# The Dictionary of Mathematical Eponymy: The Laves Graph

*A programming note: after a decade of two or more blog posts a week, I’m taking my foot off of the writing pedal a bit this year; the blog will be running one post a week (on Mondays) for the foreseeable future, likely alternating Ask Uncle Colin with regular posts, with Dictionary posts on the first Monday of the month as before.*

### What is the Laves Graph?

The Laves graph is a graph structure in three dimensions, consisting of vertices at the points:

- $(0,0,0)$
- $(1,2,3)$
- $(2,3,1)$
- $(3,2,1)$
- $(2,2,2)$
- $(3,0,1)$
- $(0,1,3)$
- $(1,3,0)$

… and any point that can be made by adding or subtracting a multiple of four to some or all of the coordinates.

Each vertex is connected to the three vertices at a distance of $\sqrt{2}$ from it.

### Why is it interesting?

It’s pretty!

In graph theory, it’s the maximal abelian covering of the complete graph $K_4$ ((In the interests of getting the post written, I have simply copied that from Wikipedia making no effort to understand.))

It’s also been hypothesised as a possible stable crystal structure for boron, and as a metastable or unstable structure for an allotope of carbon.

### Who was Fritz Laves?

Now, this site has a “No Nazis” policy ((If you find Nazis lurking anywhere here, please let me know.)), and my heart sank when I realised Laves (born Hanover, Germany, in 1906) worked for Göring during the war. It rose again when I found that Laves was known as a “protector of Jews”, and apparently valued his principles over his career.

And the circumstances of being assigned to work on an impossible project for Göring… well, they’re just bizarre:

an alchemist was assigned to look over his shoulder, and contributed to the success of his experiments by osciilating a small crystal sphere on a chain over the crucible. He also insisted on adding powdered crocodile bones to the batch, but because of Rommel’s difficulties in the land of the Nile, settled for the tail-bones of a lesser lizard

So, I’m giving Fritz the benefit of the doubt.

Before then, he studied geology at Innsbruck and Göttingen, pursued his doctorate at ETH Zürich, and took a faculty position back at Göttingen. (He unsuccessfully attempted to prevent his boss, Victor Goldschmidt, from being dismissed.)

Towards the end of the Second World War, he took up positions at Halle and then Marburg, before moving to Chicago in 1948. He returned to Zürich in 1954 and worked there until his retirement in 1976.

He died in Laigueglia, Italy, in 1978.