First Convex Polyhedron Found That Can't Pass Through Itself

So if he proves that the snub cube doesn't have the Rupert property, he could still be the first to prove that not all Archimedean solids have it.

1: https://en.wikipedia.org/wiki/Snub_cube

It's pretty easy to brute force most shapes to prove the property true. The challenge is proving that a shape does not have the Rupert property, or that it does when it's a very specific and tight fit. You can't test an infinite number of possibilities.

> The full menagerie of shapes is too diverse to get a handle on, so mathematicians tend to focus on convex polyhedra

The phrase "tend to focus on" suggests it's not an exclusive thing. However, you're right -- it appears that the Rupert property only applies to convex polyhedra, so the article title and text is at the very least incomplete given that a sphere is a shape.

The challenge is that it gets computationally intensive the more sides that you add if you don't have shortcuts like ruling out entire blocks of orientations in their parameter space (they figured out that if one shadow, projection, protrudes significantly, then you'd need a large rotation to get that protrusion into the other shadow, thus removing all of those rotational angles and reducing the number of orientations needed to check). More sides and more symmetry make it much harder to test a candidate, but you have an interesting idea.

A sphere is a surface of constant width, which the polyhedron approximation is not.

> The projected shadow has the same size as its diameter

Thus this is exactly why the sphere doesn't have the Rupert property.

Note the egg shape in the article. specifically the widest band around the equator. now imagine one passing straight down through the other. one edge ring would pass through the shadow if it has a slight rotation offset but it is blocked by the next edge ring up, which could also fit but requires a different offset, so if you could change that rotational offset while it is passing through would it fit?

Rupert's snub cube and other Math Holes - https://news.ycombinator.com/item?id=45261566 - Sept 2025 (10 comments)

Rupert's Property - https://news.ycombinator.com/item?id=45057561 - Aug 2025 (23 comments)