Flattening the Curve. . . of Spirographs

The Spirograph is an old and popular toy that produces aesthetically pleasing and fascinating spiral figures. But are spirals all it can make? In playing with a software implementation of the toy, the author chanced upon a variety of shapes that it can make that are different from its well-known repertoire of spirals, in particular, shapes that have a visible flatness and not the curved spiral geometry that we are accustomed to seeing from the Spirograph. This paper reports on these explorations by the author and his delightful discovery of very elegant and simple geometric relationships between the Spirograph’s structural parameters that enable those patterns.