Modular Multiplication Circle

Ring Size & Multiplier

Points N 180

Multiplier m 2.00

Animation speed 0.25/s

Live Readouts

gcd(N, round m)

Symmetry arms N/gcd

Lines drawn

180

m mod N

2.00

How it works

Place N points evenly on a circle, numbered 0..N-1. For a chosen real multiplier m, draw a segment from every point k to the point at index (k·m) mod N:

angle(k)  = 2πk / N
target(k) = (k · m) mod N
line k -> angle(target(k))

This is the "times tables" construction: for integer m it is literally the multiplication table of m taken modulo N, drawn as chords. Letting m vary continuously turns the discrete table into a smooth family of curves — cardioids near m=2, nephroids near m=3, and higher rose/star patterns as m grows, because each point's chord is really an epicycle of frequency ratio m:1 around the ring.

The pattern's rotational symmetry is governed by d = gcd(N, round(m)): the N chords collapse into exactly N/d geometrically distinct arms, each repeated d times around the ring. When gcd(N, round(m)) = 1 (N and m coprime) every point connects to a different point and the diagram traces a single unbroken star polygon.

Points N — how many equally spaced points sit on the ring.
Multiplier m — the modular multiplier; drag to freeze on any pattern.
Play/Pause — animates m continuously, sweeping through the whole family of curves.
Random m — jumps to a random integer multiplier and pauses there.

Real-world relevance: this exact k ↦ k·m mod N map is the core operation behind modular arithmetic, RSA-style modular exponentiation and cyclic-group generators — visualizing it as chords is how mathematicians (popularized by Mathologer/Burkard Polster) reveal the hidden symmetry of multiplication modulo N.

Ring points Chords k → k·m mod N

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