It's a wonderful universe.(The Shape of Space)(Book review)

The Shape of Space, by Jeffrey R. Weeks; Marcel Dekker, 2002 (second edition), $45.

I recall in the dim distant past playing the then new computer shoot-em-up games, where you had to navigate a spaceship on a computer screen by pressing arrow keys, and fire missiles at the alien menaces by hitting the space bar. The alien menaces, with wriggly tentacles, came down from the top of the screen and vanished off the bottom, to immediately reappear at the top. Similarly, if my spaceship went off the right of the screen it immediately reappeared at the left and vice-versa.

Forget the aliens and the missiles, and concentrate on the geometry. The "shape" or geometry of the screen had been turned into a torus, the surface of a ring doughnut, by identifying or gluing the edges together. If you don't see this, take a long strip of paper, and glue or tape the opposite long edges together to get a tube. Now bend the thing around to tape the two bounding circles together to get a torus. You will have to buckle it a bit because paper doesn't readily stretch, but this is a consequence of trying to make the object sit in the three-dimensional space in which we live. If we lived in four dimensions, it could be done without buckling, but this is of limited interest since we don't. The actual torus is the surface of the computer screen with the edges identified, so is two-dimensional. I shall call it a 2-torus to remind you that the thing itself has the same local properties as a sheet of paper. It has different global properties, since the sheet of paper has edges and vertices and the 2-torus does not.

Our inability to visualise three-dimensional analogues of the torus stems from the fact that we have a huge amount of geometric experience of living in three dimensions which is extremely local. So we can visualise a 2-torus either in terms of a flat rectangular screen with the edges identified, or as a ring doughnut sitting in three dimensions. But a three-dimensional analogue of a torus, a 3-torus, is conceivable only in the first way. We couldn't make one out of paper and display it on the credenza for our friends and relatives to admire. Still, we can imagine the possibility with only a very little thought. Attend carefully, this is your first lesson in serious geometry, and has a lot to do with the Space Invaders geometry, but with the dimension one higher.

You have to think of yourself as standing in a cubical room with doors in each wall. The room is empty of people, except you, but has lots of junk scattered around with a bed and a few chairs and a fridge in the corner.

You are going to need them. You walk to the north door, and open it; looking through you see a room remarkably like the one you are in, same junk, and a figure standing at the new north door looking into the next room beyond. Over his shoulder you see an empty room, with the same sort of junk as yours, with a door in each wall, and on the point of leaving that room is a figure that looks a lot like yours ... There is an infinite string of people like you, all stepping through an open door into a sequence of rooms disappearing into the distance.' On impulse, you turn and look over your shoulder. Coming through the southern door is a figure which has just turned to look behind, at a figure turned to look behind ... You get the idea. Best close the door quickly and sit down for a minute.

Is there a doubly, or quadruply, infinite sequence of rooms? You take a bunch of keys and use one to scratch your initials on the wall next to the western door. Then you go through the western door into the next room, momentarily unnerved by the sight of all those other figures ahead and behind you doing the same thing. You close the door behind you with relief. It is the eastern door of a new junk-strewn room. You cross the room and look at the new western door, and just next to it you see your initials scratched on the wall. All those other people are you. There is only one room.