The 17 Rules That Control Every Repeating Pattern 🧩 #math #satisfying #geometry

Added: 2026-06-03

[00:00:02]Every repeating pattern ever created follows one of exactly 17 symmetry rules.

[00:00:08]Why not more? Regular pentagons cannot tile the plane. Five-fold rotational symmetry always leaves gaps. Only two-fold, three-fold, four-fold, and six-fold rotations are compatible with a repeating lattice.

[00:00:22]Here is P1, pure translation, the simplest wallpaper group. Now add mirrors on a square grid. This is P4mm.

[00:00:30]Nearly half of all Islamic geometric patterns follow this single group.

[00:00:34]Switch to a hexagonal lattice with full symmetry, P6mm, the most symmetric wallpaper group possible. Five lattice types, four allowed rotation orders, exactly 17 groups, no more, no less.

[00:00:47]First proved by Fedorov in 1891.

[00:00:50]The artisans of the Alhambra had already encoded at least 13 of them into stone centuries before group theory existed.