Grades 3–5 · Mathematics
How does a rolling circle draw a looping pattern?
A single pen point can draw a flower-like pattern without following a flower-shaped guide. In a geared drawing toy, a small wheel rolls around the inside of a fixed ring. A pen sits in a hole away from the wheel’s center. The hole stays fixed in the wheel while the whole wheel changes position and turns.
In the diagramTwo linked motions draw a looping curveA wheel rolls inside a fixed circular ringRing radius 5Wheel radius 2Pen offset 12 center circuits3 clockwise turnsFixed ring · no slippingThe outlined pen hole stays fixed in the moving wheel.
The pictured ideal ring has radius five units, the wheel radius two, and the pen hole lies one unit from the wheel’s center. The centers stay three units apart, so the wheel remains against the ring. Rolling without slipping links its movement around the ring to its rotation; those motions are not independently chosen.
The blue trace combines both motions. For these sizes, the wheel’s center makes two complete trips around the ring while the wheel turns three times in the opposite direction. The pen and wheel then return to their starting positions and the curve closes. Other matching gear sizes or pen-hole positions give different curves rather than arbitrary decoration.
About this explanation
Starting knowledge: Learners should recognize center, radius, a complete turn and whole-number counts of repeated circuits.
Sources
- University of Cambridge NRICH — Like a Circle in a Spiral
- University of St Andrews MacTutor — Hypotrochoid