What makes the inside of a shape the inside?

Question

A question occurred to me when browsing SE this evening, just curious.

What specifies the inside of a 3D construct? If I have a hollow sphere, what's to say that the world isn't the inside, and the center of the sphere isn't the outside. Picture the Asylum in the Hitchhiker's Guide to the Galaxy series, if you have read that.

Additionally, if the inside of a 3D object is defined as what is smaller than the outside, what happens when there is a a sphere with a volume equivalent to 51% of the universe?

Answer 1

Mathematically speaking, when we talk about a two dimensional manifold embedded in 3-space (like the hollow sphere you're talking about), we have the assumption that the universe is unbounded. Thus, the surface acts as the boundary for a bounded interior and an unbounded exterior, and it's that bounded/unbounded distinction that defines which is the inside and the outside.

For the two dimensional case of this proof, and the higher dimension analogues, see http://en.wikipedia.org/wiki/Jordan_curve_theorem

Answer 2

Interesting question! I think that the boundary of the sphere is perceived differently by intelligent beings on the inside than those on the outside. For those inside, the boundary of the sphere will eventually block them, in every direction that the beings choose to travel. For observers on the outside, the boundary might present a mighty obstacle in front of them, but they can also choose to move away from the sphere. In that case they can travel arbitrarily far without ever meeting the sphere again.

Now it is possible that the beings living outside the sphere also encounter a spherical barrier if they travel far enough. Would they be able to distinguish this outside barrier from the sphere in the centre of their universe? Yes. They can paint the outside barrier blue, and the surface of the sphere red. This procedure should not result in a contradiction; in the sense that the colours red and blue can never touch each other. If the opposite occurs, then the universe has a strange topology.....