Part II — Coxeter & Dynkin Diagrams

Dynkin Diagrams & Lattices

When symmetry meets the integer grid
Based on John Baez, "Coxeter and Dynkin Diagrams" (2026)
Section 01

The Crystallographic Condition

A lattice in \(\mathbb{R}^n\) is the set of all integer linear combinations of \(n\) linearly independent vectors. When does a finite reflection group arise as the symmetry group of some lattice?

Not always. The constraint is called the crystallographic condition, and it has a beautifully simple consequence: only edges labeled \(m = 3, 4,\) or \(6\) survive. Pentagons, heptagons, and all other exotic polygons are ruled out. If you've ever tried to tile a floor with regular pentagons, you already have physical intuition for why.

Interactive · The Crystallographic Filter

Toggle the crystallographic condition to see which Coxeter diagrams survive — and which are killed.

The survivors are precisely the diagrams whose edges use only \(m = 3, 4, 6\). These are:

An,   Bn,   Cn,   Dn,   E6,   E7,   E8,   F4,   G2

Notice something: BCn from Part I splits into two diagrams, Bn and Cn. We'll see why shortly — it has to do with choosing which generating vector is longer.

Section 02

Lattice Builder

A lattice is generated by two basis vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\). Every lattice point is an integer combination \(a\mathbf{v}_1 + b\mathbf{v}_2\). Adjust the angle between the vectors and their length ratio below to see how the lattice changes.

The special angles \(\pi/3\), \(\pi/4\), and \(\pi/6\) produce lattices with reflection symmetry. These correspond exactly to the crystallographic edge labels \(m = 3, 4, 6\).

Interactive · 2D Lattice Generator
Angle between vectors
60°
Length ratio |v₂|/|v₁|
1.00
Section 03

The Three Tilings

In two dimensions, only three regular tilings exist: the triangular, square, and hexagonal tilings. Each corresponds to a crystallographic Coxeter diagram in rank 2.

Interactive · The Crystallographic Tilings
Baez's trick question: The vertices of the hexagonal tiling don't actually form a lattice! To get a lattice you need to add the center of each hexagon — but then you recover the triangular lattice. The G₂ lattice is really the same as the A₂ lattice, but considered with a larger symmetry group: the hexagon's 12-element dihedral group instead of the triangle's 6-element one.
Section 04

The Root Lattice Recipe

Here is how to build a root lattice from a crystallographic Coxeter diagram with \(n\) dots. Pick basis vectors \(\mathbf{v}_1, \ldots, \mathbf{v}_n \in \mathbb{R}^n\) according to these rules:

Reference · The Construction Rules

Why \(\pi - \theta\) instead of \(\theta\)? Because we're choosing root vectors rather than the mirror-normal vectors from Part I. If the angle between \(\mathbf{v}\) and \(\mathbf{w}\) is \(\pi - \theta\), then the angle between \(\mathbf{v}\) and \(-\mathbf{w}\) is \(\theta\). The root convention is standard and makes the Cartan matrix come out nicely, though Baez notes it can be confusing at first.

Interactive · Root Vector Angles

Select an edge type to see how the root vectors are oriented. The solid vectors are the roots; the dashed vectors show the reflection mirrors.

Section 05

Arrows & the Bn/Cn Split

When \(m = 4\) or \(m = 6\), the two root vectors have different lengths. We need to decide which one is longer. To record this, we draw an arrow on the edge pointing from the longer root to the shorter one.

For most diagrams, the two choices of arrow direction give isomorphic results (you can just flip the diagram). But for BCn with \(n \geq 3\), the two choices are genuinely different. They give two distinct lattices, called Bn and Cn.

Interactive · Choose the Arrow Direction

Click to flip the arrow on the last edge of BC₃. Watch how the lattice near the origin changes shape.

In B3, the lattice points nearest the origin lie on a cube. In C3, they lie on an octahedron. The pattern continues in higher dimensions: Bn gives hypercubes, Cn gives orthoplexes.

Section 06

Dynkin Notation

Dynkin's innovation was a cleaner way to draw the same information. Instead of numbered edge labels, we use multiple parallel lines:

— An edge labeled 4 becomes a double edge (two parallel lines).
— An edge labeled 6 becomes a triple edge (three parallel lines).
— An unlabeled edge (label 3) stays as a single line.

The arrow indicating which root is longer is preserved. The result is a purely pictorial encoding with no numbers at all.

Interactive · Coxeter vs. Dynkin Edge Notation
Section 07

The Complete Dynkin Diagrams

Putting it all together: every connected Dynkin diagram encodes a root lattice with a finite reflection group as symmetries. Here is the complete family, shown in both Coxeter and Dynkin notation.

Reference · All Connected Dynkin Diagrams
The upshot: Any Dynkin diagram with \(n\) dots describes basis vectors \(\mathbf{v}_1, \ldots, \mathbf{v}_n \in \mathbb{R}^n\) such that (1) their integer linear combinations form a lattice \(L\), (2) reflections through these vectors generate a finite reflection group \(\Gamma\), and (3) \(\Gamma\) preserves \(L\). These vectors are called roots, and \(L\) is called a root lattice.
Coming in Part III: These same Dynkin diagrams classify compact simple Lie algebras — the continuous symmetry groups that underpin all of modern physics. We'll see how a Lie group hides a lattice inside its maximal torus, and how the Weyl group acts on it by reflections.