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Generative Art & Algorithmic Patterns

Mathematics is inherently beautiful. Recursive trees, infinite fractals, noise-driven flow fields and symmetry groups — algorithms that produce breathtaking visuals from surprisingly simple rules.

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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★★☆ 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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★★☆ 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

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.