Oh, I can prove that the *only* example is the 3-ball, based on the following argument: * Assume, without loss of generality, that the light shines from above (parallel to the Z-axis). * If we rotate the object about a perpendicular axis (X-axis), the shadow must remain constantly invariant. (Otherwise, it must rotate, and this requires that the shadow has a constant radius, therefore circular.) * After 180° of rotation, the shadow will have flipped without changing. Hence, this implies that the shadow has a line of symmetry (parallel to the X-axis). * Repeat this with all possible axes perpendicular to the Z-axis, showing that there is an infinite number of mirror lines passing through the shadow. * From this we can conclude that the shadow is circular, implying that the original object is spherical. Presumably, this generalises to higher dimensions...? (Sorry, I would have preferred there to be other examples, in the same way that I would prefer there to be more than 26 sporadic groups. Unfortunately, it doesn't look as if there are any more out there.) So, the only constant-shadow solids are spheres. What about solids where the *area* of the shadow remains contant, but not the shape? Can they exist? Sincerely, Adam P. Goucher ----- Original Message ----- From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, December 16, 2010 3:12 AM Subject: [math-fun] Constant-shadow solids

Suppose that a closed bounded convex body X in 3-space, with nonempty interior, is such that the area of its orthogonal projection onto any plane is independent of the plane.

Does this imply X is a round 3-ball, or are the other examples?

What about for higher dimensions? My hunch is that beginning at some dimension, round balls will be the only examples.

--Dan

Those who sleep faster get more rest.

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