Modular Times Table
Place the numbers 0 to N−1 evenly around a circle, then draw a chord from every point i to (i×k) mod N. This single rule paints startling curves: the times table for 2 traces a cardioid, 3 a nephroid, and other multipliers weave nested petals and stars. Sweep the multiplier and the number of points, or animate the multiplier to watch one pattern morph smoothly into the next — the envelopes of the chords are the epicycloids hiding inside multiplication.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- Each point i is joined to (i×k) mod N; the curve you see is the envelope those chords trace.
- Multiplier 2 gives a cardioid, 3 a nephroid, and k gives an epicycloid with k−1 cusps.
- Raising N sharpens the envelope because more chords sample the same underlying curve.
- Runs 100% in your browser — simulations are computed locally on your device.
Related tools
3D Geometry Board
Points, lines, planes, spheres and polyhedra on an orbiting camera — slice solids with planes, unfold nets, drop the Dandelin spheres into a cone and drag everything.
Bézier Curve Playground
Drag control points and watch De Casteljau’s algorithm construct a Bézier curve step by step.
Central Limit Theorem Demo
Sample from any lopsided distribution and watch the averages pile up into a bell curve.
Conway’s Game of Life
Paint cells and watch Conway’s cellular automaton breed gliders, guns and oscillators.
Dynamic Geometry Board
Construct with points, lines, circles and transformations, then drag anything — the construction stays true, and a checker tells you which relations are theorems.
Fantasy Map Generator
Roll a seed and get a continent: coastlines, mountains, rivers, biomes and named towns, ready for your campaign.