Modular Multiplication on a Circle

Cardiod Curve

The visual experiment below is an interactive recreation of the modular multiplication circle, which generates geometry found in the Mandelbrot set and other mathematical phenomenon. The circle is based on a simple mathematical pattern wherein the circumference of a circle is subdivided by an arbitrary number of points, and then a multiplication factor is chosen that is used as one of the operands to connect each dot via multiplication. This model was inspired by this video created by Youtube channel Mathologer, which explains the mathematics in detail.

Scroll through the presets and then play around with the controls yourself. By adjusting the modulus (number of points around the circle) as well as the multiplication factor used, many interesting features and patterns appear that are reflected in our natural world, including the golden ratio, sacred geometry and many famous fractal structures. Clicking out of the view window pauses all animation.

You can right-click and save any interesting shapes you find. Please share the control settings you discover in the comments below! I’m also open to any suggestions for usability improvements of the tool. And of course, you can fork it and add improvements yourself!

Interactive simulation

Play with full-screen simulation here.

View more digital experiments @

1,451 0 0