Why Does One Rolling Circle Draw Exactly 11 Loops?

BEHIND THE WORK

Why does one rolling circle create exactly 11 interlocking loops?

A small circle rolls around the outside of a larger one while an attached point traces the curve. In this episode, we build the mechanism step by step, separate the horizontal and vertical coordinates, and show why 15 fast rotations minus 4 broad rotations produces 11 relative cycles.

x(t) = 7.5 cos(t) − 3.5 cos(15t/4) y(t) = 7.5 sin(t) − 3.5 sin(15t/4) 0 ≤ t ≤ 8π

Chapters 00:00 One point draws the pattern 00:15 The central question 00:40 How the rolling circle works 01:30 The x-coordinate 02:15 The y-coordinate 03:00 Why exactly 11 loops? 03:45 Changing the attachment distance 04:25 Building the final animation 05:00 The complete drawing 05:35 Equation and result

Music: “Helium” by TrackTribe

Artismathic turns equations into visual art—one function at a time.

#MathArt #GenerativeArt #Mathematics