darcy scheeler

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Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?

We had a great time discussing the question after our logic seminar here in New York this week, when our speaker Maryanthe Malliaris asked the spherical Earth question.

December 20, 2010: In light (or dark, as it were) of the lunar eclipse tonight, I am bumping this question, with the remark also that despite the truly outstanding answers we have received, several of the further questions stated above are not fully answered.

• mg.metric-geometry

$\begingroup$ I've often heard that we can deduce a spherical Earth by observing that its shadows on the Moon are always circular - but always without proof, and in non-mathematical sources where it wouldn't be obvious that proof is necessary. $\endgroup$

$\begingroup$ What is the "set of shadows"? Do you know which projection gives which shadow, or just an unstructured set of planar shapes? $\endgroup$

$\begingroup$ I had meant the set of planar shapes, without knowing in which direction the projection is taken. $\endgroup$

$\begingroup$ Perhaps Joel can add this to his generalizations. Suppose we have a catalogue of information which may be a proper subset of the following: for each 2-D projection of a 3-D object, we have the list of widths ( and perhaps directions ) for that projection. How much of the catalogue do we need to reconstruct the original (let's assume convex) object? Note this may be different from knowing bounding hyperplane distance in a given direction for all directions, as the catalogue might be indexed by projection. Or is it the same? Gerhard "Ask Me About System Design" Paseman, 2010.09.17 $\endgroup$

$\begingroup$ Gerhard, your suggestion is indeed in the spirit of my further questions. For the catalogue, I suppose we could imagine living on the object, and occasionally being able to observe part of a projection, from an essentially random direction, thereby simulating in part the shadow-of-the-Earth-on-the-moon situation. To what extent can we deduce the shape of the object from this information? $\endgroup$

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darcy scheeler