An infinitely repeating Universe would mean that someone could travel in a straight line and return to where they began.

Back when people thought the Earth was flat, it was almost heretical to suggest that traveling in a straight line for a long enough distance would eventually bring you back to your starting point. But it’s true: travel about 40,000 kilometers (or 25,000 miles) in any one direction — over the mountains, oceans and any other terrain you ran across — and return you would to where you began. It makes you wonder if space could be the same way. If you got in a rocket ship and traveled fast enough for long enough, and didn’t ruin things by colliding with a distant star or galaxy, could you eventually return to where you started?

A simulation of the structure of the Universe. If you leave one “edge” of the Universe and return through another, you just might live in a recurrent Universe.

It’s not as crazy as it sounds. We might think that the Universe is infinite, going on forever in all directions, but the evidence we have for the size and shape of the Universe is very much finite. For one, it’s only been 13.8 billion years since the Big Bang, and so we can only see the amount of space that 13.8 billion years of light traveling through the Universe and winding up where we are can illuminate. For another, there are hundreds of billions of galaxies, all appearing younger in the distant past the farther away we look. Is it possible that one (or more) of them is an infant version of the Milky Way we grew up in? And finally, could it be the case, just as the Earth has two dimensions we can move in on it (north-south and east-west, but not up-and-down), that the Universe might be a higher-dimensional structure like a hypersphere or a hypertoruswhere the various dimensions are closed and finite, curving back on themselves?

In a hypertorus model of the Universe, motion in a straight line will return you to your original location.

If that were the case, if you could travel in a straight line for long enough, you would wind up right back where you started. If you didn’t age, perhaps you could even wind up seeing the back of your own head just by looking for long enough, as your eyes would eventually encounter the light emitted from your own origin. If the Universe were like this, how would we figure it out?