Nice to see myself again
Suppose you're writing a program where you keep track of the position of a bug, Brilli, as she wanders around her environment looking for things to do. You don't want her to get too far from where she starts so you design the environment so that, in some directions, it repeats itself. You design your space so that if Brilli is standing on the point \(\left(x,y\right)\), it is the same as if she stood on the point \(\left(x+2\pi, y+2\pi\right)\).
This equivalence is true no matter where she stands in her environment: as long as she shifts her position in the \(x\) direction by \(2\pi\) AND her position in the \(y\) direction by \(2\pi\), she'll end up back where she started. Which of the following spaces shows the kind of environment Brilli finds herself in?
Notes and assumptions
- This is a common trick in computer simulations to avoid nasty mathematical problems and also to avoid overflowing the computer's memory.