Shape of Our Universe

Today I tried to recreate for my chemistry class a presentation that changed my life. At Governor’s School last year one of the math teachers gave an optional lectured titled Shape of the Universe. Although this wasn’t the only math lecture I attended at Governor’s School, and I probably wouldn’t have said it was my favorite at the time, it is the one that has stuck with me the most. The original Shape of the Universe presentation, along with the other math presentations at Governor’s School, really changed my perspective on math. I began to view math as a collection of concepts to be understood, rather than a set of problems to be solved. This presentation in particular helped shift my view, as it contained almost no numbers. It made me realize that numbers were just one way to explore mathematical concepts. Numbers are a tool to understand math, not the purpose of math itself. All that being said, this is my textual adaptation of the presentation I gave today:

What is topology?

Topology is a bit difficult to define with words; the definition given by a quick Google search is “the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.”. This definition is a bit difficult to understand, so I find it easier to use an image to explain.

The top row of images are all topologically the same because they contain no holes. Any one of those could be stretched and misshapen to form any of the others. The shapes in the second row, although they are visually similar, are not topologically the same. One contains a hole, while the other is simply a circle bent into a strange shape. This means they are topologically different, as you could not create the donut shape from the other shape without cutting or gluing faces. This is important to note because when I refer to the shape of the universe, I am really referring to its topology. Some of the models I’ll be showing aren’t uniformly sized. This isn’t to imply that one part of the universe would be larger than another; we are only concerned with the connectedness of the theoretical spaces. This will make more sense after we look at some visuals.

A Two Dimensional Universe

Before we explore what the shape of a three-dimensional universe might look like, it is useful to think about possible shapes of a two-dimensional universe. Let’s use the analogy of a game of Pacman.

The important thing to note here is how Pacman seems to teleport across the screen upon reaching the edge. You could think of this as teleportation through programming magic, but for the sake of this exercise, it is better to think of those two locations as physically joined. To physically join these two locations, you must bend the entire screen to form a cylinder.

This image–taken from Lev Kruglyak’s blog¹–shows that Pacman does not simply teleport across the screen, but is simply moving around the outside of a cylinder.

Of course, Pacman has no understanding of moving through a third dimension. Just as you probably can’t imagine moving through a fourth spatial dimension, Pacman can’t understand that he exists on the surface of a 3D object. Although the cylinder is certainly a more interesting shape than a flat plane, it isn’t the most interesting possible topology of the Pacman universe. It only connects in one direction, leaving a barrier at the top and bottom of the screen. If we connect those two sides, we end up with a space that looks a bit like this:

This is essentially the same operation as before, just repeated on the other side. In the first example, we created a cylinder by bending the plane of Pacman’s world such that one edge connected to its opposite, leaving two open edges on the cylinder. This process can be repeated to connect those two open edges, resulting in a torus.

One may think that a spherical universe may yield the same results. If I’m being honest, I don’t yet fully understand why it doesn’t, but I can say that for the rest of this post, it makes a lot more sense to think of this as a torus, rather than a sphere.

Something a little more Complicated

Now that we understand the basic ideas of bending 2D topologies, we’re going to try something a little more complex. We’re going to imagine Pacman’s universe as a Mobius strip. A Mobius strip is a simple shape with a lot of interesting properties, but today we’re going to look at how the topology of a mobius strip would be perceived by a 2D creature.

The Mobius strip is very similar to a cylinder, but with one key difference. For each full revolution around the strip, the surface rotates one half turn. This means that a two-dimensional object travelling along this surface would be mirrored after each full revolution.

An easy way to understand this concept is to take a thin strip of paper and draw a line down the middle with a marker, making sure the ink bleeds through to both sides of the paper. Then, draw single-barbed arrows every few inches on the line, once again ensuring that marker bleeds through the paper. Rotate one end of the strip 180 degrees, and bring the two ends together, making a Mobius strip. You will notice that, at the border, the barbs on the arrows switch sides. This demonstrates the mirroring of two-dimensinal objects travelling along a Mobius strip.

Using the Pacman metaphor, each time Pacman goes through one wall of the game, he comes out the other side mirrored. This operation would be impossible with only two-dimensional transformations. This is the equivalent of you, a three-dimensional creature, traveling around the universe and coming back to find everything a mirror image of what you remember. From Pacman’s perspective, he has simply travelled a distance, and returned to find himself mirrored.

A Step Further

What would happen if we tried to perform the same operation we used to turn the cylinder into a torus to a mobius strip? If you have a strip of paper in front of you, I encourage you to try. Quickly, though, you’ll find that it is impossible. The shape this would create is called a Klein bottle, and is impossible to create in just three dimensions. Just as a 3D shape like a cube can be drawn in 2 dimensions if you’re okay with a few lines intersecting, a 4-dimensional shape can be sculpted in three dimensions if you’re okay with a few surfaces intersecting.

So, what would Pacman’s universe look like if it had the topology of a Klein bottle? In one direction, it behaves like a cylinder, one side of the screen leads directly to the other, with no twists. The other side, though is a mobius strip, so going that side leads to coming out the other side as a mirror image.

A Three Dimensional Universe

Now that we’ve explored some possible topologies of a 2D universe, it is time to look at possible three-dimensional universes. There are many possible topologies of a three-dimensional universe that we will not explore (as well as many 2D topologies that we did not explore), but I will show the 3D allegories to the torus and Klein bottle 2D topologies we looked at above. Unfortunately, my art skills are not good enough to create a pixel-art 3 dimensional Pacman universe, but I have drawn a few diagrams.

The above image show the 3D version of our 2D torus topology. This is what our universe would look like if we existed on the volume of a 4D tours. Just as 2D shapes have 1D edges and 3D shapes have 2D surfaces, 4D shapes have 3D volumes around them. This is impossible for a human brain to fully comprehend, but it does explain how the entire volume of our 3D universe could exist on the “edge” of a 4D object. In this example, going in any direction will lead you right back where you came from, with no mirroring. It is possible, though, that we live in a 3D space in which going far enough in one direction would lead you to find a mirrored image of the world you left, this is the 3D equivalent of the surface of a Klein bottle.

Each figure is holding a sword to make it clear when one is mirrored. Going through any of the side walls leads you to exit through the opposite side, unmirrored, however the “ceiling” and “floor” are mirrored, the red figure is mirrored as it passes through the floor, and reenters the room in the opposite corner of the cube. These two sides are those connected by the equivalent of a mobius strip. Of course, the figure is not actually passing through a floor or reentering through the ceiling; from the perspective of the figure, the room never ends, it is simply connected to itself.

Conclusion, further reading, and some fun resources.

There are many possible topologies, perhaps infinite possible topologies, of our 3D universe. In my presentation, I only explained a few possible topologies because it had to be short, but there are so many I did not explore. I highly recommend playing with a piece of software called Curved Spaces². It is available on iOS, macOS, and Windows. It also runs on Linux with Wine, but Android seems to be entirely unsupported.

I also read a few pages from the textbook The Shape of Space³, which was helpful. I didn’t dive too deeply into the concepts explored in that book, but it was useful to look at some of the diagrams.

Some other interesting topologies that I would like to explore and better understand later are the real projective plane and half-turn space. I’m mostly leaving this as a reminder to my future self to read about these topics.