Wythoff symbol

Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.

However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space.

The three numbers in Wythoff's symbol, p, q, and r, represent the corners of the Schwarz triangle used in the construction, which are ⁠π/p⁠, ⁠π/q⁠, and ⁠π/r⁠ radians respectively.

The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: In this notation the mirrors are labeled by the reflection-order of the opposite vertex.

A special use is the symbol | p q r which is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored.

Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.
The eight forms for the Wythoff constructions from a general triangle ( p q r ).