It is well known that you can get all sorts of things from a Japanese department store. On my recent trip to Tokyo, I found the following two items on the same floor of one store:
This got me thinking about the contents of this article. The math here is mainly descriptive - so I hope it can appeal to a general audience.
For other articles on maths and slides for talks I have given, visit my Thoughts on Maths page.
For other articles on maths and slides for talks I have given, visit my Thoughts on Maths page.
In 1979, a yellow circle with a mouth cut out was born. Invented by legendary game designer Toru Iwatani and released by Namco, the yellow circle was given the name Pac-man. His objective was to eat all the yellow dots in a maze while avoiding the four ghosts that chase him. Different releases then followed which also involved the introduction of Pac-man’s wife (Ms. Pac-man) and son (Pac-man Jr.).
One of the tricks in his arsenal is the focus of this article: to avoid the ghosts that are chasing him, there are two passages that lead to the left and right side of the screen respectively. If Pac-man enters the passage on the left, he will emerge from the right and vice versa.
One of the tricks in his arsenal is the focus of this article: to avoid the ghosts that are chasing him, there are two passages that lead to the left and right side of the screen respectively. If Pac-man enters the passage on the left, he will emerge from the right and vice versa.
The question we consider is this: is there a mysterious portal on one side of the screen that leads to the other? Or is there another reason that explain why Pac-man can seemingly teleport from one side to the other.
Pacman on a donut
Take a rectangle made of paper. Roll up the rectangle and then glue the opposite sides together. The resulting shape is a cylinder. This is the key observation. This can be seen in the YouTube video below with steel instead of paper (actually there are several other similar videos with paper but this is one is more exciting).
If we suppose that Pac-man lives on a cylinder, then it is possible to go from one end of the maze to the other without teleportation.
You may be wondering: from an outside viewpoint, it is clear that Pac-man lives on a flat surface and not a cylinder. However, considering Pac-man’s viewpoint, stuck in the maze that he is, he would not be able to tell whether he is on a cylinder or a flat surface just by looking at the ground. Indeed it took several million years for humans to realize that they lived on a sphere and not a flat surface. Two things to note:
1) The fact that Pac-man can keep going left (or right) and end up where he started should, in fact, tell him that he does not live on a plane.
2) In fact, as we will indicate later, a cylinder is a flat surface!
You may be wondering: from an outside viewpoint, it is clear that Pac-man lives on a flat surface and not a cylinder. However, considering Pac-man’s viewpoint, stuck in the maze that he is, he would not be able to tell whether he is on a cylinder or a flat surface just by looking at the ground. Indeed it took several million years for humans to realize that they lived on a sphere and not a flat surface. Two things to note:
1) The fact that Pac-man can keep going left (or right) and end up where he started should, in fact, tell him that he does not live on a plane.
2) In fact, as we will indicate later, a cylinder is a flat surface!
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In later versions of Pac-man, there are mazes in which it is also possible to enter a passage on the top of the maze and reappear in the bottom (like in the screen shot above). If we assume that Pac-man is not teleporting, then what kind of surface does he live on?
Take a rectangular piece of paper as before. Gluing the right and left sides results in a cylinder. What do we get if we now glue the top and bottom sides together? Firstly, it is impossible to do this with paper without scrunching up the paper - you'd need a sheet of rubber. So if we suppose we had this instead, glueing the top and bottom surface of the cylinder results in a donut shape. So in the maze above, it appears that Pac-man lives on a donut shape. In maths, we call a donut shape a torus.
Take a rectangular piece of paper as before. Gluing the right and left sides results in a cylinder. What do we get if we now glue the top and bottom sides together? Firstly, it is impossible to do this with paper without scrunching up the paper - you'd need a sheet of rubber. So if we suppose we had this instead, glueing the top and bottom surface of the cylinder results in a donut shape. So in the maze above, it appears that Pac-man lives on a donut shape. In maths, we call a donut shape a torus.
Teleportation in real-life...
..is not possible. The reason is that a consequence of Einstein's theory of relativity shows that it is not possible to travel faster than the speed of light. This does not exclude the possibility that we can get from point A to point B faster than the speed of light though. Why? We might be able do what Pac-man does! Just like Pac-Man can go either "left" or "right" to get from one point to another, there might be "short-cuts" that one can take to go between two points - you don't have to be able to travel faster than the speed of light to do this. These short-cuts are called wormholes by physicists (trust the physicists to come up with cooler names than the mathematicians). They are also called Einstein-Rosen bridges - these are what Dr. Selvig and Bruce Banner keep going on and on about in the Marvel's Thor movies.
The "creatively named" Einstein-Rosen bridge depicted in Thor: Ragnorok
Another viewpoint
Here is another way of think about things. In the original case with exits on the left and right only, imagine that when Pac-man takes the left exit, he enters different copy of an identical maze from the right. Similarly, if he takes the right exit. In this way, you can imagine that we have a whole strip consisting of identical mazes glued side by side. Using this viewpoint, Pac-man doesn't teleport, he simply enters a new copy of the maze with ghosts and dots at identical positions. This seems like a huge over-complication.
In the other case with exits in all four directions corresponding to a torus, then we assume that we have a plane of Pac-man mazes such that whenever Pac-man exits one, he enters an identical copy at the corresponding place.
Again, we seem to be making things more complicated. But this idea will be very important shortly. In the torus case, the plane is called the universal covering surface of the torus. The geometry of the universal covering surface, the "standard" geometry (called Euclidean geometry) of a 2 dimensional plane can be "passed down" (or projected) to the torus. In this way, we can define lengths and angles on the torus. Evidently, planes are flat - the geometry itself determines this; this means that the torus is also (paradoxically) flat.
The fact that the geometry determines the curvature is a deep result in mathematics proved by the great mathematician Carl Friedrich Gauss which he called his Theorema Egregium (which is latin for "eminent theorem"). For comparison, the surface of a sphere has positive curvature.
Again, we seem to be making things more complicated. But this idea will be very important shortly. In the torus case, the plane is called the universal covering surface of the torus. The geometry of the universal covering surface, the "standard" geometry (called Euclidean geometry) of a 2 dimensional plane can be "passed down" (or projected) to the torus. In this way, we can define lengths and angles on the torus. Evidently, planes are flat - the geometry itself determines this; this means that the torus is also (paradoxically) flat.
The fact that the geometry determines the curvature is a deep result in mathematics proved by the great mathematician Carl Friedrich Gauss which he called his Theorema Egregium (which is latin for "eminent theorem"). For comparison, the surface of a sphere has positive curvature.
What about a pretzel?
A pretzel
Can Pac-man live on a pretzel? Well Pac-man can live wherever he wants. There is a difference if Pac-man lives on a pretzel though. Just like before, we can create the universal covering surface by cutting open the pretzel surface and glueing copies of this together at the places where cuts are made. This forms the universal covering space of the pretzel surface. Note that pretzels have more than one hole.
However, we now encounter a problem! This universal covering space of a pretzel surface cannot be assigned a flat geometry such that all angles and lengths are preserved. Hence, this Pac-man game cannot be played on a flat screen such that the whole map is visible without distortion. In fact, if we want to have this property, the only possibilities are for Pac-man to live on part of a cylinder or on part of a torus. This result is a consequence of another deep result called the Uniformization Theorem. In particular, the pretzel surface has negative curvature (like looking at a bowl from the top).
In conclusion, Pac-man is stuck to living on a cylinder or torus - if he wants the players to be able to play the game on a flat screen.
However, we now encounter a problem! This universal covering space of a pretzel surface cannot be assigned a flat geometry such that all angles and lengths are preserved. Hence, this Pac-man game cannot be played on a flat screen such that the whole map is visible without distortion. In fact, if we want to have this property, the only possibilities are for Pac-man to live on part of a cylinder or on part of a torus. This result is a consequence of another deep result called the Uniformization Theorem. In particular, the pretzel surface has negative curvature (like looking at a bowl from the top).
In conclusion, Pac-man is stuck to living on a cylinder or torus - if he wants the players to be able to play the game on a flat screen.
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