Higher dimensional versions of theorems of Euler and Fuss

TL;DR

This paper extends classical theorems of Euler and Fuss to higher dimensions, focusing on polytopes inscribed in and circumscribed about ellipsoids, using linear algebra techniques.

Contribution

It introduces higher dimensional analogs of Euler and Fuss theorems for polytopes, establishing bijections with the orthogonal group based on trace conditions.

Findings

Existence of bijections between orthogonal group and sets of simplices when trace A=1.

Families of parallelotopes and cross polytopes indexed by O(n) when trace A^2=1.

Use of linear algebra to generalize classical geometric theorems.

Abstract

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If $trace A = 1$, there exists a bijection between the orthogonal group $O(n)$ and the set of such labeled simplices. Similarly, if $trace A^2 = 1$, there are families of parallelotopes and of cross polytopes, also indexed by $O(n)$.