On the trace of polyhedra
type: Item
instance of: scholarly article
title: On the trace of polyhedra
zbMATH Open document ID: 0838.52013
DOI: 10.1007/BF02179086
published in: Geometriae Dedicata
publication date: 1996-06-03T00:00:00Z
review text: 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.
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 806220
zbMATH Keywords: convex polyhedron
zbMATH Keywords: trace
zbMATH Keywords: trace-dense
author: Norbert Hegyvári
MaRDI profile type: Publication
cites work: Q3687182
cites work: Q3257805
description: scientific article; zbMATH DE number 806220