Spirograph Online Kit
A spirograph kit in your browser — crank it by hand. Drag the disc to roll the wheel around the ring and watch the line build up stroke by stroke. Swap ring and wheel teeth, the pen hole and ink color, or let it auto-spin, then export your hypotrochoid and epitrochoid art as a PNG.
Editing gears or color never erases your drawing. Hit New layer to freeze the current curve and stack a new one on top; only Clear sheet wipes it.
Gear chain — wheel inside wheel, add as many as you like
Start from an example
Custom pen color (or tap a swatch on the canvas)
Paper
What is a spirograph?
A spirograph traces the path of a pen held inside a small toothed wheel as that wheel rolls around the inside (or outside) of a larger fixed ring. Because the wheel and the ring turn at different rates, the pen never repeats the same line until the gears come back into phase — producing the looping, flower-like curves the classic toy is famous for.
Mathematically these are roulette curves. A wheel rolling inside the ring draws a hypotrochoid; a wheel rolling outside the ring draws an epitrochoid. Three numbers control everything: the ring radius R, the wheel radius r, and the pen offset d from the wheel's center.
The equations
The pen position is swept by the parameter t. The two families differ only by a sign on the ring–wheel sum:
Hypotrochoid (wheel inside ring)
x = (R−r)·cos t + d·cos((R−r)/r · t)
y = (R−r)·sin t − d·sin((R−r)/r · t)
Epitrochoid (wheel outside ring)
x = (R+r)·cos t − d·cos((R+r)/r · t)
y = (R+r)·sin t − d·sin((R+r)/r · t)
When d = r the curve becomes a sharp-cusped hypocycloid or epicycloid; smaller d rounds the petals into smooth loops.
How many petals — and when it closes
The number of petals comes from the ratio of the two radii. If R/r reduces to the fraction p/q, the wheel must make q full trips around the ring before the pen lands back where it started, and the finished pattern shows p − q petals (hypotrochoid) or p + q petals (epitrochoid).
That is why a wheel of r = 60 in a ring of R = 96 closes quickly with a few bold loops, while a co-prime pair like R = 97, r = 60 wanders for dozens of revolutions before closing — filling the disc with dense lace. Picking nearby co-prime radii is the trick behind the most intricate patterns.
How to use this kit
Instead of a single fixed ring and wheel, this kit is a free gear chain: a stack of wheels, each riding on the tip of the one before it (a wheel inside a wheel inside a wheel). Add as many as you like — two gears draw a classic spirograph rose, three or four weave dense compound lace.
- Add & remove gears: build the chain freely. Each gear has a teeth / spin value (how many times it turns per crank; negative reverses it) and a radius. The pen sits at the very end of the chain.
- Crank by hand: drag the disc to turn the whole chain and watch the line build up, or press Auto-spin.
- Pick a color on screen: tap a swatch right on the canvas, or choose any custom pen color below.
- Layer many curves: editing gears or color never erases your drawing. Hit New layer to freeze the current curve and stack a fresh one on top for multi-color art; only Clear sheet wipes everything.
- Ink & paper: set the pen thickness, switch the paper color, or flip on neon glow for an additive light-painting look.
- Save it: export the whole layered sheet as a PNG.
A short history of the toy
The roulette curves behind the spirograph were studied by mathematicians including Albrecht Dürer and the Polish astronomer Ola Nikodym, but the household toy was patented by British engineer Denys Fisher in 1965 and became a runaway hit after Kenner brought it to the United States in 1966. The physical set used plastic gears with numbered teeth; here those teeth become the ring and wheel radius sliders, so you can dial in ratios no toy box ever shipped.