Affine unfoldings of convex polyhedra

Ghomi, Mohammad

doi:10.2140/gt.2014.18.3055

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Affine unfoldings of convex polyhedra

Mohammad Ghomi

Geometry & Topology 18 (2014) 3055–3090

DOI: 10.2140/gt.2014.18.3055

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

Volume 18, issue 5 (2014)