Math Ladder · Pictures you can play with
Math Gallery
Some ideas in math are easiest to meet as pictures. Here you can zoom into a fractal, watch two pendulums part ways, drop a thousand balls through pegs, or play a game show a thousand times. Each piece starts with what you’re seeing, then explains why it works, step by step.
24 pieces
- The Mandelbrot set One tiny rule, repeated, draws a shape with endless detail. Zoom in as far as you like.
- Julia sets Every point of the Mandelbrot set has a Julia set of its own. Move the point and watch the shape change.
- The double pendulum Two pendulums start a millionth of a radian apart. Within seconds they go their own ways.
- The Lorenz attractor Three simple equations from a weather model trace a looping, never-repeating path. Turn it around in 3D.
- The logistic map and the road to chaos A one-line population model settles down, then splits in two, then four, then turns to chaos.
- The Galton board Each ball bounces left or right at random, yet together they always build the same bell-shaped pile.
- Buffon's needle Drop needles on a striped floor and count how many cross a line. Out comes π.
- The Monty Hall problem Three doors, one car, and a host who opens a door with a goat. Should you switch? Play it, then run 1,000 games.
- Random walks: will you find your way home? Take steps in random directions. On a line or a flat grid you are sure to get back home. In space, you might not.
- The birthday problem In a room of just 23 people, it's more likely than not that two share a birthday. See why.
- The Koch snowflake and the Sierpinski triangle A shape with an endless edge around a finite area, and a triangle with holes all the way down. Plus: what is a fractal dimension?
- The golden angle and sunflower seeds Turn by 137.5° before placing each new seed, and the seeds pack perfectly. Turn by a tidy fraction, and they don't.
- Space-filling curves A single line, folded again and again, that ends up passing through every point of a square.
- The 4D hypercube We can't build a four-dimensional cube, but we can draw its shadow, the same way a drawing of a cube is a shadow of a 3D shape.
- The Möbius strip and the Klein bottle A strip of paper with only one side, and its closed-up cousin, a bottle with no inside.
- Penrose tilings Two diamond-shaped tiles that cover the whole floor with a pattern that never repeats.
- The Ulam prime spiral Write the counting numbers in a spiral and mark the primes. Diagonal lines appear, and nobody can fully explain them.
- Collatz paths Halve it if it's even; triple it and add one if it's odd. Every number tried so far falls to 1, but no one can prove they all do.
- Fourier epicycles Draw any loop. A chain of spinning circles, each turning at a whole-number speed, can trace it.
- The bridges of Königsberg Can you walk across all seven bridges, crossing each one exactly once? Euler showed why you can't, and invented graph theory doing it.
- The four color theorem Any map can be colored with just four colors so that neighbors never match. Proving it took 124 years and a computer.
- The brachistochrone What is the fastest slide from A down to B? Not the straight line. The winner is a curve called a cycloid.
- Conway's Game of Life A grid of cells and three short rules. Out come blinkers, gliders and even a gun that fires gliders forever.
- Still unsolved Four questions about whole numbers that anyone can understand and no one has answered: twin primes, Goldbach, Collatz and the Riemann hypothesis.