It is a common conjecture that all convex polyhedra must be edge-unfoldable but to date a valid proof of this has escaped discovery. This dissertation presents several new directions in the quest for the proof. Also discussed is a method which may lead to a counterexample to the conjecture through the construction of ‘hard to unfold’ polyhedra. Algorithmic solutions are discussed for the task of determining the specific set of edges which must be cut in order that an unfolding not self-intersect. A series of <i>Unfolder </i>algorithms are explored and compared, in terms of both algorithmic design and empirical performance on test data. No surface of uniformly negative internal curvature with fewer than two border curves is unfoldable. The <i>coolinoids </i>are a class of non-convex polyhedra having exactly two border curves and negative curvature at every internal vertex, which may be constructed so as to be unfoldable without overlap. The fascinating interaction between construction and overlap in coolinoids is modelled and explored.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:596583 |
| Date | January 2008 |
| Creators | Benton, P. A. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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