Living on a Torus

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You might think that life in a three-dimensional torus that measured only several metres in each dimension would be cramped. To the contrary, you would perceive space to expand infinitely around you in all directions.

It would certainly be a peculiar existence. To pass the time, you might amuse yourself by tying ropes between the many cylinders that you see around you. You tie a rope to the cylinder next to you, and start heading toward a cylinder off in the distance. Intriguingly, when you arrive at your destination, you're surprised to see that someone else has already tied a rope there. When you look around, you're shocked to see that in fact, ropes seem to have been tied everywhere on all cylinders as far as your eye can see.

While musing over this enigma, out of the corner of your eye you notice another creature that looks just like you. In fact, there are many just like you in all directions! Sadly, whenever you try to approach one, they run away, and no matter how much you try, you can never close the distance.

If you're on mobile or viewing the embedded visualization above on the desktop, you can drag around to move around the cylinder with the red sphere. Pinching or scrolling will zoom in and out. Touch or click on a cylinder to launch a rope to that cylinder.

At the top of the page there is a link to the full-screen version, which is a more immersive experience. In full-screen desktop, you have complete freedom to navigate the torus using your mouse and keyboard: WASD to move forward, back, up, and down. Arrow keys to turn, Q and E to roll, R and F to move up and down. Moving your mouse around will also turn you. Placing your mouse cursor over a cylinder will set a target and pressing space bar will launch a rope that will connect the source to the target in a straight line. If you move through the space, you will have the impression that it is never ending and that it extends infinitely in all directions.

The smaller canvas in the bottom left of the visualization demonstrates the finite reality of the torus space. What looks to be an infinitude of cylinders, cones, and ropes are in reality just single objects viewed from multiple angles.

The situation is reminiscent of how stars in the sky can be perceived in multiple directions. According to Einstein's theory of general relativity, light travels from A to B along paths that are locally shortest, which are known as geodesics. Objects can appear at different points in the sky when there are multiple geodesics connecting those objects to Earth, which happens when light from that object passes by a heavy object on its way here.

In our torus universe, it is also the case that there are multiple geodesics connecting the cylinder to our eyes. It is not due to heavy masses bending the course of light, but due to the looping geometry of the torus.

You can think of the torus geometry as a cube with opposing sides glued to one another as in the canvas at the bottom left of the demonstration. The sides of the cube are portals that instantaneously warp you across to the same point on the opposing side. The portals transport not just objects, but light as well. Because the light can loop around space, there are infinitely many geodesic paths from any point to any other, which results in many visual copies appearing all across space.

Note that the red and green spheres depicted in the scene are not real objects in the torus. If they were, we would see them duplicated all across our visual field. Instead, they are just virtual entities that are convenient for specifying a specific path through the torus.

When you launch a rope between the red and the green spheres, it seems evident that with the ends of the rope fixed to the cylinder, no amount of stretching or pulling could ever hope to gather all the rope back into one spot. The only exception is the case where the red and green spheres are the same. Contrast this with the case of a normal flat three-dimensional space. No matter how you pass a rope around in space, you can always pull the rope in and gather it all up at the base point where the ends are fixed. This difference in behaviour of the ropes is one way to prove that the torus space is distinct from normal three-dimensional space.

The ensemble of paths of rope where the ends are fixed to a given base point, are an essential object of study for identifying and distinguishing spaces from each other. They are so important that they are known as the fundamental group.

We talked about groups in the Rubik's Cube blog post. Recall that a group is just a collection of objects that can be combined with each other, that has a trivial object whose combination rule does nothing, and where each object can combine with an inverse object to give the trivial object.

So how is the fundamental group a group? To combine two paths of rope, we simply take a rope that first goes around one path and then the other. The identity path is just any path where all of the rope can be pulled in and gathered at the base point. The inverse path then, is obtained by taking a rope and tracing along the original path in the reverse direction.

If you launch a second rope within the torus, you'll get to see the fundamental group operation in action. The red and green loops are concatenated to form a new loop, which is then pulled taut into its canonical straight line configuration.

By watching how the ropes combine with each other in the torus, you'll notice that the fundamental group has the same structure as a triple of three integers that sum up the number of times the ropes loop around the \(x\), \(y\), and \(z\) axes.

I got the inspiration for this math demonstration from reading Chapter 1 of Three-Dimensional Geometry and Topology by Bill Thurston.