Can an Orthogonal Net Fold to a Nonorthogonal Polyhedron?

Title: Can an Orthogonal Net Fold to a Nonorthogonal Polyhedron?

Research Question: Can an orthogonal net (a flat, rectangular shape) be folded into a nonorthogonal polyhedron (a three-dimensional shape with faces that are not right angles)?

Methodology: The researchers used mathematical proofs and geometric reasoning to explore this question. They started by defining what an orthogonal net, folding, and polyhedron are. They then posed two related 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?

They first addressed Question 2, which they proved to be true. This meant that if a polyhedron's faces are all rectangles, it must be an orthogonal polyhedron. They then addressed Question 1, which they proved to be false. This meant that an orthogonal net can be folded into a nonorthogonal polyhedron.

Results: The researchers found that it is possible to create a nonorthogonal polyhedron by folding an orthogonal net. They provided an example of a polyhedron with rectangular faces that is not an orthogonal polyhedron.

Implications: This research suggests that the properties of a flat, two-dimensional shape do not necessarily determine the properties of a three-dimensional shape created by folding it. It also provides a new way to think about the relationship between two-dimensional and three-dimensional geometry.

Link to Article: https://arxiv.org/abs/0110059v2 Authors: arXiv ID: 0110059v2