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.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/5312