Spirograph

Layered roulette curves — hypotrochoids and epitrochoids — closed exactly by their integer tooth ratios.

What it is

The toy, in software: a pen fixed in a wheel that rolls inside (hypotrochoid) or outside (epitrochoid) a toothed ring. Integer tooth counts mean every curve closes exactly — the number of lobes is ring/gcd(ring, wheel) and the wheel makes wheel/gcd turns before the figure repeats. Several curves nest per page, scaled inward with daylight between them.

How to use

Colouring pages with a distinctive petal structure: the overlapping lobes create lens-shaped regions that alternate naturally. Kids mode uses fewer, fatter curves. The pen position sweeps a family from near-circles (pen at centre) to pointed stars (pen at rim).

Purpose

Nostalgia with mathematics attached — the one design crate whose output every adult recognises from childhood. Its development recorded two taste lessons now cited across the lane: dedup_by_key only removes adjacent duplicates (a page shipped wheels 5, 13, 22, 13), and ranking by a single metric loses variety (a 3-lobe trefoil vanished under a 96-lobe rosette) — hence banded selection with a lobe floor.

History

Mathematicians studied roulettes from the 17th century (La Hire, Euler); the 19th century built lathes and harmonographs around them. Denys Fisher, a Leeds engineer, turned the geared version into the Spirograph toy in 1965 — Toy of the Year 1967, and the name became generic.

This implementation

• Spec knobs: size, ring (tooth count), curves per page, family (hypo/epi/both), pen, stroke, kids.
• Generation: wheels are chosen by banded lobe-count selection with a taste floor of 6 lobes (plus the deliberate low-lobe band), degenerate combinations rejected (wheel ≥ ring, < 3 lobes, Tusi-couple straight lines); curves are sampled per-turn and closed exactly.
• Guarantees: every curve closes (integer arithmetic, not tolerance); colourability-gated with curve-count escalation.