Volume 18, issue 5 (2014)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Affine unfoldings of convex polyhedra

Mohammad Ghomi

Geometry & Topology 18 (2014) 3055–3090
Abstract

We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular, there exists no combinatorial obstruction to a positive resolution of Dürer’s unfoldability problem, which answers a question of Croft, Falconer and Guy. Among other techniques, the proof employs a topological characterization of embeddings among the planar immersions of the disk.

Keywords
convex polyhedron, unfolding, development, spanning tree, edge graph, isometric embedding, immersion, covering spaces, Dürer's problem
Mathematical Subject Classification 2010
Primary: 52B05, 57N35
Secondary: 05C10, 57M10
References
Publication
Received: 2 September 2013
Revised: 16 January 2014
Accepted: 16 April 2014
Published: 1 December 2014
Proposed: Tobias H Colding
Seconded: Dmitri Burago, Gang Tian
Authors
Mohammad Ghomi
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332
USA
http://www.math.gatech.edu/~ghomi