Reflections & Coxeter Diagrams

The Grand Classification

There is a small collection of diagrams — dots connected by lines, sometimes labeled with numbers — that appear again and again across mathematics and physics. They classify finite reflection groups, lattices in Euclidean space, compact simple Lie algebras, quivers with tame representation type, and even the Platonic solids (in two separate ways!).

These are the Coxeter diagrams and their close relatives, the Dynkin diagrams. In this first essay, we'll see how Coxeter diagrams emerge naturally from the question: what are all the finite groups generated by reflections?

Reflections in the Plane

Given a nonzero vector \(\mathbf{v}\) in Euclidean space, the reflection through \(\mathbf{v}\) sends \(\mathbf{v}\) to \(-\mathbf{v}\) and fixes every vector orthogonal to \(\mathbf{v}\). A finite reflection group is a finite group of transformations where every element is a product of such reflections.

Drag the blue point below to move a vector, and watch its reflection in real time. The mirror line is orthogonal to the reflection axis.

Composing Reflections

Here is the key insight: two reflections compose to give a rotation. If vectors \(\mathbf{v}\) and \(\mathbf{w}\) are at an angle \(\theta\) from each other, and \(r\) and \(s\) are the reflections through \(\mathbf{v}\) and \(\mathbf{w}\), then \(rs\) is a rotation by angle \(2\theta\).

This means that if \(\theta = \pi/m\), then \((rs)^m\) is a rotation by \(2\pi\) — the identity. So \((rs)^m = 1\). But if \(\theta\) is not a rational multiple of \(\pi\), the composition never returns to the identity, and \(r\) and \(s\) cannot both belong to a finite reflection group.

This constraint — that the angle between any two generating reflections must be \(\pi/m\) for some integer \(m \geq 2\) — is what makes the classification possible. It forces the geometry into a rigid combinatorial structure.

The Coxeter Encoding

A Coxeter diagram encodes a finite reflection group as follows. Draw one dot for each generating reflection. If two generators have their vectors at angle \(\pi/m\):

— If \(m = 2\), the reflections commute. Draw no edge.
— If \(m = 3\), draw an unlabeled edge (the most common case).
— If \(m \geq 4\), draw an edge labeled \(m\).

Conversely, a Coxeter diagram defines a Coxeter group by generators and relations:

The Classification

Not every Coxeter diagram yields a finite group. But the complete list of connected diagrams that do is known — and it is remarkably short. Click any diagram below to explore it.

Polytopes & Symmetry Groups

Each Coxeter group is the symmetry group of some beautiful geometric object. The classical families correspond to simplices, hypercubes, and cross-polytopes. The exceptionals give us the icosahedron, the 24-cell, the 120-cell, and the magnificent E₈ root polytope.

The E₈ root polytope deserves special mention. Take a sphere in 8 dimensions and pack as many equal-sized spheres around it as possible. You'll fit exactly 240. Their centers form the vertices of the E₈ root polytope, and its symmetry group is the E₈ Coxeter group with 696,729,600 elements.

Polygon Symmetry Explorer

The symmetry group of a regular \(n\)-gon is one of the simplest examples of a finite reflection group. Choose a polygon below, then apply reflections and rotations to see the group in action. The Coxeter diagram is \(\mathrm{I}_m\) (two dots connected by an edge labeled \(m\)), which for \(m = 3\) is \(\mathrm{A}_2\), for \(m = 4\) is \(\mathrm{BC}_2\), and for \(m = 6\) is \(\mathrm{G}_2\).