One knows a sphere when one sees one.
Problem is, how do you define a sphere? A simple answer might seem to be, if the diameter is constant from every direction, then it is a sphere.
Many years ago a friend told me that it is not so. There are many bodies with this property and they are not spheres.
Just think of a two dimensional projection, my circlesquare from previous notes may qualify for a constant width curve. If you take the square part, due to symmetry, any line through the center has the same size. There you have it; but the size should be maximum from every direction. So this is not a constant width curve.
Problem is, how do you define a sphere? A simple answer might seem to be, if the diameter is constant from every direction, then it is a sphere.
Many years ago a friend told me that it is not so. There are many bodies with this property and they are not spheres.
Just think of a two dimensional projection, my circlesquare from previous notes may qualify for a constant width curve. If you take the square part, due to symmetry, any line through the center has the same size. There you have it; but the size should be maximum from every direction. So this is not a constant width curve.
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