Modular Multiplication on a Circle

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Multiply the positive integers by 3 (mod 10) to get the repeating sequence . This Demonstration connects the starting values to the ending values for a chosen modulus and multiplier .


With multipliers of 2, 3, and 6, cusped epicycloids develop—the cardioid, nephroid, and ranunculoid. Other patterns develop with other moduli and multipliers.


Contributed by: Ed Pegg Jr (November 2015)
Open content licensed under CC BY-NC-SA




[1] B. Polster. Times Tables, Mandelbrot and the Heart of Mathematics [Video]. (Nov 11, 2015)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.