Developments in The Cubing Problem
Date
2024
Authors
Janssen, Noah
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Journal ISSN
Volume Title
Publisher
University of Victoria
Abstract
In our research, we set out to make progress in either developing or proving it is likely impossible to develop a polynomial time algorithm to partition orthogonal polyhedra (cornerhedra) into the minimum number of boxes possible. For our purposes, a box is either a rectangle or square shape (i.e. a rectangular prism) made of UCs that is filled in. This is a special case of the problem of rectangular partition of polyhedrons which has applications in various fields and industries. In our research, we proved that cornerhedra can either have certain qualities or cannot. Examples of these are that some cornerhedra can be partitioned without splitting 2 layers and others cannot. We have also shown that the minimum number of boxes some cornerhedra can be partitioned into is greater than the number of outward-facing corners of that cornerhedron. Outside of this, we formulated conjectures about possible cornerhedra. An example of a conjecture would be whether there exists a cornerhedron such that when any two of its layers are split at least 2 new outward-facing corners are created.
Description
Keywords
computational geometry, orthogonal polyhedron, decomposition, rectangular partition