Physics you can
touch with a cursor.
A living notebook of small, precise simulations and engineering experiments. Everything here runs in your browser — no installs, no nonsense. Move your cursor to reshape the interference field above.
Experiments
Barnsley Fern
An iterated function system constructed from four affine transformations. By repeatedly applying these simple rules to a single point, a mathematically precise, self-similar structure of a fern emerges.
↗Lotka-Volterra Model
Explore the classical predator-prey equations. Watch the cyclical struggle between species unfold in time series and trace the limit cycles in phase space.
↗Spirograph
Simulate hypotrochoids and epitrochoids by rolling one circle inside or outside another. A pen attached to the rolling circle traces out beautiful, intricate, continuous mathematical curves.
↗Lindenmayer Systems
An interactive string-rewriting system initially conceived to model the growth of plants. Generate fractal plants, Koch curves, and dragon curves using simple replacement rules.
↗Chaos Game
A simple geometric rule generating intricate fractal patterns. Plot a point, pick a random vertex, and step a fixed fraction towards it. Repeat to watch perfect structure emerge from randomness.
↗Langton's Ant
A two-dimensional universal Turing machine running on a grid. Initially, simple rules generate completely chaotic and seemingly random behavior, but eventually, an emergent ordered structure—a "highway"—is reliably built.
↗Harmonograph
Four damped pendulums coupled to a single drawing point. Tune frequencies to musical ratios and watch complex Lissajous figures slowly decay.
↗Lorenz Attractor
Integrate the three equations that shattered determinism. Tune σ, ρ, β and watch the butterfly shed its wings.
↗Double Pendulum
Two rods, one Lagrangian, infinite sensitivity to initial conditions. Drag the bobs and watch the ghosts fan out.
↗Fourier Drawing Machine
Decompose any closed curve into a chain of rotating epicycles. Draw your own, or pick a preset and watch Parseval's identity keep score.
↗Wave Packet
A Gaussian state evolving under the time-dependent Schrödinger equation. Tunnel through barriers, oscillate in a well, watch unitarity hold to machine precision.
↗Turing Patterns
Two chemicals, two PDEs, and the spontaneous symmetry-breaking that Turing proposed explains leopard spots. Paint your own initial conditions and watch order emerge from noise.
↗N-Body Gravity
Newton's law in full: binary stars, Keplerian solar systems, and the improbable figure-8 choreography where three equal masses trace one closed curve forever. Add your own body and break it.
↗Mandelbrot Set
A deceptively simple rule: z → z² + c. Iterate from z=0, and color by escape time. Zoom into the boundary between order and chaos and discover infinite self-similar detail.
↗Fluid Dynamics
Incompressible 2D Navier-Stokes solved in real time. Paint dye, seed vortices, and watch the Kelvin-Helmholtz instability roll up a shear layer into spirals before your eyes.
↗Quantum Double-Slit
Two slits, one wave packet, and the impossibility of saying which path was taken. Split-operator FFT integration of the 2D Schrödinger equation, with live fringe spacing and phase portrait.
↗Ising Model
Metropolis Monte Carlo on a 200 × 200 spin lattice. Watch ferromagnetic domains form and coarsen below the Onsager critical temperature, dissolve above it, and fractalise at Tc ≈ 2.27.
↗Black Hole Geodesics
Photon and particle paths in exact Schwarzschild spacetime. Light bends past the photon sphere, massive orbits precess into rosettes, and anything crossing the ISCO plunges to the event horizon.
↗Buoyancy & Stability
Sutherland-Hodgman polygon clipping computes the submerged volume on every frame to simulate exact buoyant forces and torques. Discover why some shapes float flat while others capsize.
↗Bifurcation & Chaos
One equation, one parameter, infinite structure. Watch the logistic map period-double its way to chaos, zoom into self-similar Feigenbaum cascades, and read the Lyapunov exponent flip sign at the edge of order.
↗Chladni Figures
Two thousand grains of sand driven by the gradient of the time-averaged displacement field settle onto the nodal lines of a vibrating plate. Dial in mode (m, n), watch the Chladni pattern reconstruct, mix two modes for richer superposition geometry.
↗Magnetic Pendulum
A damped pendulum above three magnets always settles — but which magnet it chooses depends with fractal sensitivity on where it started. Every boundary point is simultaneously on the edge of all three basins. Click the map to trace a live trajectory to its attractor.
↗Harmonic Oscillator
The ubiquitous model of physics: a mass on a spring. Whether describing molecules, pendulums, or quantum fields, the mathematics of restoring forces are universal. Watch the phase space spiral inward to an attractor or lock onto a driving force resonance.
↗Kuramoto Oscillators
N phase oscillators with random natural frequencies coupled through a mean field. Below Kc = 2γ they drift independently; above it a macroscopic locked cluster crystallises and the order parameter jumps — a continuous phase transition with exact mean-field theory. The same equations govern fireflies, cardiac pacemakers, and power-grid stability.
↗Cellular Automata
Wolfram's 256 elementary 1D rules — a three-cell neighbourhood, a single byte of logic, and the full spectrum from uniform order to fractal self-similarity to proven Turing universality. Rule 30 is cryptographically unpredictable; Rule 90 is a Sierpiński triangle; Rule 110 can simulate any computation that can be computed.
↗Game of Life
Conway's classic zero-player game on an infinite 2D grid. From just three rules governing survival, birth, and death, watch complex self-organising structures like gliders, oscillators, and guns emerge from the primordial soup.
↗Electric Field Lines
Place positive and negative point charges and watch Coulomb's law draw itself: field lines radiate from sources, terminate on sinks, and are everywhere perpendicular to the equipotential contours. Superposition, Gauss's law, and saddle-point topology all visible in a single click.
↗Boids Flocking
Three steering rules — avoid collisions, match headings, stay with the group — produce flocking indistinguishable from starling murmurations. No leader. No global plan. Watch the Vicsek order parameter jump as coherent motion crystallises out of random noise.
↗Pendulum Wave
N pendulums tuned to complete successive integers of oscillations in a fixed period. Released in phase they produce a travelling wave that decays into apparent randomness, then reforms, reverses, and breathes — all from simple harmonic motion and arithmetic. Nothing nonlinear needed.
↗Wave Interference
Coherent point sources emit circular waves whose superposition paints hyperbolic fringe patterns in real time. Drag sources to reshape the field; switch between in-phase and anti-phase configurations; add up to six sources for phased-array topology. Young's double-slit is just the default scene.
↗Julia Set Explorer
Every point c in the complex plane generates a distinct Julia set Jc for the map z → z²+c. Click the Mandelbrot navigator to pick c and watch Jc switch from a connected fractal to a Cantor dust the moment c crosses the Mandelbrot boundary — the dichotomy theorem, live.
↗KdV Solitons
The Korteweg–de Vries equation balances nonlinear steepening against cubic dispersion, producing solitary waves that pass through each other without deformation. Pseudospectral RK4 integration on a periodic domain with 2/3-rule dealiasing. The x–t space-time diagram reveals their particle-like worldlines and the phase shift at each collision.
↗Spacetime & Lorentz Boosts
Minkowski diagrams with live Lorentz boosts. Drag β = v/c and watch the coordinate axes tilt toward the light cone — the geometry of simultaneity breaking, time dilation, and length contraction made visible. Three scenarios: two observers, simultaneity of distant events, and the twin paradox with proper-time accounting.
↗Diffusion-Limited Aggregation
Release a Brownian walker. When it brushes the cluster, it freezes forever. Repeat. What grows is not random but a fractal crystal — the same geometry as lightning, coral, electrodeposition dendrites, and snowflake arms, all from two rules: walk randomly, stick on touch.
↗Hard Disk Gas
N elastic disks. Two conservation laws. Nothing else. Watch Maxwell-Boltzmann emerge spontaneously from Newtonian mechanics — the live speed histogram converges to the exact 2D Rayleigh distribution as collisions drive the system to maximum entropy. Dial temperature up and the peak shifts as √T; pack the box and glassy correlations appear.
↗Standard Map
Chirikov's twist of the torus — the cleanest example of Hamiltonian chaos in two dimensions. Below Kc = 0.971635 invariant KAM tori partition phase space into nested rings of quasi-periodic motion; above it the last golden-mean torus shatters and the stochastic sea connects globally. A live tangent-vector estimator paints the largest Lyapunov exponent orbit-by-orbit. Click anywhere to launch a new trajectory.
↗BTW Sandpile
Drop grains on a 192 × 192 lattice; whenever a column hits four it topples to its neighbours and an avalanche cascades. With nothing to tune, the pile organises itself onto the critical state — and the live log–log histogram of avalanche sizes locks onto the famous P(s) ∼ s−τ power law. The same statistics govern earthquakes, neuronal cascades, forest fires, and 1/f noise.
↗Spring Pendulum
A mass suspended from an elastic spring. Two degrees of freedom, one Lagrangian, and a deeply nonlinear coupling that allows energy to slosh continuously between pendular swinging and radial bouncing. Drag the bob and watch chaotic trajectories unfold from the simplest of setups.
↗Cloth Simulation
A grid of point masses connected by structural springs, integrated using Verlet integration. This provides an incredibly stable and fast numerical method for simulating soft bodies and fabrics. Drag the fabric with your pointer to tear constraints or observe the dynamic rippling of the mesh.
↗Galton Board
Drop a ball into a triangular array of pegs. At each peg, the ball bounces left or right with equal probability. What seems like chaotic tumbling at the micro scale transforms into elegant order at the macro scale. As thousands of balls accumulate in the bins below, the binomial distribution emerges—a live demonstration of the Central Limit Theorem.
↗Articles
Spirograph: The Mathematics of Rolling Circles
An introduction to the Spirograph, explaining how rolling circles create complex hypotrochoids and epitrochoids, explained for high schoolers.
↗Barnsley Fern: Nature's Math
An introduction to the Barnsley Fern, explaining how a simple Iterated Function System generates a mathematically accurate model of a fern, explained for high schoolers.
↗Magnetic Pendulum: Unpredictability and Chaos
An introduction to the magnetic pendulum experiment, explaining chaos theory and fractals, explained for high schoolers.
↗Lotka-Volterra: The Dance of Predators and Prey
An introduction to the Lotka-Volterra predator-prey equations, explaining non-linear dynamics and ecological balance, explained for high schoolers.
↗Pendulum Wave: The Dance of Time
An introduction to the pendulum wave experiment, explaining how simple harmonic motion and careful timing create mesmerizing, dancing patterns, explained for high schoolers.
↗Chladni Figures: Seeing Sound
An introduction to Chladni figures, explaining how sound waves and vibrations create beautiful geometric patterns on metal plates, explained for high schoolers.
↗Harmonograph: Drawing with Pendulums
An introduction to the harmonograph, explaining how coupled damped pendulums trace complex Lissajous curves based on musical ratios, tailored for high schoolers.
↗Lindenmayer Systems: The Geometry of Growth
An introduction to Lindenmayer Systems, explaining how simple string-replacement rules and turtle graphics can model the complex, emergent growth of fractal plants.
↗The Chaos Game: Order from Randomness
An introduction to the Chaos Game, explaining how a simple geometric rule involving randomness generates perfect fractal structures like the Sierpinski Triangle, tailored for high-schoolers.
↗Diffusion-Limited Aggregation: Growing Fractals
An introduction to the mathematics of diffusion-limited aggregation, explaining how random walks and simple sticking rules create complex fractal geometries like lightning and coral.
↗Ising Model: The Mathematics of Magnets
An introduction to the Ising Model, explaining how spontaneous magnetization, domains, and fractal phase transitions emerge from a simple grid of spins.
↗Langton's Ant: Chaos to Order
An introduction to Langton's Ant, explaining how a simple set of rules leads to complex, chaotic behavior that eventually builds an ordered highway.
↗Conway's Game of Life: Emergence and Complexity
An introduction to Conway's Game of Life, explaining how infinite complexity, gliders, and Turing completeness emerge from three incredibly simple rules.
↗Kuramoto Oscillators: Spontaneous Synchronization
An introduction to the mathematics of spontaneous synchronization, explaining how fireflies, heart cells, and metronomes manage to find a shared beat without a leader.
↗Turing Patterns: How Chemistry Paints Biology
An introduction to reaction-diffusion systems and Turing patterns, explaining how simple math creates complex biological shapes for high schoolers.
↗The Harmonic Oscillator
An introduction to the simple harmonic oscillator, the fundamental model describing restorative forces and periodic exchange of energy in physics.
↗The Double Pendulum: A Study in Chaos
An introduction to deterministic chaos and Lagrangian mechanics through the lens of the double pendulum, explained for high schoolers.
↗Self-Organised Criticality: The BTW Sandpile
An introduction to self-organised criticality through the lens of the Bak-Tang-Wiesenfeld sandpile model, explained for high schoolers.
↗What is Eureka?
A living notebook of small, precise simulations and engineering experiments. Learn what this website is and how to use it.
↗Fluid Dynamics: The Mathematics of Flow
An introduction to fluid dynamics, the Navier-Stokes equations, and the Kelvin-Helmholtz instability for high school students.
↗Emergence: Unpacking the Boids Algorithm
An in-depth look at Craig Reynolds' boids algorithm and how three simple rules produce complex emergent behavior.
↗Understanding Electric Fields: An Invisible Web
An introduction to electric fields, Coulomb's law, and equipotential lines for high school students.
↗Hard Disk Gas: Understanding Molecular Dynamics
An introduction to statistical mechanics and the Maxwell-Boltzmann distribution through the lens of a simple 2D hard disk gas, explained for high schoolers.
↗The Butterfly Effect: Understanding the Lorenz Attractor
An introduction to deterministic chaos, the butterfly effect, and the Lorenz Attractor, explained for high school students.
↗N-Body Gravity: The Cosmic Dance
An introduction to gravitational fields, Newton's laws, and the complex celestial mechanics of N-body systems for high school students.
↗Cellular Automata: Computation from Simplicity
An introduction to Wolfram's 256 elementary 1D cellular automata, emergence, and how simple rules can create infinite complexity.
↗The Mandelbrot Set: The Boundary of Chaos
An introduction to the Mandelbrot Set, complex dynamics, and fractals, explained for high schoolers.
↗Black Hole Geodesics: Navigating Curved Spacetime
An introduction to general relativity, curved spacetime, and black hole geodesics for high school students.
↗Fourier Series: Drawing with Circles
An introduction to the Fourier Transform and how complex curves can be decomposed into a chain of rotating epicycles.
↗Cloth Simulation: Verlet Integration in Practice
A grid of point masses connected by structural springs, integrated using Verlet integration. This provides an incredibly stable and fast numerical method for simulating soft bodies and fabrics.
↗Galton Board: The Mathematics of Chance
An introduction to the Galton Board, the binomial distribution, and the Central Limit Theorem, explained for high schoolers.
↗Wave Interference: When Waves Collide
An introduction to classical wave superposition, wave interference, and the geometry of fringes, explained for high schoolers.
↗Wave Packet: Quantum Dynamics
An introduction to the quantum wave packet and the time-dependent Schrödinger equation, explained for high schoolers.
↗Quantum Double-Slit
An introduction to the quantum double-slit experiment, matter waves, and quantum interference, explained for high schoolers.
↗About
Eureka is a workshop, not a portfolio. Each entry is an attempt to take an idea from physics, mathematics, or engineering and render it interactive in as few lines as the problem permits. The goal is clean tone, low noise — the equivalent of a well-tuned amplifier.
New experiments land irregularly. Old ones get refined. Nothing ships rough.
The project is maintained by Archimedes, acting as the Chief Orchestrator. Our overarching goal is truth in simulation, elegance in presentation, and absolute reliability across our pantheon of experiments.
Technically, Eureka is built using Vanilla HTML, CSS, and JavaScript. There is no build step and no framework. It is served as static files. Every simulation runs entirely in the browser, ensuring nothing leaves the device. Mathematical equations are beautifully rendered using KaTeX.