Demos & Visualizations

Newton's method finds a root by following the tangent line. Run it from every point in the complex plane at once, color each by which root it lands on and how fast, and the boundaries between basins turn out to be fractal — infinitely detailed, for a purely deterministic method. Pan and zoom across four polynomial presets.

A pendulum hanging from another pendulum. The single pendulum is predictable; the double pendulum is chaotic. Drag the bobs to set a starting position, adjust masses and lengths, and watch the lower bob trace wild, unpredictable paths. Tiny changes in starting angle produce completely different trajectories within seconds. Integrated with RK4 and explained with the full Lagrangian equations of motion.

A grid of cells, each updating based only on its immediate neighbors. No cell knows the global state. Yet from purely local rules (modular arithmetic, XOR, rock-paper-scissors, Conway's Game of Life, Wolfram Rule 30), large-scale patterns emerge: spirals, fractals, gliders, and structures of striking complexity. Parameterized with configurable rules, states, grid size, initial patterns, update modes, and color schemes.

Place particles with a mass and a charge—positive, negative, or neutral—and watch gravity and the electromagnetic force act simultaneously. Gravity always attracts; charge attracts or repels depending on sign. Click-drag to launch a particle like a slingshot, or load a preset: a gravity-only binary orbit, a three-body dance, an orbit held together by opposite charges instead of mass, or a mixed cluster where both forces compete at once.

The same rule, z → z² + c, produces both pictures: the Mandelbrot set on the left (c varies, z starts at 0) and a Julia set on the right (c is fixed, z varies). Click anywhere on the Mandelbrot to pick a new c and watch its Julia set change instantly. Pan and zoom either one independently.

The Lorenz attractor is a famous example of chaos theory. Watch as simple deterministic equations create beautiful, complex patterns. Adjust parameters to see how tiny changes can lead to wildly different outcomes.

A 2×2 matrix defines a linear transformation of the plane. Manipulate matrix entries directly or choose from preset transformations—rotation, scaling, shear, reflection, projection—and watch the effect on a shape and basis vectors in real time. See the determinant, eigenvalues, and eigenvectors update live. Compose transformations to discover that matrix multiplication is not commutative.

Draw an arbitrary probability distribution—as weird as you like—and watch the distribution of sample means converge to a Gaussian in real time. Adjust the sample size n from 1 to 100 and see convergence speed. Presets include uniform, exponential, bimodal, dice roll, and U-shaped distributions. The theoretical normal curve is overlaid for comparison, with live statistics showing how the standard deviation of means approaches σ/√n.

Draw a signal or choose a preset waveform (square, sawtooth, triangle, chirp) and watch it decompose into individual sine waves in real time. The frequency spectrum shows which components are present; the reconstruction canvas shows how adding them back together approximates the original. Control the number of terms to see convergence and the Gibbs phenomenon at discontinuities.

Drive the same 3D object with roll/pitch/yaw sliders on one side and the equivalent quaternion on the other. Push pitch toward ±90° and watch gimbal lock happen live — roll and yaw collapse onto the same axis. Then animate between two orientations and compare an Euler-angle path against a quaternion SLERP, side by side.

Visualize eight sorting algorithms side by side: Bubble, Selection, Insertion, Merge, Quick, Heap, Shell, and Radix sort. Watch comparisons and swaps happen in real time on arrays up to 300 elements. Compare O(n²) algorithms against O(n log n) on random, reversed, and nearly-sorted data. The speed difference becomes visceral at scale.

Train a neural network on 2D classification problems: circles, XOR, spirals, moons, and clusters. Adjust the number of hidden layers, neurons per layer, activation function (ReLU, tanh, sigmoid), learning rate, and noise level. Watch the decision boundary evolve in real time as gradient descent runs. See the network architecture with weights visualized as colored lines.

Visualize complex functions through domain coloring, where hue encodes argument and brightness encodes magnitude. Drag a contour across poles and watch the integral jump by exactly 2πi times the enclosed residues. Nine preset functions with known poles, zeros, and residues let you verify the residue theorem interactively.

Change of basis in 2D, then the leap to infinite dimensions. Rotate a basis in R² and see how the same vector gets different coordinates. Below, watch how Fourier coefficients are the coordinates of a function in the sine/cosine basis—the exact same mathematical operation, generalized to infinite dimensions.