Eisenstein integers and equilateral ideal triangles

Abstract

We discuss the relationship between Penner's $\lambda$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k + 1$ is the norm of an Eisenstein integer that is can be written as $a^2 - ab + b^2$ for some $a,b \in \mathbb{Z}$.