Patterns — a mathematical pattern generator

Every generator and effect

Each layer has its own page: how it works, where it comes from and what to tweak.

Tilings and mosaics

• Archimedean tilings All 11 ways to tile the plane with regular polygons so that every vertex looks the same — and their duals, the Laves tilings.
• Islamic star patterns Girih by Hankin's method: rays leave the edge midpoints of an Archimedean tiling at one angle and meet as stars and rosettes, and the straps weave over and under.
• Quasiperiodic tilings Penrose, Ammann–Beenker and their relatives: order without repetition. De Bruijn's multigrid method and Robinson-triangle deflation.
• The aperiodic hat A single 13-sided tile that covers the plane yet never repeats (2023). Built by substituting the H, T, P and F metatiles.
• Truchet tiles One tile in random orientations builds endless paths and mazes.
• Maze A uniform spanning tree by Wilson's algorithm: exactly one path between any two cells. Seamless on a torus.
• Voronoi and Delaunay Every point claims the region closest to it: honeycombs, stained glass, low-poly mosaics. Bridson's Poisson disk and Lloyd relaxation.
• Circle packing Circles that grow until they touch fill the plane densely; the Apollonian gasket nests circles forever by Descartes' theorem.
• Celtic knots Knotwork on a Mercat grid: bands run diagonally, strictly over and under, and walls in the grid turn them back in smooth loops.
• Weaves Cloth from a weave draft: plain, twill, satin, basket, herringbone, houndstooth and tartan — rounded threads dipping under each other.

Fields and noise

• Fractal noise Octaves of lattice noise summed up: clouds, marble, mountains, smoke. Seamless and looping.
• Domain warping Noise that displaces itself twice: smoke, marble, liquid swirls, marbled paper. Inigo Quilez's method. Seamless and looping.
• Cellular noise Distances to the nearest random points (Worley noise): bubbles, facets, crackle, grouted mosaic, pebbles, scales.
• Topographic map A map of an imaginary land: contour lines with bolder index contours, hypsometric tints and hillshading.
• Mesh gradient Soft blobs of the palette colours flow into each other in OKLab — like the wallpapers of new phones. Seamless and looping.
• Quasicrystal A sum of plane waves with a symmetry no crystal can have: the pattern is ordered yet never repeats.
• Wave interference Circular waves from several sources add up and cancel out, as in a ripple tank; drawn as thin rings they make a moiré.
• Op art Checkerboards, stripes, Bridget Riley's waves, concentric squares and rings — swollen by lenses like Vasarely's «Vega».

Simulations and growth

• Reaction–diffusion Two chemicals: one feeds the other, both spread at different rates — and Turing's spots, stripes and coral appear.
• Physarum Hundreds of thousands of slime-mould agents follow their own trails, sense them ahead and turn toward the strongest scent — and weave a transport network.
• Cellular automata A grid of cells, each looking at its neighbours by one rule over and over: spirals, mazes, coral and Wolfram's triangles grow out of random noise.
• Venation Veins grow toward scattered growth points and use them up on the way — the way leaf veins, coral and river beds branch.

Fractals

• Mandelbrot and Julia Every point of the plane starts an orbit z → z² + c; the colour tells how fast it runs away. Zooms to 10⁻¹².
• Fractal flames An iterated function system with non-linear «variations»: a million particles play the chaos game and their trails glow by log density.
• Strange attractors A simple formula repeated millions of times draws the cloud a chaotic orbit is pulled into.
• Hyperbolic tilings Regular {p, q}: p-gons, q at every vertex — they only fit on the hyperbolic plane. Poincaré disk, half-plane, band.
• Circle inversions and Kleinian groups Groups of Möbius maps: where their orbits pile up, the limit set appears — a lace of circles («Indra's Pearls»).
• L-systems A word rewritten by rules generation after generation and read as turtle moves: space-filling curves, snowflakes, dragons, plants.

Curves

• Guilloché and spirograph The engraving of banknotes and watch dials: hundreds of fine lines woven into rosettes, trochoids and ornamental borders.
• Harmonograph Damped pendulums steer a pen across paper: the musical interval between them draws a figure that slowly turns as it fades.
• Phyllotaxis How a sunflower packs its seeds: each one a golden angle further round, and spirals in Fibonacci numbers appear by themselves.
• Gielis superformula One equation by Johan Gielis draws circles, polygons, stars, flowers and shells — as nested outlines, a grid or a table of shapes.
• Flow lines Evenly spaced streamlines of a vector field after Jobard and Lefer: swirling noise, whirlpools, dipole field lines — as if drawn by a pen plotter.
• Ridgelines The Unknown Pleasures look: dozens of lines one behind another, each hiding the ones behind it. From noise — or from sound: a portrait of a track or ridges flowing with the music.

Sound

• Spectrogram Sound as a picture: time along one axis, frequency along the other, loudness as colour. A portrait of a whole track or a live scrolling strip.
• Waveform A track's loudness over time as rounded bars, a mirrored skyline, an envelope, rings or a spiral. Live, an oscilloscope trace of the sound right now.
• Chromagram How much of each of the twelve notes — C, C♯ … B, octaves folded together — sounds at every moment: harmony you can see.
• XY oscilloscope The left channel against the right, like a studio goniometer: mono is a vertical line, wide stereo a cloud, pure intervals Lissajous figures. The beam glows like phosphor.
• Chladni figures Sand on a sounding plate runs off the shaking parts and gathers on the nodal lines, where the plate stands still. A note picks the figure; a chord adds several.
• Faraday waves Shake a dish of liquid at an audio frequency and standing waves rise on its surface: squares, hexagons, 8-, 10- and 12-fold quasi-patterns. The note sets the symmetry and the wavelength.

Effects

• Wallpaper symmetry All 17 ways a pattern can repeat on the plane. The motif is the picture below.
• Kaleidoscope A rosette made of the picture below: n rotations with or without mirrors, plus optional endless Droste recursion — in rings or Escher's spiral.
• Distort The picture below flows, swirls, bulges, wraps into a «tiny planet» or shows through fluted glass.
• Mosaic The picture below laid in tesserae: squares, bricks, hexagons, triangles or Voronoi cells, set in grout and bevelled.
• Halftone The picture below as a newspaper or magazine print: dots, lines, diamonds or crosses, in one ink or four CMYK screens.
• Dither The picture below in chunky pixels and a few colours: a Bayer matrix, blue noise or gradient noise turns halftones into a dot pattern.
• Palette recolour Recolours the picture below with the document colours: a gradient map by lightness, the nearest palette colour or a hue rotation in OKLCH.
• Relief The brightness of the picture below becomes height, lit by a lamp: emboss, a shaded picture or polished metal.
• Blur Blurs the picture below: Gaussian, motion, a camera zoom or a spin around a centre. The radius is in frame units.
• Glow Bright parts of the picture below get a soft halo, like in a lens or on film (bloom).
• Grain Film grain over the picture: sized in frame units, strongest in the midtones, mono or colour; in animation it changes every frame and loops.
• Vignette The picture fades gently towards the edges — into darkness, light or a palette colour.
• RGB shift Chromatic aberration: the channels spread from the centre like in a cheap lens, or slide one way like a misregistered print.