Math Simulations

Spin a spinner, flip a coin or roll a die and watch two sets of bars: the chances the shapes predict, and the results you actually got. Ten spins prove nothing; a few hundred and the two are hard to tell apart.

Type any function and see its shape straight away. Plot up to three at once in three colours to compare them, drag the trace point to read a value off a curve, and see where each one crosses zero.

Two bodies under gravity trace an ellipse forever and you can write the formula down. Add a third and no such formula exists. Run the figure-eight orbit, nudge one mass, and watch a stable pattern become unpredictable while the total energy stays put.

Approximate the area under a curve with strips and choose how each takes its height: left edge, right edge, middle, or a sloping top. Push the strip count up and watch the error fall, with the running total plotted as its own curve.

Move m and watch the line tilt; move b and watch it slide. Or drag the two points on the line and read the equation it writes back. A rise-over-run staircase climbs the line so slope becomes something you can count.

Two paths start a hair apart and follow the same three equations. Watch them agree, then part onto opposite wings of the butterfly. Drag the start, spin the shape and pull rho down until chaos stops.

Pick sin x, cos x, eˣ, ln(1 + x) or 1/(1 − x), raise the degree one term at a time, and watch a polynomial creep outward until it hugs the curve. A shaded strip shows where it is inside your tolerance, and where the radius of convergence stops it dead.

Two identical double pendulums are released a tenth of a degree apart. They move as one, then separate completely, while a log-scale graph shows the gap doubling at a steady rate. Start it gently and it is not chaotic at all.

Add sine waves one at a time and watch a square, sawtooth or triangle wave appear. The target sits behind the running total so you can see what is still missing, including the overshoot at a jump that never goes away.

A pendulum you can launch rather than just release: choose the starting angle and the starting spin. Beside it the phase plane plots angle against angular velocity, so a swing draws a closed loop and a full rotation draws an open track.

A differential equation drawn as a field of little dashes becomes a landscape you can read. Drop a starting point and watch Euler's method walk it, corners and all, beside a finely stepped reference curve. Halve the step and halve the error.