Every Combinatorial Polyhedron Can Unfold with Overlap
DOI:
https://doi.org/10.57717/cgt.v4i2.54Abstract
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net, a non-overlapping planar polygon. One can view his result as establishing that every combinatorial polyhedron P has a metric realization P that allows unfolding to a net.
Joseph Malkevitch asked if the reverse holds (in some sense of “reverse”): Is there a combinatorial polyhedron P such that, for every metric realization P in ℝ3, and for every spanning cut-tree T of the 1-skeleton, P cut by T unfolds to a net? In this paper we prove the answer is NO: Every combinatorial polyhedron has a realization and a cut-tree that edge-unfolds the polyhedron with overlap.
Downloads
Published
How to Cite
Issue
Section
Categories
License
Copyright (c) 2025 Joseph O'Rourke
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
