Modular Multiplication on a Circle

modulus and number of points

100

multiplier

2

Multiply the positive integers by 3 (mod 10) to get the repeating sequence

3,6,9,2,5,8,1,4,7,10,…

. This Demonstration connects the starting values

1,2,3,…

to the ending values

1k,2k,3k,…

for a chosen modulus and multiplier

k

.

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