Local overlaps in special unfoldings of convex polyhedra
type: Item
instance of: scholarly article
title: Local overlaps in special unfoldings of convex polyhedra
zbMATH Open document ID: 1168.52010
author: Brendan Lucier
published in: Computational Geometry
publication date: 2009-06-18T00:00:00Z
review text: It is an old conjecture of \textit{G. C. Shephard} [Math. Proc. Camb. Philos. Soc. 78, 389--403 (1975; Zbl 0312.52011)] that the boundary of a convex polyhedron (\(3\)-polytope) can be cut up along a spanning tree of its edges and unfolded into the plane without overlaps. In general, unfoldings will have overlaps. Call such an overlap \(k\)-local if there are at most \(k\) vertices in the shortest edge-path between the two overlapping faces. In this paper, the author develops conditions for the occurrence of the minimal case \(k = 2\); as the name suggests, these conditions are purely local. However, the author then applies his results to normal unfoldings, determined by the (increasing, say) angles between the normals to the faces and a fixed direction, giving examples of polyhedra for which all normal unfoldings have overlaps.
reviewed by: Peter McMullen
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 5566430
zbMATH Keywords: convex polyhedron
zbMATH Keywords: unfolding
zbMATH Keywords: overlap
zbMATH Keywords: \(k\)-local unfolding
zbMATH Keywords: normal unfolding
MaRDI profile type: Publication
full work available at URL: j.comgeo.2007.07.008
OpenAlex ID: W2077123238
cites work: Edge-unfolding nested polyhedral bands
cites work: Q2728851
cites work: Convex polytopes with convex nets
DOI: 10.1016/J.COMGEO.2007.07.008
description: scientific article; zbMATH DE number 5566430