In the last article, we saw how mathematically speaking, Pacman takes place on the surface of a cylinder, and Asteroids is on the surface of a doughnut (or a Torus as less sweet-toothed mathematicians call it).
When discussing Pacman, we assumed that when Pacman goes through a ‘warp tunnel’, he isn’t flipped over by the experience, which means his sense of ‘up’ stays the same as the up on the screen. But we don’t know this. Classic Pacman has no eyes, or any other feature that may allow us to determine his orientation. What if when he goes through a warp tunnel, he is flipped upside down?
This changes the setup – or topology – of the situation completely. As the sense of up is switched when Pacman goes through the tunnel, the left hand side’s up direction is opposite to the right hand side’s up direction. This leads to the diagram looking like this, with reversed arrows to before:
If we try and fold it around into a cylinder like we did last time, then the arrows no longer line up. This just won’t do! Instead we have to twist the strip as we join it together, as this video of my lovely hands shall demonstrate:
I encourage you to have a go at making this shape.
The surface we have created is known as the Möbius loop, which was independently discovered by German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858. This famous surface has some curious properties. Firstly, it only has one side. Try for yourself. Take a pen, and put it on your Möbius loop. Then draw a line along the curve. Eventually you will get back to the point you started drawing at, only to find that in the process you’ve drawn along the whole surface of the loop.
Contrast this with a regular loop of paper, where if you were to do the same thing, you would only draw along the outer side of the loop.
Another curious thing occurs if you try to cut the loop in half. If you cut along the line you’ve just drawn, the resulting loop now has two twists in it, which means they cancel out. It no longer has its one sided property. In fact, this new loop is equivalent to a cylinder.
To see this, let’s return to the diagram, and draw a ‘cutting line’ across the middle.
I’ve given the different halves of the sides different coloured arrows, as the top left side is what joins with the bottom right side (green), and the bottom left joins with the top right (blue) when forming the original Möbius loop. We now cut along the red line to get two separate strips:
We then flip the second strip face over to get this:
Finally, we attach the strips together along the blue arrows, to get back to:
This look familiar? It’s the original Pacman diagram – the one that gave us the cylinder! This means the cylinder and the cut Möbius loop have the same diagram, and so in the eyes of Topologists (mathematicians that study the properties of surfaces), they are the same. This is just one example of the strange way surfaces and shapes can be equivalent in Topology.
So far we’ve seen that by messing about with the arrows in Pacman, we can create a completely new, and interesting shape. What if we similarly play with the rules of Asteroids?
In the original game, if we leave the screen at the top of the left-hand side, we come out at the top of the right side of the screen. What if we came out at the bottom of the right side of the screen instead? Like in the above Pacman example, the sense of up and down is changed as we pass from one side to the other. Crucially though, we will keep things the same as the original when going off the top or bottom of the screen. This gives us this diagram:
Note how this diagram is subtly different to the one we drew last time (below). The right hand arrow has flipped. We’ll soon see how this small change marks a big difference.
Returning to our new diagram, we can join the top and bottom sides by rolling it up into a tube. Like when we made the Torus, we then want to join the ends of the tube together. But unlike the Torus, we have to join the ends in such a way that the orientation of one of the end circles is reversed, since we want the arrows to line-up with one another.
If you have some bendy tube at home, try to make this work. You’ll soon discover that no matter which way you twist and bend the tube, you’ll never be able to get the arrows to line up. To quote another famous bit of space-related media; Houston, we have a problem.
So what do we do? We enter the fourth dimension of course!
Physicists often like to say the fourth dimension is time. Mathematicians are more flexible. If 2D space (i.e. the xy-plane) can be described with 2 coordinates (an x and a y), and 3D space can be described with 3 coordinates, (x, y and z), then 4 dimensional space is just a space described with four coordinates (usually x,y,z and w). If we allow the surface we want to make to exist in 4D space, then the extra dimension gives us enough wiggle room to allow the ends of the tube to join in the desired way. The resulting shape is the famous Klein Bottle.
Sadly, humans are not very good at thinking about 4D space, by virtue of not living in it. However, like how a 3D object creates a 2D shadow, a 4D object like the Klein bottle creates a 3D shadow, which we can visualise, and even make.
Sadly, the process of taking the 3D shadow of the Klein bottle means we have a self-intersection, that is a point where the tube has to pass through itself. However, as you can see the resulting object is still very cool. The Klein bottle is in many ways the equivalent to the Möbius loop in a higher number of dimensions. For example, you may be able to see from the picture that it too has only one face.
Thus, from an action as simple as flipping one direction in Asteroids, we’ve created a surface that cannot exist in our universe, and requires us to think about 4D space. It may seem daunting that something so simple can lead to seemingly complicated, higher dimensional surfaces, but I prefer to see it another way. Strange shapes that exist beyond human perception and comprehension, that defy the properties of our physical universe, can still be understood, using no more than a retro video game and a rectangle with some arrows on.
That to me is out of this world.
In the final part of this series, we will approach the ‘final boss fight’, and think about the wide range of other surfaces we can create by joining edges of polygons.