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Cusp catastrophe. Smooth parameter changes cause sudden jumps in equilibrium state. Hysteresis traces the geometry of discontinuous bifurcation.

Chaos control. Tiny parameter perturbations stabilize unstable periodic orbits hidden in the logistic map's chaos — the OGY method tames disorder into order.

Stochastic resonance. A paradoxical phenomenon where adding noise amplifies a weak signal by helping it cross a threshold.

Arnold tongues. Tongue-shaped mode-locking regions fill the circle map's parameter space, and the devil's staircase marks each winding number.

Random Boolean network. N nodes control each other via Boolean functions. A phase transition between order and chaos emerges near K=2.

Strange attractor explorer. Discover chaotic orbits from Sprott's quadratic map parameter space, rendered as log-density accumulation.

Arnold's cat map. An area-preserving chaotic transformation shears and mixes a color gradient, yet it perfectly recurs after finite steps.

Visual feedback loop. The screen redraws itself scaled and rotated, creating infinite regress patterns.

Lyapunov fractal. Mapping the Lyapunov exponent across logistic map parameter space reveals fractal boundaries between stability and chaos.

Chirikov standard map. KAM tori float in phase space; as perturbation grows, they dissolve into a sea of Hamiltonian chaos.

Phase portrait. Visualize 2D dynamical system trajectories as flowing particles. Map of fixed points, limit cycles, and separatrices.

Hénon strange attractor. Fractal orbits of a two-dimensional chaotic map. Tune parameters to cross between order and chaos.

Chaos billiards. The stadium table is an ergodic chaotic system; ellipse and circle are integrable. Shape determines the fate of trajectories.

Basin-of-attraction fractal from a 3-magnet pendulum. Tiny differences in starting position lead to completely different outcomes.

Logistic map bifurcation diagram. Period-doubling cascade leads to chaos as r increases. Zoom reveals infinite self-similar structure.

Clifford strange attractor. Two simple equations, four parameters, infinite patterns emerge as millions of points accumulate.

Explore multiple Lorenz trajectories and a Poincare section in Canvas.

Manipulate single pendulums, double pendulums, and phase-space traces.

Ballistic deposition. Particles rain down and stick to neighbors, growing a rough surface whose roughness obeys KPZ universality scaling laws.

Aeolian dune simulation. Wind saltation, selective deposition, and avalanche self-organize crescent-shaped barchan dunes.

Polymerization simulation. Monomers collide and bond into chains that undergo a gelation phase transition into spanning networks.

Cellular Potts Model. Extended cells on a lattice self-organize through surface tension and differential adhesion — two cell types spontaneously sort apart.

Self-avoiding walk. A random walker that never revisits a site traps itself in its own trail. The foundational model of polymer physics.

Brittle fracture in a spring lattice. Stress concentrates at crack tips while random bond strengths create branching patterns.

Lennard-Jones molecular dynamics. Attraction and repulsion drive crystallization, melting, and vaporization phase transitions.

Space colonization algorithm. Branches grow toward attraction points, self-organizing into leaf-vein-like structures.

Metaballs. Implicit surface isovalues merge and split organically.

Heat diffusion simulation. Paint hot and cold sources, draw insulating walls, and watch temperature spread by the heat equation.

Hex-grid crystal growth. Diffusion and freezing produce six-fold symmetric dendrites.

Hydraulic erosion. Raindrops carve terrain, and rivers and valleys emerge naturally.

Differential growth. A closed curve subdivides and repels itself into coral-like organic forms.

Spinodal decomposition. A conserved field separates into phases and coarsens into labyrinthine domains.

L-system branches grow slowly into a luminous garden and sway with wind.

Physarum algorithm. Particles follow chemical trails and slowly grow a mycelial network.

Draw diffusion-limited aggregation crystals through particle random walks.

Grow recursive trees from branch angle and wind-like variation.

Kelvin-Helmholtz instability. Two fluid layers sliding past each other roll up into spiral vortices at the shear interface.

二成分プラズマ。電子とイオンがクーロン力で相互作用し、デバイ遮蔽とプラズマ振動が創発する。

Torsion wave machine. Flick the chain and watch pulses propagate, reflect at fixed/free/absorbing boundaries. A spacetime diagram paints the wave's entire history.

Coriolis effect. Straight-line motion on a rotating disk appears to curve. Switch between inertial and rotating frames to experience fictitious forces.

Rayleigh-Bénard convection. Heated fluid spontaneously forms convection cells; increasing the Rayleigh number drives the flow from ordered rolls to chaos.

Optimal transport. Particles flow between distributions via greedy assignment and 2-opt refinement, minimizing total displacement.

Granular flow dynamics. Particles pour through a funnel, force chains glow under load, and neck width governs the jamming transition.

Hele-Shaw viscous fingering. A low-viscosity fluid displaces a high-viscosity one, and the interface destabilizes into branching finger patterns.

Ferrofluid simulation. Place magnets and watch spike structures self-organize from the competition between magnetic force and surface tension.

Kármán vortex street. Lattice Boltzmann simulation of alternating vortices behind an obstacle in fluid flow.

2D lattice vibration. Wave equation on a discrete grid. Tap to create pulses, pin nodes to build barriers.

Particles follow a Perlin noise flow field, tracing organic paths through an evolving vector landscape.

Ink drops fall into fluid and spiral. Mathematical marbling simulation.

Solitons from the KdV equation. Waves pass through each other intact after collision — a nonlinear miracle.

Fluid simulation. Shape the flow with your fingertips and watch luminous ink vortices drift slowly.

Chladni patterns. Particles gather on nodal lines formed by vibration modes, with gentle sound.

Practice Morse code as timing waves and signal pulses.

A granular cell simulation for falling, flowing, and fixed matter.

Touch and shape a Canvas wave field where multiple sources interfere.

A quiet phase-wave breathing visualization drawn on Canvas.

Active Brownian particles. No attraction, yet clusters form spontaneously from density and persistence alone — motility-induced phase separation.

Braitenberg vehicles. Simple sensor-motor wiring produces fear, aggression, love, and exploration from the same light sources.

Termite wood-chip sorting. Simple agents pick up isolated chips and drop them near clusters. Self-organization without central control.

Swarmalators. Particles that swarm in space and sync in phase. Five collective states emerge as coupling parameters change.

Crowd simulation. The social force model drives pedestrians autonomously — lane formation and bottlenecks emerge naturally.

Schelling segregation model. Mild same-type preference produces strong segregation — emergent social physics.

Nagel-Schreckenberg traffic flow model. Traffic jams emerge spontaneously from four simple rules.

Attraction and repulsion rules between particle species generate self-organizing clusters, vortices, and creature-like emergent behavior.

Spatial prisoner's dilemma. Cooperators and defectors copy strategies on a grid, generating complex emergent patterns.

Predator and prey. Fish breed, sharks hunt, and Lotka-Volterra oscillations emerge from three simple rules.

Ant colony optimization. Pheromone trails reinforce over time, spontaneously revealing shortest paths.

Kuramoto model. N oscillators weakly coupled — past the critical point, they spontaneously synchronize.

SIR epidemic model. Waves of infection sweep across a grid. β controls spread speed, recovery shapes the wave.

Observe genetic-algorithm generations through motion trails and fitness.

Boids visualization where separation, alignment, and cohesion create flocking.

A Canvas particle clock where points flow into seven-segment time fields.

Lifelike cellular automata. Explore 262,144 birth/survival rules on one grid. Each rule string unlocks a completely different universe of emergent patterns.

Cyclic cellular automaton. N states prey on each other in a ring, and spiral waves emerge spontaneously from noise.

Coupled map lattice. A grid of logistic maps with diffusive coupling spawns spatiotemporal chaos and pattern selection.

Random domino tiling of the Aztec diamond. The shuffling algorithm generates uniformly random tilings, and the arctic circle theorem's sharp boundary emerges.

Excitable medium. Click to ignite waves — when a wavefront breaks, self-sustaining spiral waves emerge. The math governing cardiac rhythm.

Toothpick sequence. Perpendicular branches grow from exposed endpoints, revealing a self-similar fractal.

Turing machine. A head reads, writes, and moves on an infinite tape — the archetype of computation.

Continuous cellular automaton. Life-like creatures emerge from convolution and a growth function alone.

Electron flow in a 4-state cellular automaton. Draw wires and inject signals.

Ising model. Spins on a lattice flip via Metropolis algorithm, forming beautiful domain structures near the critical temperature.

1D Cellular Automaton. Complex patterns emerge from simple rules.

Open grid cells by probability and observe connected clusters near the critical point.

Visualize Abelian sandpile avalanches and self-organized criticality.

Grow Gray-Scott reaction diffusion patterns directly on the canvas.

Explore Langton's Ant as multi-state cellular-automaton trails.

Explore Conway's Game of Life as a field of density and generations.

Dyson Brownian motion of random matrix eigenvalues. Logarithmic repulsion and thermal noise produce the Wigner semicircle law and level-repulsion statistics.

Domain coloring of the Riemann zeta function. Non-trivial zeros align on the critical line, revealing deep number-theoretic structure.

Gaussian primes on the complex plane. The fourfold symmetric pattern reveals deep number theory.

Continued fraction geometry. The Euclidean algorithm decomposes a rectangle into squares, and the golden ratio produces the perfect spiral.

Discrete Markov chain on a state graph. Walkers hop by transition probabilities and the empirical distribution converges to the stationary distribution.

Monte Carlo estimation. Random points rain down to approximate area. Watch convergence as sample count grows.

Ford circles. Every reduced fraction p/q becomes a circle tangent to the x-axis. Two circles touch iff the fractions are Farey neighbors.

Pascal's triangle. Color binomial coefficients by mod N and watch self-similar fractals emerge.

Ulam spiral. Integers on a spiral — primes form mysterious diagonal patterns.

Recamán sequence. Subtract if possible, otherwise add — a simple rule that draws semicircular arcs on the number line.

Modular multiplication. Connect N points on a circle by multiplier k — cardioids and nephroids emerge.

Lévy flight. Luminous trails shaped by short wanderings and sudden long leaps.

Galton board. Balls falling through pegs reveal the normal distribution.

Draw Collatz sequences as branching trajectories and density fields.

A Fourier epicycle visualization where rotating vectors assemble waveforms.

Explore prime gaps as luminous rings, record-gap traces, and residue-wheel classes.

Phyllotaxis, spirals, and growth fields shaped by the golden angle.

Draw digits of pi as walks, spirals, and chords.

Geodesics on curved surfaces. Sculpt a height field and watch shortest-path particles bend where Gaussian curvature warps space.

Pedal curves. The locus of the foot of perpendicular from a fixed point to the tangent of a base curve traces a transformed shape.

2x2 linear transform. Drag basis vectors and watch the unit circle, grid, eigen directions, and flow update.

Catenary curve. A chain hanging under gravity traces the hyperbolic cosine. Compare it to the parabola.

Circle inversion. Lines become circles, circles become circles. Place inversion circles and watch geometry transform.

Winding number. A closed curve topologically partitions the plane — each region colored by how many times the curve winds around it.

Torus knots. Adjust p and q to wind a curve on a torus, projected with depth-sorted crossings.

Maurer rose. Connect evenly-spaced points on a rose curve to reveal intricate star-like geometric patterns.

Harmonograph. Delicate geometric patterns drawn by damped pendulums. Frequency ratios and phases create infinite curves.

Conformal mappings via complex functions. Watch a grid distort smoothly through mathematical transformation.

Real-time projection of a 4D hypercube. Six rotation axes reveal geometry beyond three dimensions.

Superformula. Four parameters — m, n1, n2, n3 — generate flowers, leaves, shells, and snowflakes from a single equation.

N-point pursuit polygon. Each vertex chases the next at constant speed, and logarithmic spirals emerge.

Bézier curves. Interpolation of interpolations — visualizing the de Casteljau algorithm.

Particles flowing on a Möbius strip. Luminous trails drift slowly across a twisted surface.

Spirograph curves drawn by rotating circles and pen offsets.

Connect points on a circle with formulas and let thread density draw curves.

Schottky group circle fractal. Four circles define Möbius transformations whose iteration nests circles within circles, converging on a Cantor-set limit.

The Multibrot set. Smoothly vary the exponent d in z^d+c and watch the Mandelbrot set morph into (d-1)-fold symmetry.

Generalized Sierpinski carpet. Design your own removal pattern on an N×N grid and watch the recursive fractal emerge with continuously varying fractal dimension.

Burning Ship fractal. Absolute value operations break the symmetry of the Mandelbrot formula, producing an asymmetric fractal resembling a ship ablaze.

Pickover biomorphs. Apply a biomorphic bailout test to complex iteration — organism-like fractals emerge from pure mathematics.

Buddhabrot. Overlaying escape trajectories from the Mandelbrot iteration reveals a nebula-like luminous structure.

Fractal flames. Nonlinear IFS variations produce luminous organic forms via log-density histogram rendering.

Symmetric chaos icons. Enforce n-fold rotational symmetry on a chaotic iterated map, and crystal-like structures emerge from chaos.

Iterated Function System. Probabilistic affine transforms produce self-similar fractals. Mutate parameters to explore unknown forms.

Dragon curve. A self-similar fractal that emerges from repeated paper folding.

Hilbert curve. A 1D curve that fills 2D space without gaps — the convergence of a space-filling curve.

Apollonian gasket. A circle-packing fractal where every gap between tangent circles is recursively filled. Governed by Descartes' Circle Theorem.

Newton fractal. Visualizing Newton-Raphson convergence to roots of z^n−1=0 in the complex plane. The basin boundaries are fractal.

Explore the Mandelbrot set through zoom and layered color fields.

Drag through Julia-set parameter space and watch the field morph continuously.

Explore recursive Koch curve generation with adjustable angle and depth.

Polarization of light. Polarizing filters select oscillation direction by Malus's law. Insert a 45° filter between crossed polarizers and watch extinct light revive.

Stereographic projection. Sphere patterns—grid, loxodromes, icosahedron—map to circles on the plane, morphing as the sphere rotates.

Hydrogen electron orbital probability density. Cross-section and Monte Carlo particle cloud visualization of wavefunctions for quantum numbers (n,l,m).

Fraunhofer diffraction patterns. FFT transforms aperture shapes into far-field interference figures.

CT-scan Radon transform. Rotating projections form a sinogram, then backprojection reveals the hidden density field.

Optical refraction simulation. Light rays bend at medium boundaries following Snell's law. Beyond the critical angle, total internal reflection traps the light.

Charged particle tracks. Particles spiral through a magnetic field, lose energy, and branch at decay vertices. A bubble chamber reimagined.

Doppler effect. A moving source compresses and stretches wavelengths. Mach cones appear at supersonic speed.

Dielectric breakdown model. Discharge paths branch and race, leaving Lichtenberg figures behind.

2D raycasting. Cast rays from light sources to compute visibility polygons and paint light and shadow in real time.

Gravitational lensing. Mass warps spacetime, bending starlight. Einstein rings emerge.

Kaleidoscope. Particle trails reflect and rotate under dihedral symmetry D_N, forming geometric patterns.

Double-slit experiment. Particles pass through slits one by one, and an interference pattern emerges.

Moiré fringes. Two periodic grids overlap — angular difference generates beat-frequency stripes.

Quantum tunneling. A wave packet splits into reflected and transmitted parts — experience the Schrödinger equation.

Optical chaos. Surface ripples bend light and paint slowly drifting caustics.

Forced oscillation and resonance. When the driving frequency matches an oscillator's natural frequency, amplitude grows dramatically — the Lorentzian peak sweeps across an array of oscillators.

Gear train simulation. Meshing gears transmit rotation ratios through interlocking tooth profiles.

Roche limit. When tidal force exceeds self-gravity, a satellite is torn apart and becomes a ring.

Minkowski spacetime diagram. Lorentz boosts shear the grid while the light cone stays at 45 degrees. Place events, draw worldlines, and watch simultaneity tilt.

Lagrange points of the restricted three-body problem. Effective potential and zero-velocity curves in the co-rotating frame reveal L1-L5 equilibria.

Coulomb crystal. Charged particles in a harmonic trap repel each other and self-organize into concentric shell structures.

Brownian ratchet. Asymmetric sawtooth potential and thermal noise create directed transport — the principle behind molecular motors.

Magnetic field visualization. Flowing particles trace closed field lines from bar magnet dipoles.

Spiral galaxy density wave simulation. Thousands of stars in differential rotation form emergent arm structures.

Rope simulation. Verlet integration and distance constraints for chain physics. Grab, swing, and cut.

FABRIK inverse kinematics. Articulated arms reach toward a target through forward-backward iteration.

Pressure-based soft body simulation. Throw, bounce, and squish spring-mass bodies inflated by internal pressure.

Kinetic theory of ideal gas. Particle speeds converge to the Maxwell-Boltzmann distribution through elastic collisions.

Electric field visualization. Place point charges and watch field lines reveal spatial structure.

Four-bar linkage kinematics. A rotating crank drives a coupler whose traced point carves closed algebraic curves.

Pendulums of different periods create collective phase patterns. Full resync every 60 seconds.

N-body gravity simulation. Particles attract each other and trace orbits.

Touch an elastic cloth mesh built from points and constraints under gravity and wind.

An orbital race simulation showing ground tracks, laps, and relative lead without external APIs.

Touch a physics field of gravity, attraction, collisions, and chains.

Polyomino exact cover. Watch backtracking search fill a rectangle with polyomino pieces, branching and pruning its way to every solution.

Aperiodic monotile. A single shape tiles the plane without ever repeating. The 2023 einstein tile discovery visualized.

Wallpaper groups. Particle trails replicated by rotation, reflection, and translation tile the plane with crystallographic symmetry.

Signed distance field. Combine geometric primitives with CSG boolean operations and render the pulsing iso-distance contours.

2D foam dynamics. Surface tension curves cell walls, von Neumann coarsening shrinks small bubbles, and topology rearranges as cells pop.

Weighted Voronoi stippling. Lloyd's algorithm drives point clouds toward density-matched equilibrium — self-organization from randomness to order.

Delaunay triangulation. Place points and watch a mesh form where no circumcircle contains another point. Voronoi duality.

Quasicrystal interference. N plane waves superpose to reveal quasiperiodic order — neither periodic nor random.

Truchet tiles. Randomly oriented quarter-circle arcs form labyrinthine flowing patterns.

Wave Function Collapse. Tiles resolve one by one through constraint propagation, spreading order from minimum-entropy cells.

Poincaré disk model. Hyperbolic tilings spread infinitely in non-Euclidean space.

Circle packing. Growing circles slowly fill the space.

Draw nearest-point space partitions with moving seeds and distance fields.

Explore Penrose-style aperiodic tile orientation and layered patches.

Cascading failure on networks. One node falls, its load spills to neighbors, and collapse propagates. Scale-free networks are robust to random failure but fragile to targeted attack.

Hopfield network. Patterns stored by Hebbian learning are recalled from noisy states through energy minimization.

PID controller dynamics. Proportional, integral, and derivative gains shape the step response. Five presets draw five different trajectories toward the same target.

Scale-free network. Preferential attachment draws new nodes to hubs, and a power-law degree distribution emerges.

Kohonen self-organizing map. A neural grid deforms itself to match data topology, drawing a map that preserves neighborhood structure.

Particle swarm optimization. A swarm explores a fitness landscape — individual memory and collective knowledge guide it toward the global optimum.

Steiner tree. A minimum-length network finds its own 120° junction points to connect terminals optimally.

Hough transform for line detection. Points become sinusoidal votes, and peaks reveal hidden lines.

Vietoris–Rips complex. As the distance threshold grows, points connect into edges and triangles — a simplicial complex revealing topological features.

Q-Learning. An agent learns optimal paths through trial and error. The value function heat map spreads slowly from the reward.

Convex hull algorithms visualized. Watch Graham Scan, Jarvis March, and QuickHull trace the outer boundary of a point set step by step.

Quadtree spatial partitioning. Space recursively subdivides by particle density, accelerating range and nearest-neighbor queries.

Minimum Spanning Tree. Prim's algorithm continuously reconnects drifting nodes at minimum cost. A living network.

Neural network classification boundary in real time. Place data points and watch the network learn.

Gradient descent visualized. Watch SGD, Momentum, and Adam descend optimization landscapes.

Simulated annealing solves the Travelling Salesman Problem. Watch a hot chaotic path cool into the shortest route.

Pathfinding algorithms visualized. Watch A*, Dijkstra, and BFS search for the goal.

Random graph phase transition. Nodes connect and suddenly become one.

Watch sorting comparisons and swaps as luminous bar fields.

Navigate a generated maze through luminous exploration trails.