So far, we’ve seen how the regular game of Asteroids gives us a Torus shaped world, and how switching one side around gives us a Klein bottle world that only really exists in 4D. But, what if we switch more sides around? What if we say we don’t want to play Asteroids on a regular rectangular screen like everyone else, but on a hexagonal screen? Or an octagonal screen? What if the edges of our screen were connected in stranger ways? What if our game of asteroids looked like this*?
This seems very weird, and quite different to what we had before. What strange surface would this game of Asteroids be played on? Surfaces like this, that are created by joining together edges of polygons, and where every edge is connected to one other edge only, are called closed connected surfaces. Our earlier Möbius loop is not a closed connected surface, as the top and bottom edges are not connected to other edges. However, our Klein bottle and the Torus are closed connected surfaces.
Ambitious though it may seem, it would be very interesting if we could understand what all possible games of asteroids, or closed connected surfaces, look like. Well, it turns out we can. And even more than that, it requires only a Torus, a sphere, and one other new surface which we’re about to be introduced to…
This new shape, like the Klein bottle, only truly exists in 4D. It is called the cross cap. The cross cap comes about by taking the surface of a half sphere, and gluing opposite points on the boundary circle together. This wonderful animation by Jos Leys demonstrates what it looks like, accompanied by some strangely creepy music.
Like with the Klein bottle, in order to make the cross cap in 3D space, we need the surface to cross through itself. On the graph below which shows the most important part of the cross cap, this intersection is along the z axis (interactive version here).
In the form of our earlier diagrams that show which edges are attached together, the cross cap looks like this:
The dots here represent the end of one ‘side’ of the circle, and the start of the other.
Now that we have all of the 3 building blocks we need to construct any surface, the next step is figuring out how to glue them together. To do this, we cut out a circle from each surface, then glue the surfaces together along these circles.
Following this method, the effect of attaching a torus to a sphere looks like this:
Which is called adding a handle, for fairly obvious reasons.
We can also glue together multiple toruses: 2,3,5, as many as you want, to give what I call ‘Mega Doughnuts’.
Finally, we can also glue cross caps together. This is a little more difficult to visualise so let’s look at this step-by-step to see what happens. We start with the diagram we had a moment ago, as it turns out these diagrams showing which edges are connected are very powerful when studying surfaces, as they allow us to represent quite complex surfaces in a simple way. Let’s start with our two cross caps:
As we’ve discussed above, to glue two surfaces together we need to cut a circle in both, and then glue those circles together. The result of this is shown in the diagrams below, where the red arrows show how the new circles we’ve cut out need to be connected together.
We can unfurl these circles at the bottom dot to get two triangles. We get triangles, as each of our cut ‘circles’ have 3 ‘sides’, the two ‘sides’ on the outside (purple/green arrows with the dot representing a corner), and the ‘side’ given by the inner circle we have cut out (the red arrow).
We join the triangles together along the red arrows to get this square.
Now we’re going to split our diagram into 2 different triangles along a new line, shown with the orange arrow below. It’s important to realise that we are not cutting the surface, as the sides of the line will still be connected (represented by the arrow along them), we are just changing how we are representing our surface.
Next we are now going to take the second triangle, flip it face over, and move it horizontally until the purple arrows align. Here’s another ‘handy’ video of me showing how this works.
The result is:
If we then stick the triangles together along the purple edge we get:
Finally, straightening up the shape into a rectangle, we have:
Which is our familiar Klein Bottle from the modified version of Asteroids in the previous article! Gluing two cross caps together gives you something that is equivalent to a Klein Bottle.
So far we’ve looked at gluing a torus to a sphere, and a cross cap with another cross cap, but what happens when you glue a torus to a cross cap? Well, it turns out that attaching a handle to a cross cap is the same as attaching a Klein bottle to a cross cap. A Klein bottle is itself two cross caps put together, so adding this to another cross cap is the same as adding 3 cross caps together!
So, we can glue handles to a sphere, we can glue cross caps to a sphere, but if we do both of them together we just end up with more cross caps. Topologists have shown that in fact any closed connected surface is either just a sphere by itself, or can be made out of only toruses glued together or only cross caps glued together. No matter how many sides we have, and in what weird ways we pair them off, we will still end up with either a mega doughnut, or something that resembles some Klein bottles glued together.
How cool is that? No matter how complicated the surface looks, it can be constructed out of a few basic building blocks – and only one type at a time! Complicated structures being built out of simpler blocks is a common feature across mathematics – the whole numbers (integers) can all be built from prime numbers for example. Yet we shouldn’t lose sight of how remarkable it is that mathematics allows us to make things simpler, to break objects down into smaller things we can more easily understand.
I hope you’ve enjoyed this journey into the field of Topology, via some arcade classics of the 1980s. Topology is a fascinating area of maths that often just gets summarised as “coffee mug = doughnut” or “geometry with Play-Doh”. I’ve tried in these articles to offer a different perspective to the one you’ll often find elsewhere, and to show what it’s like to actually do Topology, to manipulate these surfaces in order to understand them better.
One final observation. Over the course of our Topological journeys together we have done a lot of Maths, yet we’ve hardly used any numbers. At school it can seem like all Maths must involve numbers and algebra. Lots of algebra. But the kind of Maths you learn at school is just one narrow part of the much broader, stranger, more awe-inspiring universe of Mathematics that lies out there. Topology is just one blazing light among a whole sky of glorious stars, waiting to be discovered. I encourage you to get out there, to be like our intrepid astronaut, and explore it all.
This paper details a rather different, but also very intuitive, proof of much of what has been discussed here that was formulated by the late, great John Conway (there are also pictures).
*turns out this is a Torus as well, the adjacent red arrows pointing in opposite directions cancel out, to give us a familiar diagram. If we flip the direction of one of the red arrows, we get a cross cap.