  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 $O(n)$.
