Ununfoldable polyhedra with convex faces
type: Item
instance of: scholarly article
title: Ununfoldable polyhedra with convex faces
zbMATH Open document ID: 1021.52013
DOI: 10.1016/S0925-7721(02)00091-3
author: Yanyan Li
published in: Computational Geometry
publication date: 2003-04-28T00:00:00Z
review text: A classic open question in geometry is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge cutting of the polyhedron, and the resulting simple polygon is called an edge unfolding or net. The authors investigate the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. Two examples of polyhedra are given, one with 24 convex faces and one with 36 triangular faces that cannot be unfolded by cutting along edges. This proves, in particular, that the edge-unfolding conjecture does not generalize to topologically convex polyhedra. It is shown that cuts across faces can unfold some convex-faced polyhedra that cannot be unfolded with cuts only along edges. This is the first demonstration that general cuts are more powerful than edge cuts. The problem of constructing a polyhedron that cannot be unfolded even using general cuts is also considered. Finding a ``closed'' ununfoldable polyhedron is an intriguing open problem. As a step towards this goal, the authors present an ``open'' polyhedron with triangular faces that cannot be unfoldable no matter how it is cut.
Mathematics Subject Classification ID: 52C45
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 1902830
zbMATH Keywords: convex closed polyhedron
zbMATH Keywords: open convex-faced polyhedron
zbMATH Keywords: topologically convex polyhedra
zbMATH Keywords: edge cutting
zbMATH Keywords: edge unfolding
zbMATH Keywords: vertex unfolding
zbMATH Keywords: convex unfolding
zbMATH Keywords: unfolding polyhedra
zbMATH Keywords: cuts across faces
zbMATH Keywords: general unfolding
zbMATH Keywords: tree
zbMATH Keywords: polyhedron with negative curvature
zbMATH Keywords: middle vertices
zbMATH Keywords: gluing hats together
zbMATH Keywords: spike triangle
zbMATH Keywords: spiked tetrahedron
zbMATH Keywords: nets
zbMATH Keywords: discrete geometry
reviewed by: Serguey M. Pokas
describes a project that uses: UnfoldPolytope
MaRDI profile type: Publication
Wikidata QID: Q56970670
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full work available at URL: s0925-7721(02)00091-3
OpenAlex ID: W2148125855
description: scientific article; zbMATH DE number 1902830