## What is the shape of the universe? (Intermediate)

I have read most of the popularizations of physics and astronomy Hawkings, Weinberg, ect. The big bang is often explained using the image of a two dimensional universe (surface of a balloon) expanding in three dimensions. I read once that our universe is a "three dimension surface of a four dimensional balloon" or maybe orange, I don't recall for sure. Is this correct? I realize that 4 and up dimensional spaces can be described mathematically although we can't really imagine them. To the extent possible, I would like to know the "shape" of the universe.

Most astronomers would like to know the shape of the universe too! There are three general possibilities. First, like your balloon, the universe might have what we call positive curvature, like a sphere. In this case, which we call a "closed" universe, the universe would be finite in size but without a boundary, just like the balloon. In a closed universe, you could, in principle, fly a spaceship far enough in one direction and get back to where you started from. Closed universes are also closed in time: they eventually stop expanding, then contract in a "Big Crunch." All the geometry that is true on a sphere is also true in a closed universe: parallel lines eventually converge (e.g. longitude lines are parallel at the equator, but converge at the poles), large triangles have more than 180 degrees, etc.

The second possibility is that the universe is flat. This kind of universe you can imagine by cutting out a piece of your balloon material and stretching it with your hands. The surface of the material is flat, not curved, but you can expand and contract it by tugging on either end. Flat universes are infinite in spatial extent, and have no boundaries. Parallel lines are always parallel and triangles always have 180 degrees. Flat universes expand forever, but the expansion rate approaches zero.

Finally, the universe might be "open," or have negative curvature. Such universes are sort of saddle-shaped. They are also infinite and unbounded. Parellel lines eventually diverge, and triangles have less than 180 degrees. Open universes expand forever, with the expansion rate never approaching zero.

What determines the shape of the universe is its density (and the Cosmological Constant, a sort of anti-gravity force allowed by General Relativity). It is difficult to figure out what the density of the universe actually is, but it seems that the universe is probably flat.