On the trace of polyhedra
A fixed plane \(\sigma\) and a convex polyhedron \(P\) having one face in this plane being given, the polyhedron \(P\) is rotated around any of its edges lying in \(\sigma\) into a similar position. Such rotations are performed indefinitely, and the author defines the trace of \(P\) as the set of all points \(A\) of \(\sigma\) which coincide with some vertex of \(P\) in some position of \(P\). He then establishes the following results: Theorem 1. Among the regular polyhedra exactly the dodecahedra are trace-dense (i.e. the trace is everywhere dense in \(\sigma)\). Theorem 2. A rectangular parallelepiped is trace-dense if and only if it has at least two edges with irrational ratio. Theorem 3. Let \(x\) be a vertex of the convex polyhedron \(P\), and denote by \(\Sigma (x)\) the sum of all angles which occur at \(x\) on the faces. If \({\Sigma (x) \over \pi}\) is irrational for at least one vertex \(x\), then \(P\) is trace-dense. The proofs are straightforward.
scientific article; zbMATH DE number 806220