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3 Lunar eclipse tonight; added 2 characters in body

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth by measuring a difference in shadow length between Alexandria and Syene. These observations, however, suggest merely a curved Earth rather than a spherical one.

Another ancient argument specifically suggesting a spherical Earth is the fact that the shadow of the Earth on the moon during a Lunar eclipse is circular.

My question is: is it true that the sphere is the only surface all of whose projections are disks?

It surely seems to be true. The corresponding fact, however, is not true in two dimensions. The Reuleaux triangle pictured below is a figure of constant width, meaning that every projection of it in the plane is a line segment of the same length.

There are also surfaces of constant width in higher dimensions, meaning that any two parallel bounding set of hyperplanes (touching the boundary) have constant separation. But all of the non-spherical examples of such surfaces I have seen have obviously non-circular projections.

It also seems clear that finitely many circular projections is insufficient, since intersecting finitely many cylinders would produce a surface having corners and containing some straight line segments.

The fact that you can spin such a surface with all circular projections inside any bounding cylinder is suggestive, but it is also true that you can spin the Reuleaux triangle inside a square, even though it isn't circular.

Further questions would include:

• To what extent are other surfaces determined by their projections? That is, which other shapes can we recognize by the set of their shadows?
• In particular, can we recognize the cube and other regular solids by their shadows?
• Which sets of shadows are realizable as projections of a surface? Is there some way to characterize these sets? Clearly they must be continuously deformable to one another and obey several other obvious conditions.

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.

2 typo in a name

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth by measuring a difference in shadow length between Alexandria and Syene. These observations, however, suggest merely a curved Earth rather than a spherical one.

Another ancient argument specifically suggesting a spherical Earth is the fact that the shadow of the Earth on the moon during a Lunar eclipse is circular.

My question is: is it true that the sphere is the only surface all of whose projections are disks?

It surely seems to be true. The corresponding fact, however, is not true in two dimensions. The Reuleaux triangle pictured below is a figure of constant width, meaning that every projection of it in the plane is a line segment of the same length.

There are also surfaces of constant width in higher dimensions, meaning that any two parallel bounding set of hyperplanes (touching the boundary) have constant separation. But all of the non-spherical examples of such surfaces I have seen have obviously non-circular projections.

It also seems clear that finitely many circular projections is insufficient, since intersecting finitely many cylinders would produce a surface having corners and containing some straight line segments.

The fact that you can spin such a surface with all circular projections inside any bounding cylinder is suggestive, but it is also true that you can spin the Reuleaux triangle inside a square, even though it isn't circular.

Further questions would include:

• To what extent are other surfaces determined by their projections? That is, which other shapes can we recognize by the set of their shadows?
• In particular, can we recognize the cube and other regular solids by their shadows?
• Which sets of shadows are realizable as projections of a surface? Is there some way to characterize these sets? Clearly they must be continuously deformable to one another and obey several other obvious conditions.

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

1

# Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth by measuring a difference in shadow length between Alexandria and Syene. These observations, however, suggest merely a curved Earth rather than a spherical one.

Another ancient argument specifically suggesting a spherical Earth is the fact that the shadow of the Earth on the moon during a Lunar eclipse is circular.

My question is: is it true that the sphere is the only surface all of whose projections are disks?

It surely seems to be true. The corresponding fact, however, is not true in two dimensions. The Reuleaux triangle pictured below is a figure of constant width, meaning that every projection of it in the plane is a line segment of the same length.

There are also surfaces of constant width in higher dimensions, meaning that any two parallel bounding set of hyperplanes (touching the boundary) have constant separation. But all of the non-spherical examples of such surfaces I have seen have obviously non-circular projections.

It also seems clear that finitely many circular projections is insufficient, since intersecting finitely many cylinders would produce a surface having corners and containing some straight line segments.

The fact that you can spin such a surface with all circular projections inside any bounding cylinder is suggestive, but it is also true that you can spin the Reuleaux triangle inside a square, even though it isn't circular.

Further questions would include:

• To what extent are other surfaces determined by their projections? That is, which other shapes can we recognize by the set of their shadows?
• In particular, can we recognize the cube and other regular solids by their shadows?
• Which sets of shadows are realizable as projections of a surface? Is there some way to characterize these sets? Clearly they must be continuously deformable to one another and obey several other obvious conditions.

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