Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection). I am wondering if each of the five regular polyhedra in 3D are shadows of regular 4-polytopes. The 4-simplex can project to a regular...

Joseph, in your response to Igor in the comments you allow holding the candle that's casting the shadow very close to a face. This is precisely the picture that Ian gave in the comments. But in your question you require that the candle be infinitely far away, so that it is a parallel projection...

Let $M$ be the Mandelbrot set. Question: Is the interior of $M$ dense in $M$? My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and moreover that this is known (as it is not very close to the Hyperbolicity Conjecture and thus not ...