Mathematics has several underrated properties, of which I am writing about the Rupert Polyhedrons. Quoting wikipedia “A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P”. This property has led to a conjecture stating that every 3-dimensional convex polyhedra has this property. This article will not prove that, but it will provide several examples of the property holding for different polyhedra.
Before finding any examples, I had to find a source library for polyhedra. At first, this was challenging, and I ended up going down the wrong path, leading toward getting false results. Eventually, after hours of searching, I found the site Visual Polyhedra. This site had a complete index of the vertices and faces of the polyhedron. Using some pretty basic code, I converted these vertices and faces into the obj files I needed to prove the property in some shapes.
Next, I needed to find a way to prove the property true. I first converted complex 3d shapes into several 2d representations using the viewport. Next, I calculated the area of these representations and sorted them from largest to smallest. I then overlaid the smallest one on top of the largest. If the perimeter of shape A did not touch the perimeter of shape B, I found the shape. If not, I repeated the last step until it either failed to find one with every pair or the shape was proven.
Now, let's get to the interesting stuff, data. The linked data shows the 2d intersection of the Rupert Polyhedrons along with other data like how well the shape overlapped itself. Overall, that's pretty much it, but if you're still interested here's another article that goes into the depth of the problem.