Notes

Hypotrochoids: the maths behind Spirograph

JAEV Notes · October 2026 · 2 min read

A Spirograph set in its red plastic tray, seen from above: two large transparent yellow rings with teeth inside and out, two toothed bars, eighteen round toothed wheels of different sizes full of small pen holes, and four ballpoint pens.
Spirograph set made by Palitoy, bought in the UK in the early 1980s. Photo: Multicherry, CC BY-SA 3.0, via Wikimedia Commons. Source

A hypotrochoid is the curve drawn by a point on a disc that rolls around the inside of a fixed circle. It is the curve of the Spirograph, the drawing toy in which a toothed plastic wheel with a pen in one of its holes is rolled round the inside of a toothed ring.1, 2 The British engineer Denys Fisher made the first prototype from Meccano parts, and the toy went on sale in Leeds in March 1965.3

The toy

According to his obituary in The Times, Fisher "spent the Christmas of 1962 building, with his family, various drawing machines from Meccano pieces", and made his prototype "from a Meccano cog and a geared ring". Fisher then reached an agreement with the toy company Kenner, and the obituary puts total sales at more than 100 million sets in 85 countries.3

The curve

Let R be the radius of the ring, r the radius of the wheel, and d the distance from the wheel's centre to the pen. As the wheel's centre goes round the ring by an angle t, the pen is at

x = (R − r) cos t + d cos((R − r)/r · t) y = (R − r) sin t − d sin((R − r)/r · t)

In each line, the first part is the wheel's centre travelling round the ring, and the second is the pen turning about that centre as the wheel rolls.2 If the pen sits exactly on the rim of the wheel, the curve has sharp points and is called a hypocycloid.

Counting the teeth

Because the parts have teeth, the radii are in the same proportion as the numbers of teeth, and those numbers are printed on the parts. With N teeth on the ring and n on the wheel, the drawing closes after LCM(n, N) / N trips round the ring and has LCM(n, N) / n points, where LCM is the least common multiple.2 A wheel of 60 teeth in a ring of 96: the least common multiple is 480, so the pen goes round the ring 5 times and the figure has 8 points. Where the pen sits, d, changes the shape of the loops but not their number.

Live sketch by JAEV: a hypotrochoid drawn from the formula above. Each new drawing picks a ring (96, 105, 144 or 150 teeth, JAEV's choice, not the toy's parts list), a wheel of 24 to 84 teeth and a pen distance at random, and draws until the curve closes. Below, the number of trips round the ring and of points, from the least common multiple. Code CC BY 4.0.

A machine for repetition

A Spirograph drawing is made by one rule repeated exactly, with a single number deciding the whole result before the pen moves. Other drawing machines worked with pendulums instead of gears, see Lissajous figures and the harmonograph. For rules that make images, see Generative art.

Film by JAEV, 37 seconds: a hypotrochoid, the curve of the Spirograph, drawn as a pen plotter would. The sound is a felt-tip pen on paper. CC BY 4.0.

Sources

  1. “Spirograph”. Wikipedia
    Curves are “hypotrochoids and epitrochoids”; Bruno Abakanowicz invented a spirograph device between 1881 and 1900; the Marvelous Wondergraph in the Sears catalog (1908); with the pen on the rim of the wheel (l = 1) the curves are hypocycloids.
  2. Ashleigh Ratcliffe and Jenny Power, “On the cover: Spirographs”. Chalkdust, 8 November 2024
    Hypotrochoids are “roulette curves which are created when a circle rolls around the inside of a fixed circle”; x(t) = (R − r) cos t + d cos((R − r)/r · t), y(t) = (R − r) sin t − d sin((R − r)/r · t); with N teeth on the ring and n on the cog: rotations LCM(n, N)/N, spokes LCM(n, N)/n.
  3. “Denys Fisher”, obituary. The Times, 26 October 2002 (transcribed by the Doctor Who Cuttings Archive)
    Born in Leeds 11 May 1918, died 17 September 2002. “spent the Christmas of 1962 building, with his family, various drawing machines from Meccano pieces”; “created his prototype Spirograph from a Meccano cog and a geared ring”; “In March 1965 the first Spirograph went on sale in Schofield's department store in Leeds”; agreement with Kenner; “more than 100 million units in 85 countries”.

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JAEV Notes, October 2026.
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JAEV Notes, “Hypotrochoids: the maths behind Spirograph”, jaev.gallery/notes/spirograph-hypotrochoid/, 2026.
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