Date of Submission

Spring 2024

Academic Program

Mathematics

Project Advisor 1

Charles Doran

Abstract/Artist's Statement

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

In this project we study the folding properties of several special classes of cubical complexes. First, we look at polyominoids, which are arrangements of congruent squares in 3-space, glued edge-to-edge at 90° and 180° angles. We construct and analyze the mechanical configuration space for n-cell polyominoids, which is a graph with vertex set given by all n-cell polyominoids, where two vertices are connected by an edge if you can transform one into the other by one hinge movement. For n = 4, we provide a complete characterization, and we also prove some structural properties for all n. Next, we study fenestrated polyominos, which are arrangements of congruent squares in 2-space, glued edge-to-edge, in such a way that fenestrations (i.e. holes) are contained in the figure. We enumerate folding patterns which, when assigned to the edges of a fenestrated polyomino, collapse its fenestrations by folding it into a 3D polyominoid. Lastly, we realize 4D hypercubes as surfaces by following precise coloring rules and removing faces incompatible with those rules. We find that under a specific quotienting operation relevant to supersymmetry physics, the 4-cube becomes un-embeddable in 3-space using non-intersecting
quadrilateral flats.

Open Access Agreement

Open Access

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Download

This work is protected by a Creative Commons license. Any use not permitted under that license is prohibited.

Share

COinS