Simulations
Interactive simulations, grouped by area. Everything runs live in the browser, drag, tune and watch the maths move.
Calculus
Limits, derivatives, integrals and the geometry behind them.
Derivative as a Limit
Slide the secant line as h shrinks to zero and watch it settle into the tangent and the derivative born from a limit.
Partial Derivatives
Slide a vertical plane through a surface and what it cuts is one ordinary curve. The slope of its tangent is a partial derivative; turn the plane for the other.
Riemann Sums
Approximate the area under a curve with rectangles, then add more and thinner ones and watch the staircase converge onto the integral.
The Sum Rule
Add two functions and their graphs stack. Stack the heights and you stack the growths, so the slopes simply add, with no leftover term anywhere.
The Product Rule
A product is the area of a rectangle. Growing it adds two strips and one small corner, and only the corner is small enough to vanish in the limit.
The Chain Rule
Each stage stretches a small step by its own factor, so stretching twice multiplies the factors. Watch why f′ has to be read at g(x) and nowhere else.
Integration by Parts
Two areas that tile one rectangle: the integral you were asked for and the one you are handed back. Choosing u is choosing which of the two you would rather do.
Differential Equations
An equation for the slope instead of the value. Draw it everywhere and the plane fills with segments; a solution is any curve that stays tangent to them the whole way.
Taylor Series
Add polynomial terms one by one and watch the approximation hug a curve near a point, then fall apart past its radius of convergence.
Fourier Series
Stack rotating circles (epicycles) to build a square, sawtooth or triangle wave from pure sine harmonics.
Laplace Transform
Plot |F(s)| as a landscape over the complex plane, where poles become spikes and the Fourier transform is the slice above the imaginary axis.
Scalar Fields & Gradients
Drag a tiny probe through a 3D density cloud. An arrow shows the direction of steepest increase, with the gradient magnitude alongside.
Lagrange Multipliers
Optimize along a constraint and watch the objective contour kiss the constraint curve, exactly where the two gradients line up.
Line Integrals
Ride a bead along a curve through a field in 3D, adding up only the part of the field that points your way. Then bend the path and see whether the total cared.
Green's Theorem
Chop a wobbly region into little squares, each carrying its own spin. Every shared edge is walked twice in opposite directions, so only the rim survives.
Stokes' Theorem
Tile a surface with little loops and watch every shared edge cancel in pairs until only the boundary is left. Bend the surface and the answer refuses to move.
Gauss's Divergence Theorem
Fill a lumpy region with little boxes and count what leaks out of each. Every inner wall is shared by two boxes and cancels, leaving only the outer skin.
Divergence
Place a tiny material patch in a vector field and watch it expand near sources, contract near sinks, or deform at zero divergence.
Curl
Drop a paddle wheel into a vector field and drag it around. It spins fast in the core of a differential rotation and barely at all out at the rim.
Linear Algebra
Vectors, matrices and the transformations they encode.
Linear Transformations
Drag the matrix entries and watch the plane deform with them, carrying the grid and the two basis vectors along for the ride.
The Determinant
One number for what a matrix does to area. Squash the parallelogram to nothing and the map has thrown a dimension away; push past that and the plane comes back inside out.
Change of Basis
The same arrow, watched through two different grids and written down as two different pairs of numbers. The dictionary between them is a matrix, and a transformation translated is the sandwich.
Eigenvectors & Eigenvalues
Spin a vector around the circle and watch where the matrix sends it. When the input and output line up, you have found an eigenvector.
The Dot Product
One length times the shadow the other casts on it, with the component formula falling straight out of that one picture.
The Cross Product
Two vectors span a patch of area. Its size is the length of the answer, its tilt is the direction, and each component is a shadow of that patch.
Span
Drag two vectors and see everywhere their scaled sums can reach, the whole plane, or a single line when they line up.
Singular Value Decomposition
Watch any matrix turn a circle into an ellipse as rotate, stretch, rotate, the geometry of A = U Σ Vᵀ.
Statistics
Randomness, distributions and the patterns that emerge from them.
Central Limit Theorem
Average samples from any distribution and watch the means pile up into a Gaussian, faster as the sample size grows.
Random Walk
Thousands of aimless walkers spread from a point into a Gaussian whose width grows like the square root of time, the particle view of diffusion.
Joint, Marginals & Conditionals
Every pair of values gets a square whose area is its probability. Drag a box over the table and read the joint, the marginals and a conditional off one selection.
Probability Density
No single value has any probability, only stretches do, so a density is the height whose area is that probability. Drag an interval and read off the integral.
Monte Carlo
Throw random darts at a square with a circle inside and count the hits. The fraction inside estimates π, converging slowly as 1/√n.
Bayes' Theorem
See why a 90% accurate test for a rare disease is usually wrong when it says positive. 1,000 people as dots make the posterior obvious.
Maximum Likelihood
Slide a Gaussian over sample data and watch the likelihood rise and fall. The best fit lands exactly on the sample mean and standard deviation.
Least Squares
Drag data points and fit the best line live. The residuals are drawn as literal squares, least squares minimises their total area.
Wasserstein Distance
Measure the distance between two distributions as the least work to reshape one into the other, the area between their CDFs.
Machine Learning
Clustering, model fitting and the ideas that turn data into predictions.
k-Means
Assign points to the nearest center, move each center to its mean, repeat. Watch clusters snap into place, and sometimes into the wrong place.
DBSCAN
Density-based clustering that finds arbitrary shapes and flags outliers as noise, with no need to choose the number of clusters up front.
Hierarchical Clustering
Merge the closest groups over and over to build a dendrogram, then slice it at any height to read off clusters at that scale.
Linear Models
Regression and classification are the same straight line wx + b wearing two costumes. Change what the score turns into, and one gradient descent fits both.
Overfitting
Raise a polynomial’s degree until it threads every training point yet fails on new data, then tame it with regularization.
The Curse of Dimensionality
Add axes and space stops behaving: data thins out, a ball keeps its volume in a paper-thin crust, and near and far collapse into the same distance.
Activation Functions
Compare sigmoid, tanh, ReLU and friends alongside their derivatives, and see where flat slopes make gradients vanish.
Deep Learning
Optimization, neural networks and the math that trains them.
How Layers Bend Space
Every hidden layer is two units wide, so the space it makes can be drawn. Step through the layers and watch the grid fold until one line separates the classes.
Gradient Descent
Tune the learning rate and the starting point, then watch optimization roll downhill into a minimum, or overshoot it and diverge.
Optimizers
Race SGD, Momentum, NAG, RMSProp and Adam down the same loss, drawn either as a surface you can click on or as a volumetric cloud in three parameters.
Neuron Activations
A network trains itself in front of you, then you draw a shape and watch it go through: every wire lit by what it carries, every neuron opened up to show the picture it looks for.
Embeddings & Cosine Similarity
Every word is an arrow and meaning is the direction it points. Rank the same words by cosine, by raw dot product and by straight-line distance, and watch the level sets turn from rays into rings.
Attention as a Soft Lookup
The mechanism on its own: a dictionary returns one value, attention returns a blend of all of them, weighted by how well each key matches the query. Sharpen it, flatten it, mask a key out.
Self-Attention
Where the queries and the keys come from: three projections of one X, so the sentence looks itself up. The full matrix, one head at a time, down to every multiplication.
Image Convolution
Slide a kernel over an image to detect edges, blur or sharpen, and see how stride, padding and dilation reshape the output feature map.
Generative AI
How models learn the shape of data and generate new samples from it.
The Data Manifold
Explore the enormous space of possible pixel arrays and see how real images occupy only a thin, structured region where probability concentrates.
Byte Pair Encoding
Glue the commonest pair of neighbouring symbols into one, over and over, and watch a vocabulary grow from bare letters into whole words.
Autoregressive Models
A 40-word vocabulary with a probability for every word at every step, and temperature, top-k and top-p deciding which of them survive to be sampled.
Physics
Fields and the equations that move energy through space.
Electric & Magnetic Fields
Orbit a 3D scene of charges and currents, switching between electric field lines and the magnetic field around a wire.
Electromagnetic Waves
Drag charges through space and watch retarded E and B fields ripple outward. Build a dipole antenna and tune the medium to change wave speed and impedance.
Entropy
Count the arrangements behind what you can see. Entropy is a census, S = k ln Ω, and that counting alone is enough to make time run one way.
Heat Equation
Watch temperature diffuse across a room from a radiator and a cold window, and paint in your own heat.
Wave Equation
Pluck a string and watch the pulse travel and reflect. The contrast with heat: waves keep their shape and carry energy instead of smearing out.
Phonons
Every mass obeys Hooke’s law and pulls only on its neighbours. Out comes a pulse travelling at a speed nobody wrote down, and a shortest possible wavelength.
Wave Interference
Two sources of ripples cross and combine into bright and dark fringes. Drag the sources to see the pattern behind the double-slit experiment.
Ideal Gas
A few hundred molecules bouncing in a box. Pressure is the drumming of their impacts, temperature is how fast they are, and PV = NkT is what that adds up to.
Stationary Action
Bend the path a particle takes between two fixed events and watch its action get worse. Rolling that downhill turns out to be Newton in disguise.
Band Theory
Bring a line of atoms together and every level splits into a band. Which bands are full, and how wide the gap is, decides metal, semiconductor or rock.
Schrödinger Equation
Launch a quantum wave packet at a barrier taller than its energy and watch part of it tunnel through, the wave nature of matter in action.
Stern-Gerlach Experiment
A beam of atoms should smear into a band and arrives as two spots. Chain the analysers, turn the next one, and measuring x wipes out what was known about z.
Quantum Gates
Build a circuit on two qubits and watch the amplitudes, their phases and both Bloch arrows move together. Then build the one circuit whose arrows shrink to nothing.
Nuclear Fission
One neutron splits one nucleus and two more fly out. Turn the enrichment, the size and the control rods and watch the chain die, hold steady, or run away.
Nuclear Fusion
Slam light nuclei together hard enough to tunnel through their own repulsion, and chase the one line every star has to cross: power out beating power lost.
Special Relativity
Boost to a moving frame on a spacetime diagram and watch its axes skew: relativity of simultaneity, time dilation and length contraction, all at once.
Four-Vectors
One arrow in spacetime, read by many observers. Boost and both shadows change while the length refuses to: proper time, the speed of light, the rest mass.
General Relativity
Fall down the curved-space funnel of a black hole. Clocks run slow deep in the well and a proper meter swallows less coordinate distance near the horizon.
Electronics
The components a circuit is built from, opened up: charge, fields and carriers doing the work.
Resistor
The field pulls an electron along and the lattice knocks it back to random, over and over. The average of that stop-start crawl is the current, and the winnings the lattice keeps are the heat.
Diodes & Transistors
Pick a diode or a transistor and watch electrons and holes move through it: a one-way valve, then a valve with a knob, both built from a barrier you can raise.
Capacitor
Every charge is a ball you can count. Drive the source by hand and watch electrons get dragged off one plate, round the circuit, and onto the other one.
Inductor
A flywheel for charge. Shove the source and the electrons will not jump, only ramp, and cutting them off makes the collapsing field fight to keep them going.
Memristor
A resistor that remembers the charge that has been through it. Watch the boundary inside the film move as you drive it, and stay put when you stop.
RC, RL and RLC Circuits
One store of energy makes a circuit slow; two make it ring. Watch the transient, the damping and the frequency response all follow the same pair of poles as you move R.
MOSFET & CMOS Logic
A plate of metal that conjures a wire underneath itself and never touches it. Then stack a p-type one on an n-type one and get a gate that costs power only while it changes its mind.