#### Where do you come from, where do you go?

For those without a deep education in mathematics or physics, these concept can be hard to explain. It does not help that some of the core terms can have multiple meanings.

If you don't mind the very "of the times" language and imagery, the book Flatland, by A. Square is an extraordinary exposition of the subject of higher dimensions in the form of an engaging story. My explanation is going to take a very different tack, so if why I write does not help, you might look there.

As hinted, part of the problem is the need to be very clear about the terms in use. Let's start with "dimension". Consider a Cartesian plane. Such a thing is not real. Specifically, there is no corresponding physical entity to a Cartesian plane. Nevertheless, we are comfortable working with it to solve mathematical problems. As the problems get more interesting, we create more interesting mathematical objects to address them. Cartesian 3-dimentional space. Cartesian 4-dimentional space. Spherical spaces. Hyperbolic spaces. Whatever.

Most of the time, we find it useful to examine entities "from the outside". If we are considering the set of points x, y such that x * x + y * y = 1, we don't think of our selves as inside the curve--indeed it would be really difficult to do so. For the points x, y, z such that x * x + y * y + z * z = 1, however, we often talk as if we can. Because there is a (very rough) approximation that forms our lives, namely the surface of the earth.

Suppose you met with someone from an isolated tribe. In their experience, the world is flat. You inform them that the earth is round, and perhaps there is a fortuitous eclipse that even allows you so show the Earth's shadow. They hear your explanation, see your demonstration, and laugh at you. Why? "Because surely everything on the other side falls off!" In their minds, "down" is not "whatever direction gravity pulls", but is a coordinate in their Cartesian understanding of nature of the world.

This is key. Their mathematical model of world is a Cartesian 3-space. In such a space, a finite world has to have an edge. And a center that one might reach by walking or sailing. Of course, their model is wrong.

This kind of thinking is hard to overcome. I read a philosophy paper written in 2010 which began, "We know with probability one that the universe is infinite." That's a whole lot of wrong for so few words. Nevertheless, the author, and whomever he consulted for the paper, all educated people, thought such a sentiment true.

Every direct observation we can make tells us that the Space is at least roughly a Cartesian 3-space. Perhaps as a result of this, our brains are not wired to think about other possibilities. But our physicists tell us that it is almost certainly not the case. Recall that a circle (S1) is one-dimension. But to look _at_ this one-dimensional thing, you think of it as sitting inside a two-dimentional Cartesian system--specifically you think of it as the surface of a two-dimensional ball. Likewise, the sphere (S2) is a two-dimensional thing, but we think of it as sitting in a three-dimensional Cartesian system--again as the surface of a three-dimensional ball. Physicists tell us that space is probably shaped like the surface of a four-dimensional ball (S3).

But our brains don't really go there. At all.

So for your question, "where is the center"? There is in fact an answer. Just as the center of S1 is not in S1, and the center of S2 is not in S2, the center of S3 is not in S3. If we model Space as sitting in some four-dimensional space, however, we can specify it. But never go there. So the obvious follow up question, "Is the center 'real'?" becomes a matter of philosophy.

Unless further observations force us to consider forces operating on the Universe from the "outside" ...