Spirograph Gears

About this simulation

This simulation recreates the spirograph toy. A gear rolls without slipping inside a 96-tooth ring, and a pen fixed to the gear draws a hypotrochoid curve. Students pick a gear from 24 to 63 teeth, drag the red pen dot closer to or farther from the gear center and set the rolling speed. Options stack each layer of the drawing in 3D, snap the pen to holes, loop the drawing or redraw it. Readouts show the gear to ring ratio, the pattern, the pen distance and how much is drawn. The number of petals equals the ring teeth divided by the greatest common divisor, linking art with ratios and number theory.

What students can explore

Choose 40 teeth, wait for the drawing to finish and read the Pattern readout for the number of petals.
Change Gear teeth to 45 and compare the petals with the 40 tooth gear.
Drag the red pen dot toward the gear center and see the loops get smaller.
Turn Snap pen to holes on, then pick different holes and press Redraw.
Turn Stack layers off to flatten the drawing, and raise Rolling speed to draw faster.

Learning objectives

1Predict the number of petals from the ring and gear tooth counts.
2Explain how a pen's distance from the gear center changes the curve.
3Use the greatest common divisor to find petals and laps.
4Describe how rolling circles create hypotrochoid curves.

Questions teachers ask

Is it free?

Yes. Spirograph Gears is free, needs no sign-up and runs in any modern browser, including whiteboards, tablets and phones.

How many petals will I get?

Divide the ring teeth, 96, by the greatest common divisor of 96 and the gear teeth. A 40 tooth gear has a divisor of 8, so the pattern has 12 petals.

How do I change the pattern?

Pick a different gear in Gear teeth or drag the red pen dot to a new place on the gear. The drawing restarts using the new setting.

How long until the curve closes?

The curve closes after gear teeth divided by the greatest common divisor laps of the ring. For a 40 tooth gear in the 96 ring that is 5 laps.

Embed this simulation

Drop it into your LMS, blog, or class site — free.

<iframe src="https://kiwibee.io/en/simulations/spirograph-gears-3d-simulation/embed" title="Spirograph Gears | KiwiBee" loading="lazy" allowfullscreen style="width:100%;height:min(70vh,620px);min-height:420px;border:0;border-radius:16px"></iframe>

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From model to evidence

Use a worksheet to capture predictions, observations, and explanations after students explore.

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