Notes from Around the Sundial: Pac-Man and the Universe

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Gnomon's column image, showing a sundial surrounded with the words Notes From Around the Sundial'

And I think to myself, what a wonderful world!

Pac-Man and the Shape of the Universe

You may be old enough to remember Pac-Man. It was a computer game of the 1980s. Search Engine Google recently featured a playable version of Pac-Man as its logo for a day.In those days, computers were too expensive for most of us to have them at home, so Pac-Man was available only in the amusement arcades. The little Pac-Man character looked like a yellow disc with a segment taken out of it to form a biting mouth. Pac-Man wandered around a maze, gobbling up food as he went and scoring points. He had to avoid the ghosts that also wandered around the maze. Every so often Pacman would eat a particular type of food which scared the ghosts. He then would turn on the ghosts and chase them, getting lots of extra points for every ghost he gobbled up.

Pac-Man is interesting from a cosmological point of view, because the maze has one passageway which goes off the left hand side of the screen and reappears at the right hand side of the screen. Pac-Man may move down this passageway and instantly reappear at the other side of the screen. One can imagine that the first time Pac-Man does this, he thinks he is in a new part of the maze, but a bit of exploration shows that this maze is identical to the one he has just left. Pac-Man doesn't have any hands, pockets or any other way of carrying stuff, but one can imagine that he might drop something in the maze, go through the mysterious passageway and find the thing he dropped in the same position, telling him that he is back in the same maze.

A Pac-Man cosmologist would notice that he can in effect continue in a straight line for ever (ignoring a minor detour around the central room of the maze) without ever reaching the end of his universe, but that the area of his universe is finite. He won't have any way of knowing where the edge of his universe is, as it behaves identically at all points along his path. In effect his universe does not have an edge. It is finite but unbounded in the east-west direction. It is reasonable to assume that if the walls were removed from the maze, the little gobbler would be able to head off the left side of the screen at any point and reappear at the right. An astronomer Pac-Man with a telescope or even a good pair of binoculars could look to the east and see the back of his own head.

We have no idea what happens when Pac-Man goes off the top of the screen, as in the normal maze there is a wall to prevent him doing it. It's not a great stretch of the imagination to assume that he would reappear at the bottom of the screen. So Pac-Man's world is infinite in the North-South direction as well as the East-West: he can continue as far as he likes without ever reaching the end of the world. And yet the total area of the world is finite; he could take a paint-can and paint the entire world.

Pac-Man's world is small, and so was our Universe just after it came into existence (whether that was spontaneously or by being created by some super-being is not relevant to this discussion). We can imagine that Pac-Man's world could expand. Perhaps the screen is 10 inches wide and it increases to 20 inches wide, leaving Pac-Man the same size. Now he has more room to move. Now suppose that the Pac-Man world is expanded to billions of light-years across, but still with the curious feature that it 'wraps around'. A Pac-Man living in this late stage of his universe would have difficulty believing the stories passed down from his ancestors that the world wraps around.

Astronomers now believe that our own universe wraps around on a grand scale, although it does it in three dimensions rather than two. The looping around is not done at any one place but gradually every time you pass a piece of matter. The total amount of space around each point mass in the universe is slightly less than you would expect and the total sum of all these little bits of missing space 'bends' space around into a curious shape so that, like the Pac-Man, we could with a big enough telescope see the back of our own head. Our universe is finite but unbounded.

There are complications, of course. Light travels slowly on a cosmological scale. The universe hasn't been around long enough for light to travel the whole way across it yet, and since it is still expanding, it may never be. But the principle is the same - we may think our universe is describable by simple geometry, but it is in fact stranger than we can imagine.

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