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Geometry › Make something
Spiral artist
A small wheel rolls inside a big one with a pen attached. Twist the parameters and out come spirograph patterns, snail shells and galaxies.
Spiral artist
A small circle rolls inside a big one with a pen attached. Move the sliders — change the pattern.
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The maths
The shapes you're drawing are hypotrochoids — the path traced by a point on a wheel rolling inside a larger circle. The three numbers control:
- R — the big circle's radius.
- r — the small wheel's radius.
- d — how far the pen sticks out from the wheel's centre.
Pretty patterns from a ratio
The number of 'petals' depends on the ratio R/r. If R/r is a simple fraction (like 5/2 or 7/3), you get a closed pattern. If it's irrational, the path never quite closes — you can keep tracing forever and never repeat.
Real spirograph
The Spirograph toy (invented 1965) does exactly this with plastic gears. The toothed gears guarantee R/r is rational, so you always get a closed, repeating pattern.