Picture a bobbin resting in your hand. Your task is to wind a skein of thread around the bobbin. If you wind the thread along the shortest path about the bobbin, tracing a conic section orthogonal to the bobbin’s major axis, you will shortly find yourself confounded by an accretion of layers of thread one atop the other. What you want to do, what you will naturally tend to do, is to wind the thread along the longest possible path along the surface of the bobbin by wrapping it in a helix, each turn of the skein tracing a conic section that diverges from orthogonal to the bobbin’s major axis by an angle given by the arccos of the triangle formed by the diameter of the bobbin and that of the thread.
Now imagine a vast, warmly lit space, like a zendo but orders of magnitude more capacious, the walls receding from view in all directions. Picture a lattice, like the one we marked out in the first though experiment a week ago save that this one is marked out in gaff tape on the floor. Place a volunteer at one tape mark. Imagine that the space is characterized by periodic boundary conditions, so that if you set off along one of the principal axes at length you will encounter your volunteer again. The same thing will happen if you set off on a diagonal—it will simply take longer, like winding a skein of thread around a toroidal bobbin.
In fact, we can imagine an infinite lattice with periodic boundary conditions, so that whatever path you took, if you continued in a fixed direction, at length you would return to your volunteer—but here at length means in the limit, as the time you spent walking, t, or better yet the number of paces you’d taken, n, approached infinity. You can check that this converges.
This is the kind of thing I do at night to fall asleep.