Algorithm Dithering
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★★☆ Moderate
L-System Fractals
Generate fractal plants, trees and curves with Lindenmayer grammar rewriting and turtle graphics. Koch Snowflake, Dragon Curve, Barnsley Fern, Hilbert Curve and more.
Turtle Graphics Lindenmayer Fractals
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★★★ Advanced New
Islamic Geometric Patterns
Generate stunning {n/k} star polygon tessellations from classical Islamic geometric design. Explore Girih tiles, 8-pointed windows, Ottoman stars and quasi-periodic symmetry groups.
Star Polygon Girih Tiles Tessellation Canvas 2D
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★★☆ Moderate
Truchet Tiles
Discover emergent patterns from randomly oriented quarter-circle tiles. Adjust grid size and explore the mathematics of tiling theory — classic arcs, Smith 4-way, triangle fill.
Truchet Tiling Theory Emergence Canvas 2D
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★★☆ Moderate New
Diffusion-Limited Aggregation
Witten–Sander DLA algorithm grows fractal clusters one random-walk particle at a time. Live fractal dimension estimate D_f ≈ 1.71. Four colour modes (age, distance, heat, mono) and three seed shapes.
DLA Fractal Dimension Random Walk Canvas 2D
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★★☆ Moderate New
Cyclic Cellular Automata
Watch beautiful spiral patterns emerge from simple cyclic rules. Each cell advances to the next state only if a neighbour already has that state. Tune N states, threshold and colour palette.
Cellular Automata Spiral Waves Excitable Media Canvas 2D
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★★☆ Moderate
Random Growth
Compare Eden model (uniform surface growth) and DLA (diffusion-limited aggregation). DLA creates fractal structures with D≈1.71.
Eden Model DLA Fractal Dimension Canvas 2D
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New ★★★ Advanced
3D Curl Noise Flow Field
Thousands of particles follow a 3D curl noise velocity field rendered on canvas. Curl noise (∇×F) is...
Curl Noise 3D Particles Canvas 2D

Key Concepts

Mathematical foundations behind the visuals

Fractal Dimension
A non-integer measure of self-similarity. The Sierpiński triangle has dimension log(3)/log(2) ≈ 1.585 — more than a line, less than a surface.
Lindenmayer Systems
Context-free string grammars that model plant branching. A simple rule F→F[+F]F[-F]F generates a realistic bush structure in four iterations.
Perlin / Simplex Noise
Gradient noise with smooth spatial correlation. Used in flow fields, terrain generation and procedural textures — coherent randomness with tunable octaves.
Iterated Function Systems
A set of affine contractions whose repeated application converges to a fractal attractor. The classic chaos game for Sierpiński uses three IFS maps.

Learning Resources

Deepen your understanding with these articles

Article Fractal Geometry & Dimension Hausdorff dimension, self-similarity ratios, Julia sets and the Mandelbrot escape algorithm explained. → Article Lindenmayer Systems Formal grammars, turtle graphics interpretation, stochastic and parametric extensions for realistic plant models. → Article Perlin Noise & Procedural Textures Gradient noise, octave layering (fBm), domain warping and applications from terrain to marble shaders. → Article Voronoi Diagrams Fortune's sweep-line algorithm, Delaunay duality, weighted Voronoi and applications in stippling and cell noise. →

Explore adjacent mathematical worlds

About Generative Art Simulations

Procedural patterns, algorithmic drawing, L-systems, and creative code

Generative art simulations use mathematical rules and algorithms to produce visual art. L-system plant generators rewrite axiom strings through branching rules and interpret the result as turtle-graphics commands, growing fractal trees, ferns, and coral formations with just a handful of parameters. Reaction-diffusion canvas simulations evolve the Gray–Scott equations on a pixel grid, producing the leopard spots, zebrafish stripes, and labyrinthine coral patterns predicted by Turing in 1952.

Voronoi diagram generators, flow-field particle systems, and noise-based texture synthesisers explore the intersection of mathematics and aesthetics. Every parameter change — from diffusion ratio to F&k values in Gray–Scott, or from branching angle to segment length in an L-system — produces visually distinct and often surprising results. These tools are used by creative technologists, generative artists, and data visualisers who work at the intersection of code and visual design.

Each simulation in this category is built with accuracy and interactivity in mind. The underlying mathematical models are the same ones used in academic research and professional engineering — just made accessible through a web browser. Changing parameters in real time and observing the results is one of the most effective ways to build intuition for complex scientific and engineering concepts.

Frequently Asked Questions

Common questions about this simulation category

What generative art algorithms are covered?
Flow fields (Perlin noise vector fields), L-system fractal plants, kaleidoscopes, reaction-diffusion (Gray-Scott), space colonisation branching, fractal flames, and stipple drawing.
What is a flow field in generative art?
A flow field assigns a direction vector to every point in the canvas, derived from a noise function. Particles follow these vectors, leaving trails that form organic flowing patterns.
Can I download or share the artwork these simulations produce?
Most simulations include a Save PNG or Copy Image button so you can export the generated artwork. The canvas renders at full resolution.

Each Generative Art simulation on this page turns abstract mathematics into a hands-on visual experiment you can run instantly in any modern browser. Whether you want to learn Generative Art online from scratch or tune an interactive Generative Art model down to its noise octaves and branching angles, these tools reveal the algorithms behind fractals, flow fields and procedural patterns. The same methods drive real-world applications such as procedural terrain and texture generation in video games and film, making this a practical bridge between creative coding, design and computer graphics.