Imagine a point, which could be understood to be a zeroth of a Planck-length folded in on itself infinitesimally. A universe existing in a point would have no scale everything would be the same size because that size is the same size as the the universe itself. This shape is not just non-Euclidean, its non-Newtonian, non-Einsteinian,and non-Hawkingian. It can’t exist except as a misunderstood notion in the mind of a human.
So now that you have this bit of nonsense in your mind imagine that you extend this point out giving it length. Now we will also consider this line to be a universe, and it’s a universe that has as much in common with our universe as possible, in keeping with Copernican principle. If it is anything like our universe, as it extends into infinity it curvesarount the second dimension forming a circle. Now to the inhabitants of this line universe there would only be length extending forever, but to the outside observer it can be seen that their universe actually exists in two.
Going another step out we extend this circular dimension out in the second dimension, width, forming a plane wrapped in on itself, a cylinder, that extends forever. Once again this infinite expansion forms a curve in the third dimension, depth, creating a torus. So here we are in the three dimensional universe observing what seems to be a three dimensional shape, but to the inhabitants of this planar universe, there are only two dimensions in which to exist and travel.
So here’s where the visualization starts to break down again, because its simple enough to picture one, two and three dimensions, but the zero, four and more than four are extraordinarily hard to imagine accurately. But lets give it a go. So in order to give the planar universe a third dimension we have to extend the plane in the third dimension, depth, infinitely so that it wraps around on itself in the fourth dimension, this ensures that no matter what direction you travel you come back around. The shape in question would be similar to a torus but at every point along its axis it would be made of… toruses? tori? Tori, (thank you google) forming some kind of fourth dimensional hypertorus … I know, right? But because we inhabit the three dimensional universe we can only see the three dimension in which to exist and travel.
If a fourth dimensional being were to look at us he would see that we obviously form a hypertorus. And to travel in this new dimension would be a strange experience, like moving in and out… of yourself… that’s what she said. I am…sorry. that will not happen again. So back to the point, or rather the hypertorus. Though now that I think of it, the Hypertorus seems to have a lot in common with the point, it doesn’t make any sort of physical sense, it folds in on itself infinitely and can only exist as a weird misunderstood motion in the mind of a human.
I asked the folks over at NASA’s “Ask an Astrophysicist” if this was a fair interpretation of the theory and the was there response in full:
Hi and thanks for your question! It can be extremely difficult, if not impossible, to visualize geometry in higher dimensions. Usually we have to make analogies to two and three dimensions that we can picture and then let the mathematics guide us as we go to higher dimensions. As you point out, the surface of a torus that we are familiar with (doughnuts would be my favorite) is two-dimensional, although the curvature is in a third dimension. One can mathematically describe a three-dimensional torus that has similar properties, with the curvature in an additional dimension or dimensions. It is this “3-torus” shape that some researchers suggest for the shape of the universe based on CMB data. To make a 2-D torus, you essentially multiply two circles in two orthogonal dimensions. Similarly, to make a 3-torus, you combine three circles in three orthogonal dimensions.
Hope that Helps
-Ira & Bernard
for “Ask an Astrophysicist”
What I gather from this answer is that while the truth is slightly more complicated, my visualization, though a bit less useful for getting the idea across, is a valid way of thinking about the universe. Yay Science! I am glad that I found my thought process to be sound, because the progression of dimensions that i described as a sort of symmetry the appeals to my love of beautiful mathematics, and instills in me an appreciation for the shape of the world. I hope it does the same for you.
And that’s my take on the shape of the Universe.