Pentagramma Mirificum and Elliptic Functions

In this talk I shall give an exposition of fragments from Gauss
where he discovered,
with the help of some work of Jacobi,
a remarkable connection between Napier pentagons on the sphere and Poncelet
pentagons on the plane.

This gives rise to a parametrization of the variety of Napier pentagons
using
the division by 5 of elliptic functions.

As a corollary we will find the classical five-term relation for the
dilogarithm in a somewhat
exotic disguise, and discuss some open questions.