The other day I was describing my little novel visualizing tool to a colleague, and I told him that, like the old game Asteroids, it’s on the surface of a torus. That is, in world that is topologically equivalent to a donut. Or, if you are from New York, a bagel.
In Asteroids, when something goes off the right side of the screen, it reappears on the left side. And when something goes off the top edge of the screen, it reappears from the bottom edge. In other words, the screen wraps both ways.
“Doesn’t that mean it’s on the surface of a sphere?” he said.
I needed a really easy way to explain that it couldn’t be on a sphere, so I came up with this: “If you start in the middle of the screen and draw a horizontal line across the screen, you get a loop, because the left and the right ends of the line connect. Also, if you start at that same point in the middle of the screen and draw a vertical line from top to bottom, you get another loop, because the top and bottom ends of that line also connect.
So far so good. Then I said “the two lines only cross once (at that starting point in the middle of the screen). If Asteroids were on a sphere, the two lines would have to cross twice.”
My colleague (who is very quick) got the point immediately.
The next day I was describing this little exchange to Vi Hart, who said to me “Yes, that’s actually the definition of a torus.”
Huh. When you think about it, that makes a lot of sense.