Modular Arithmetic Patterns

Multiplication mod n creates stunning chord diagrams — mod 10 × 2 = pentagram, × 3 = triangle

Modulus n

Multiplier k

Line opacity

Presets:

Each number 0..n-1 placed around
a circle. Line from i to (i×k) mod n.

Symmetry group: ℤ/nℤ
GCD(k,n)=— → —

Modular arithmetic is the mathematics of remainders — clock arithmetic generalized. Here each integer 0…n−1 sits on a circle; a chord connects i to (i×k) mod n. When GCD(k,n)=1, k is a unit mod n and the map is a bijection (a symmetry of ℤ/nℤ). The stunning visual patterns are direct signatures of the group structure: GCD(k,n)=d means the chords split into d identical subpatterns. This is the foundation of RSA cryptography, the Fast Fourier Transform, and much of modern number theory.