Interactive Maths & CS Visualizations
Hand-built, browser-native applets — nothing is a canned plot. Every curve, field and simulation runs live from the real equations, from calculus and probability through to machine learning, dynamics and distributed systems.58 labs
Arc Length as a Rolling Odometer
Roll a wheel along a curve and its trip-counter reads the arc length — the exact value of the integral. Every tiny turn of the wheel adds a hypotenuse: flat stretches barely move the dial, steep climbs spin it fast. Drag the curve into hills and valleys and watch the odometer update live.
Open lab →A synth voice — the same knobs nature turns
Every sound is a wiggle in air pressure. Here you build that wiggle from scratch and hear it live: a raw oscillator is shaped by an ADSR envelope, carved by a filter, then thrown into delay and reverb. The whole chain is the real Web Audio API graph running in your browser.
Open lab →Bayes' Theorem as Shrinking Areas
A doctor tells you a test for a rare disease is 99% accurate and yours came back positive. How scared should you be? Draw the whole population as a unit square and watch the posterior fall out of the shrinking areas.
Open lab →The Hidden Geometry of Number Theory
Fifteen guided workbenches. Each tab walks you through one idea step by step, then hands you the sliders — drive the parameters yourself and watch the structure hidden inside the arithmetic come out of the noise.
Open lab →Collatz Orbits as a Growing River-Delta Tree
Take any whole number: if even, halve it; if odd, triple it and add one. Run the rule backwards from 1 and the branches form a vast tree — drawn outward it looks exactly like a river delta seen from orbit.
Open lab →A Complex-Function Playground
A complex function is best understood not as algebra but as a motion of the plane. Type any f(z) and watch a grid of little squares bend into a lattice of curved cells — the picture is the function.
Open lab →Complex Functions as Domain Colouring
A function of a complex variable maps a 2D input to a 2D output. Domain colouring squeezes both onto one canvas by painting phase as hue and magnitude as brightness — zeros become vortices, poles blaze white.
Open lab →Complex Manifolds & the Riemann Hypothesis
How a conjecture about a sum ends up living on a curved space. Glue charts into manifolds, turn lattices into cubic equations, count points over finite fields — and cross the bridge where RH is already a proven theorem.
Open lab →Consensus algorithms: how strangers agree without a boss
Thousands of computers that have never met, some of them lying, need to agree on one shared history with no referee anyone trusts. Run Proof of Work, Proof of Stake and Byzantine Fault Tolerant voting live.
Open lab →Continued Fractions & the Most Irrational Number
Every real number has a unique fingerprint: peel off its whole part, flip the leftover, repeat. The golden ratio's coefficients are all 1s — no lucky shortcuts, making it the most irrational number of all.
Open lab →Control theory for robots: closing the loop
A robot never moves perfectly. Five genuine simulators — PID/LQR, Model Predictive Control, sliding-mode control, trajectory planning and null-space redundancy — show how the loop actually gets closed.
Open lab →Conway's Game of Life: complexity from three rules
A grid of cells, each alive or dead, obeying a rule you could teach a child. No memory, no plan, no leader — and yet gliders crawl, guns fire, and in principle a working computer emerges.
Open lab →Eigenvectors — the axes a matrix doesn't rotate
A 2×2 matrix shears, spins and stretches the plane. A special few arrows come out pointing exactly along the line they started on — only longer or shorter. Drag the matrix entries and watch the eigenlines snap into place.
Open lab →Entropy as a Guessing Game
I've hidden one symbol. You may only ask yes/no questions — how many will it take? The average number a perfect strategy needs is exactly Shannon's entropy, measured in bits.
Open lab →The ε–δ Machine
The definition of a limit, made into a game. Pick a tolerance ε, the machine tries to respond with a δ narrow enough to fit inside your band. Shrink ε and watch δ chase it down.
Open lab →Euler's Formula — V − E + F = 2
Count corners, edges and faces of any solid ball-like shape and the answer never budges from 2 — until you punch a hole through it. A topological fingerprint that only cares about holes.
Open lab →Fourier Epicycles — draw your name
Any closed curve is just a sum of circles turning at whole-number speeds, riding tip-to-tail. Draw your own with the mouse, or type a word, and watch a blur of circles resolve into your handwriting.
Open lab →The Fourier / uncertainty trade-off
A signal lives in time and frequency at once. Squeeze the pulse sharp in time and its spectrum bulges wide — you can't make both sharp together. That floor is the mathematics behind Heisenberg's uncertainty principle.
Open lab →A Gallery of Visual Proofs
Some theorems are true because you can see that they must be. Drag, slide and reassemble classic "proofs without words" while a running area readout confirms nothing was created or destroyed.
Open lab →The Galton Board → Gaussian
Drop a bead onto a triangle of pegs — pure chance at every row. Pour thousands through and a smooth bell curve assembles itself: the Central Limit Theorem made of falling marbles.
Open lab →Mechanical Advantage — you never get force and speed for free together
Gear trains, block-and-tackle and levers all do one trick: trade force for speed or speed for force, never both. Drag the tooth counts, pulleys or fulcrum and watch the work-in = work-out meter refuse to move.
Open lab →Visual Teaching Ideas — Number Theory Menu
Number theory reads as the least visual branch of mathematics — no shapes, no space, just arithmetic. That's exactly why it's such fertile ground for interactive teaching: the structure is hidden inside the arithmetic.
Open lab →The Hidden Geometry of Number Theory (v2)
Fifteen guided workbenches, revisited. Each tab walks you through one idea step by step, then hands you the sliders — drive the parameters yourself and watch structure come out of the noise.
Open lab →Gradient Descent — Skiing a Loss Landscape
Training a model is a ball rolling downhill. Pick a landscape — a gentle bowl, a rugged field of minima, or the Rosenbrock banana — tune the learning rate and momentum, and watch it stall, diverge, or find the floor.
Open lab →Graph theory: colouring & the seven bridges
Euler answered the Königsberg bridges puzzle by throwing the map away — keep only what touches what. Walk the bridges, colour a graph down to its chromatic number, and four-colour a map that seems to demand five.
Open lab →Group Theory as a Symmetry Puzzle
The complete list of moves that leave a shape looking untouched is its symmetry group. Drive the moves yourself, trace a Cayley graph, and fill in the multiplication table.
Open lab →The Heat Equation Smoothing a Jagged Bar
Draw a spiky temperature profile onto a metal bar and let the heat equation run. The jagged edges melt away — underneath, a chorus of sine eigenmodes fades, the sharp high-frequency ones fastest.
Open lab →The Hopf Fibration
The 3-sphere can be swept out entirely by circles, one over every point of the ordinary sphere. Project them down from 4D and every pair turns out linked, like rings in chain mail.
Open lab →Predict the next token, very well, at scale
A live byte-pair tokenizer, a skip-gram embedding map, scaled dot-product attention and a real softmax classifier — all training in your browser. Press TRAIN ONE EPOCH and watch the loss drop.
Open lab →Hyperbolic Geometry in the Poincaré Disc
The whole infinite hyperbolic plane, squeezed into one finite circle. Straight lines curve into arcs and triangle angles add up to less than 180°. Drag anywhere to glide through a space that never runs out of room.
Open lab →Integration & Differentiation — the two are one machine, run backwards
Pick a curve, watch its derivative and its accumulated area draw themselves live. The slope of the area curve is exactly the height of the original — the Fundamental Theorem of Calculus, watchable.
Open lab →A Glowing Lattice Knot
A torus knot rendered live in WebGL. Tune the winding numbers p and q, the tube thickness and segment count, and watch the same closed curve rebuild itself into a different knot every time — with real bloom post-processing lighting the lattice.
Open lab →Linear Algebra Simulation Lab
Drag vectors, edit matrices and watch the core machinery of linear algebra respond live — spans, transformations and determinants explored as motion, not static proofs on a page.
Open lab →Same shape, different scale
Bacteria, earthquakes, loudness, money: quantities that grow so violently ordinary graph paper can't hold them. Switch to a log scale and watch the impossible curve snap into a straight line.
Open lab →The Lorenz Butterfly — Sensitive Dependence
Two trajectories start a millionth apart, obeying the same deterministic equations. Without any randomness they peel apart onto opposite wings of the butterfly. Watch the Lyapunov clock count down to unpredictability.
Open lab →Four models, one idea: descend the error
A hand-rolled perceptron, a skiing loss landscape, a tiny two-layer network and a polynomial overfit demo — no libraries, no faked plots. Press TRAIN ONE EPOCH and watch the loss fall.
Open lab →Matrices — machines that move space
A matrix is a verb: an instruction that grabs the plane and stretches, shears, spins or flips it. Drag the entries of A and watch the coloured grid deform; the shaded square's area is the determinant.
Open lab →Seven hard questions, seven small pictures
Seven problems, a million dollars each, one solved in 25 years. Each tab is an honest visual teaser that computes the real object live, then explains plainly why the obvious picture isn't a proof.
Open lab →The Möbius Strip & Orientability
A strip with a single half-twist has only one side and one edge. Send a bug crawling along it and it returns mirror-flipped. Cut it down the middle and get one longer, double-twisted loop.
Open lab →Modular Arithmetic as a Clock & the Discrete-Log Trap
Wrap the numbers around a circle and multiplication draws cardioids and rose windows. Swap in repeated multiplication and the tidy pattern shatters — that hardness is what guards your bank login.
Open lab →Monte Carlo Estimating π
Throw darts, blindfolded, at a square board with a quarter-circle painted on it. The fraction that land inside, times four, is π. Watch a deterministic constant assemble itself out of pure chance.
Open lab →Non-Euclidean Geometry — the parallel postulate was a choice
Euclid's fifth postulate isn't a truth, it's a dial. Turn it and get the sphere, where triangles bulge past 180°, or the hyperbolic plane, where they pinch below it. Switch tabs to switch worlds.
Open lab →The Hidden Geometry of Number Theory — Full Tabbook
A complete, self-contained edition of the number-theory tabbook — every guided workbench, slider and structure bundled into one page, nothing to load but this file.
Open lab →The p-adic / Dyadic Spectrum
How divisible is a number by 2? Flip your whole idea of distance: two numbers are p-adically close when their difference is richly divisible by p. Drive the spectrum, then walk the p-adic tree.
Open lab →Quantitative finance: pricing and risk you can drag
Black–Scholes turns volatility and time into an option's price and Greeks; portfolio theory traces the efficient frontier; Value-at-Risk runs a live Monte-Carlo of tomorrow's P&L.
Open lab →Quaternion Lab — 4D Numbers That Rotate 3D Space
Master 3D rotations the way they're actually done under the hood — unit quaternions, not Euler angles or gimbal-locked matrices. Drag the axis, spin the angle, and watch composing rotations become quaternion multiplication.
Open lab →Random Walks & the √n Law
A thousand drunkards leave the same lamppost, each stepping randomly. One walker is unpredictable; the whole swarm spreads as the square root of time — the reason ink blooms slowly in water.
Open lab →Raytracing — light as a tree of bounces
A photorealistic image is computed backwards: one ray per pixel, splitting into reflected, refracted and shadow rays at every surface. Watch the tree of light grow through mirrors and glass.
Open lab →Raytracing: light as a tree of bounces
A raytracer shoots one ray from your eye through each pixel and asks what it hits first. Mirrors and glass spawn child rays — a whole scene raytraced live on your GPU, one bounce at a time.
Open lab →The Riemann Hypothesis
A hands-on research lab for the most famous unsolved problem in mathematics. Find the zeros yourself on the Riemann–Siegel curve, watch primes assemble out of them, and test their statistics.
Open lab →Robot Arm Kinematics — Forward, Inverse & the Jacobian
Forward kinematics is one honest trigonometry answer; inverse kinematics is a negotiation — several arms give the same hand, some hands can't be reached, and the Jacobian goes deaf at singularities. Drag the target and watch the arm argue with the mathematics live.
Open lab →Spectral analysis — every signal is a sum of sinusoids
Build or draw a signal once, then view it through four lenses: its live spectrum, the effect of a window, a scrolling spectrogram, and what a filter does to it. Every FFT is computed for real.
Open lab →Taylor Series — Peeling an Apple
Peel thin polynomial layers around a single point and they wrap a smooth curve perfectly, locally. Drag the expansion point, add terms, and watch the radius of convergence hold the line.
Open lab →Transformations — the space between two poses
Give a computer a start and an end and ask for the in-between. Lerp walks the straight chord, slerp swings the true arc, easing bends time. How you interpolate is a design decision, not a law of nature.
Open lab →Trigonometry — every identity is a triangle in disguise
Sine and cosine are the shadows of a point walking around a circle. A single shared angle dial drives the wave, the identity and the circle at once — nothing is a canned plot.
Open lab →The Ulam Spiral Lighting Up Primes
Wind the whole number line into a square spiral and shade in the primes. What should be a random dusting instead snaps into diagonal streaks — the tracks of astonishingly prime-rich quadratics.
Open lab →Vector Fields & the Flow of an ODE
A first-order system assigns an arrow to every point of the plane — a wind map. Edit the field, seed particles with a click, and watch the applet find nullclines, fixed points, and their stability.
Open lab →Where Am I? — The Kalman Filter as a Shrinking Bet
A robot never knows its exact position — it keeps a Gaussian bet that odometry pushes forward and landmark sightings pull back. Poison the sensors with noise, hit kidnap, and watch a confident filter go wrong, then reel itself back in.
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