Simplices whose dihedral angles are all rational multiples of \(\pi\), and related topics (Q722346)

This interesting paper deals with the dihedral angles of \(n\)-simplices. It is proved (Theorem 2.5) that for every \(n \geq 2\), there are infinitely many, mutually non-similar \(n\)-dimensional Hill-simplices whose dihedral angles are all rational multiples of \(\pi\). The following problem is raised. Problem 2.6. If the dihedral angles of an \(n\)-simplex are all rational multiples of \(\pi\), is this simplex then a space-filler? Using the theorem concerning fairly-inscribed \(n\)-symplices (Theorem 4.2) and the outer normal transformation among others, it is showed (Theorem 5.3) that for every \(n \geq 3\), there are uncountably many, mutually non-similar \(n\) simplices whose dihedral angles are \(\pi\)-independent.