Introduction

Spirographs are a toy and mathematical tool for creating beautiful looking curves, stars, and patterns. Though they only became famous in the 1960s, the math that describes them was studied when the Ancient Greeks looked into the motions of the planets. So why don’t we look into what makes these beautiful patterns? We’ll play around with different wheels and different tracks.

But first, let’s formalize what a spirograph does. In mathematics, when we trace a point on a curve that rolls on another curve (so they do not slide), the resulting curve is known as a roulette (french for “little wheel”). If we trace a point on a circle as it rolls along a straight line, the resulting curve is given the special name of a trochoid (greek for “wheel”!). And when both the rolling curve and the fixed curve are circles, the resulting curve is called an epitrochoid if the rolling circle is outside the fixed circle, and an hypotrochoid if the rolling circle is inside the fixed circle.

Spirographs can be used to make all sorts of these curves! But for now, we’ll just be looking at hypotrochoids, epitrochoids, as well as playing around with elliptical tracks.

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