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Can I draw a net for a Steinmetz solid from two cylinders with a compass?

That is, can we flatten the net? I often make a model using paper and a compass-- it looks about right... is it really a valid method of construction?

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I didn't know the term Steinmetz solid. For others in the same position: . – Joseph O'Rourke Nov 20 '10 at 15:43
up vote 2 down vote accepted

Imagine keeping one of the cylinders intact, but merely marking on it the curves that would have been the edges of the Steinmetz solid. If you unfold the cylinder, the curves must be periodic in the azimuthal direction, so they cannot be circles.

If you solve $(x,y,z) = (\cos \theta, \sin \theta, z) = (x, \cos \phi, \sin \phi)$, you'll find that the curves are actually sinusoids, $z = \pm \cos \theta$.

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