Ok, so by that definition a geodesic sphere has the Rupert property, as the sphere is an approximation made up of equilateral triangles. What if we perform isotropic subdivision on the equilateral triangles, such that each inserted point lies on the sphere, centred on each base triangle. We then subdivide each base triangle by constructing 3 new triangles around the inserted point. Thus at each iteration, geodesic sphere of N triangles is subdivided into 3*N triangles. If we continue with the subdivision, each iteration is a refinement of the geodesic sphere, and the geometric approximation gets closer to the shape of a true sphere. As N approaches infinity, the Rupert property holds true (according to the definition). What happens at infinity?

Yes!

The "pass through itself" criteria is the same as "has one shadow that fits entirely inside another shadow". If you allow "one shadow equals another shadow" then it's trivially true for every shape because a shadow equals itself.

Note that this "shadow" language assumes a point light source at infinity, i.e. all the rays are parallel.

I think Sphere is a outlier for this context.

Yeah I’m confused