### Lie Theory Through Examples 2

#### Posted by John Baez

In this week’s seminar on Lie Theory Through Examples, we’ll move on up to the $A_3$ lattice.

You’ve seen this lattice before: when you stack spheres in a triangular pyramid, their centers lie at points that form a lattice of this sort:

But to derive this lattice starting with the Lie group $SU(4)$, we’ll need to talk a bit about the Killing form. That’s what allows us to measure angles and distances in the Lie algebra.

- Lecture 2 (Oct. 6) - The Killing form and the $A_3$ lattice.

A different view makes it clear why the A_{3} lattice is also called a "cubic close packing":

You can also think of the A_{3} lattice as built
from octahedra and tetrahedra. This figure is from Buckminster
Fuller’s patent for the "octet truss", now widely
used in architecture:

In the $A_3$ lattice, each point has 12 nearest neighbors.
These form the vertices of a cuboctahedron:

## Re: Lie Theory Through Examples 2

Wait, was the lecture really on Monday?