Math Is Fun Forum

I'm convinced the Universe is a 3-dimensional manifold embedded in 4-dimensional hyperspace.

Mathematically, a manifold is a space embedded in the space of the next higher dimension in which every point is locally "flat". For example: the surface of the sphere is a 2-dimensional manifold (embedded in 3 dimensions). Close up, the surface appears "flat". The technical definition is this: An n-dimensional (real) manifold is a topological subspace

of

such that given any point x in

there exists a real number

(depending on x) such that the open ball

is homeomorphic to

.

Let us use the surface of the sphere as an analogy. As I've said, the surface looks flat over a small area – so much so that to a flatlander living in that area, the surface appears to be an infinite 2D plane extending boundlessly in all directions. But the surface is not boundless: it is finite. In the same way, the Universe may appear boundless on a small scale – we think we see space as infinitely stretchable in every direction we look – but the Universe (so I believe) is finite in the way the surface of a sphere is finite.

Lines that are "straight" on the surface of a sphere are actually curved in the 3D space the surface is embedded in; they're arcs of great circles. Travelling on a straight line on the surface means moving along the circumference of a great circle such as the equator. If you keep travelling along this "straight" line, you will eventually end up where you started from. Indeed, no matter how you move on the surface, you can never escape from it. In the same way, I'm convinced that straight lines in the Universe are actually curved in the 4D hyperspace in which I believe the Universe to be embedded. If, starting at a point in the Universe, we could keep travelling on and on in a straight line, we would eventually end up where we started! This is because lines that are straight in 3D are actually curved in 4D hyperspace. And no matter how we move around in the Universe, we will never escape from it.

Back to the sphere analogy. If the radius of the sphere increases, the surface area also increases: the world of the flatlander expands. This, I believe, is also why our 3D Universe is expanding: because the hyper-radius of 4D hyperspace is increasing. The Big Bang was the moment at which this hyper-radius was zero.

And this is my metaphysical theory of the Universe – which I will never be able to prove. I don't expect anyone to agree with me anyway (or even understand what I'm saying in the first place).