Spherical Properties

I was reading the Wikipedia article about the eleven properties of a sphere (I can’t remember why, but I do stuff like that a lot) and I noticed the first property, which I’ll quote:

The points on the sphere are all the same distance from a fixed point….

The article then says:  “[This] is the usual definition of the sphere and determines it uniquely.”

The fixed point described is, of course, the center of the sphere, and the article says that only the sphere has the property described there. What I was wondering was, what if there is a fixed point for every shape from which the distance between it and any point on the shape is the same? I’d imagine this would have to involve curving space and some sort of 4th spatial dimension (not including the ones predicted by string theory). This would have to include everything from planes toruses (doughnut shaped, and yes, I Googled “what shape is a doughnut”. The Wikipedia article has some cool pictures in it :).

~ George

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