The Mathematics of the Spirograph
A spirograph curve is a hypotrochoid: the path traced by a point on a small circle rolling inside a larger circle. Two parametric equations define every curve: x(t) = (R−r)cos(t) + d·cos((R−r)t/r) and y(t) = (R−r)sin(t) − d·sin((R−r)t/r). Three numbers control everything: R (the outer radius), r (the inner radius), and d (the pen distance from center). Randomize all three and a unique mathematical drawing emerges.
Gear Ratios and Petals
The ratio R/r determines the pattern's complexity. When R/r reduces to p/q in lowest terms, the curve produces p distinct petals and closes after q full revolutions of the inner circle. A ratio of 5/3 yields five petals. A ratio of 8/3 yields eight. Ratios with small denominators close quickly and produce simple, bold patterns. Ratios with large denominators create intricate lace. The number displayed in the Petals statistic below reflects this ratio.
From Toy to Mathematics
The Spirograph drawing toy was patented by Denys Fisher in 1965 and became one of the best-selling toys in history. The mathematical principles date much further: Bruno Abakanowicz built a mechanical spirograph for computing integrals in the 1880s, and the study of epicycloids traces back to ancient Greek astronomy. Ptolemy used nested circular motions to model planetary orbits, the same parametric structure that generates the curves on this page.
Explore Award-Winning Generative Design
The spirograph curves demonstrated above at dice83 show how elegant mathematical formulas can produce intricate visual art. By tracing the path of a point on a circle rolling inside another circle, hypotrochoid equations generate patterns of petals, loops, and lace-like geometries. Small changes to the underlying parameters—radius ratios or pen distance—create entirely new compositions. This blend of geometry, motion, and computation reflects a broader tradition in generative design, where mathematical systems become creative tools. Designers and digital artists frequently explore parametric curves, algorithmic drawing, and rule-based geometry to produce visual structures that are both precise and expressive. If our generative exploration of mathematical curves interests you, the A' Generative, Algorithmic, Parametric and AI-Assisted Design Award category features projects where designers harness algorithms, mathematics, and computational processes to create innovative generative works, projects and products.
Above is today's featured Generative Design. Discover more works at the A' Design Awards.
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Every gear ratio on this page is selected using crypto.getRandomValues(). The curve is computed and rendered entirely in your browser using the HTML5 Canvas API. No image data leaves your device. The downloadable SVG is generated locally from the mathematical coordinates.
