Built from Rectangles

Title: Built from Rectangles

Research Question: Can an orthogonal net ever fold to a nonorthogonal pol yhedron?

Summary: This research study aims to answer the question above by exploring the relationship between orthogonality in R2 and R3. The study focuses on two main questions:

1. If a polyhedron is created by an orthogonal folding of an orthogonal polygon, must it be an orthogonal polyhedron? 2. If a polyhedron's faces are all rectangles, must it be an orthogonal polyhedron?

The study first investigates the second question, which suggests that orthogonality in R2 forces orthogonality in R3. It then presents an example of a nonorthogonal polyhedron made entirely of rectangle faces, proving that the answer to the second question is no.

The study further demonstrates that the answer to the first question is also no for polyhedra of genus seven or above. However, it is yes for polyhedra of genus zero and one. This is achieved by exploring the concept of nonoverlapping unfoldings and using the Euler characteristic to prove the result.

Significance: This research provides a resolution to the question posed by Biedl, Lubiw, and Sun [BLS99] and contributes to the understanding of the relationship between orthogonality in R2 and R3. It also presents a new method for constructing nonorthogonal polyhedra using rectangle faces, which may have applications in various fields such as computer graphics, architecture, and mathematics education.

Link to Article: https://arxiv.org/abs/0110059v1 Authors: arXiv ID: 0110059v1