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Pacman lives on the 2D cylinder, $S^1 \times I$, where $I \subset \mathbb{R}^1$ is an interval of the real line. So that we can play the game on a flat surface, we portray Pacman's world on a fundamental domain, a rectangle, and identify two boundary edges in the appropriate way. Let's say the width of the screen is $2\pi$.

Suppose instead, Pacman lived on the Mobius strip. Would we need to double the screen width to $4\pi$? Or would that introduce a 2-1 redundancy in configurations on the Mobius band? In other words, is a reversed orientation at the same place considered a different configuration? Has his world gotten twice as big, or is it the same size with peculiar properties? Would we see Pacman reverse or would we see his world be reversed? For example, if Pacman had a mole on his right cheek while traveling rightward, would we now only see the mole on his left cheek while traveling leftward in the section of the screen that is $2\pi < \theta < 4\pi$?

Help me understand Pacman on the Mobius strip. Thank you.

Johnver
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    Depends on what you mean - are the maze walls the same on both sides? You can, in theory, do it with just the $2\pi$ width in that case, and just have pacman re-appear on the other side. But if "on the strip" means that potentially there are different mazes on the other side, then there is less of interest - you are essentially playing on a $4\pi$ cylinder. – Thomas Andrews Sep 26 '16 at 20:48
  • (Many years back I wrote an article explaining asteroids on a Klein bottle. http://www.thomasoandrews.com/klein/explanation.html ) – Thomas Andrews Sep 26 '16 at 20:49
  • Well there is no thickness to the Mobius band, so the maze walls would have to be the same on either side, correct? – Johnver Sep 26 '16 at 21:53
  • Very nice article by the way. I am still confused, however, about the orientability. Suppose you are a flatlander. You let your dog out while you stay fixed in the space. The dog makes one $2\pi$ circuit around the space. Can you interact with your dog? Is it considered to be in a "different configuration" or is it the "same configuration" with different orientation? – Johnver Sep 26 '16 at 22:07
  • It depends on your definitions, but there is no reason why you couldn't interact with your dog in that circumstance. – Thomas Andrews Sep 26 '16 at 22:22
  • If I was the dog, then is it true that I would not see myself reversed, but rather my environment reversed? In contrast, if I am the owner, I see my dog reversed, by not the environment. – Johnver Sep 26 '16 at 22:24
  • Yes. A three-dimensional example - if you and a friend were separated, and your friend went around the loop once, and if you were both right-handed to start, when you meet again, you will think your friend has become left-handed, and you've stayed right-handed. Similarly, for your friend, he'll still think of himself as right-handed, and you'll appear left-handed. – Thomas Andrews Sep 26 '16 at 22:28
  • So if you and your dog always stayed close together for all his formative years, and your right hand always had the treats, then when the dog got twisted, he'd expect treats from what to you seems like your left hand. – Thomas Andrews Sep 26 '16 at 22:29

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A flatlander who lives on a Möbius strip has a 'local' orientation. The stuff that is on his left can, if he travels in a loop that passes through the twist once, be later seen to be on his right, standing at the same spot looking in the same direction.

Consider if you and your dog stick together for all of his training, and you always give him treats with what you perceive as your left hand.

If you then tie up your dog, and go around the strip, when you get back, your dog will expect treats from the wrong hand.

Another way to imagine this is to consider the universe as consisting of two exact copies of you on a cylinder, with one oriented the opposite way. So when you are at $(x,y,z)$ then your duplicate is always at $(-x,-y,-z)$ with $x^2+y^2=1$ and $z\in [-1,1]$. Everything in this universe is similarly duplicated. So, if you travel around to the opposite side and find your "other" dog there, he is oriented differently than your first dog.

You could emulate this in a video game by showing two PacMan on a standard rectangle $[0,1]\times[0,1]$ such that when PacMan 1 is at $(x,y)$ then PacMan 2 is at $(x\pm 1/2,1-y)$. The maze and the ghosts would all have to also be similarly duplicated, and the joystick controls would be mapped from exactly one of the Pac Men - so you'd keep your eye on only one. This is an interesting example, because you'd actually not need to render the "other" you, because if the "other" you is about to die, so are you. So you can actually visualize it entirely as you on a cylinder, with the ghosts having a duplicated behavior, and the maze having a certain inverted symmetry. That might be the best way to render the game for ease of play.

A final way is to give the Möbius strip "depth." This is like my Klein bottle maze, where, when you (a 3D person) get back to a point where you started, you are "flipped" - your feet are on the surface the ceiling used to be on. The Möbius strip in this case is the midpoint between the floor and the ceiling. (There is another way to give the Möbius strip depth, where you just invert orientation like above, but that is essentially the same as before.)

The point of this example is that "left" and "right" invert when you turn upside down. If you face the same direction, but upside down, what was left is now right.

In that case, if you traveled once around the twist, you'd find your dog appeared to be on the ceiling, rather than the floor.

Thomas Andrews
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