Harmonic and Circular Oscillation by quantumaniac via dataanxiety

I can’t find enough illustrations explaining phase space. Phase space is a space people make up with their minds. The fact that a real thing bobbing up and down harmonically along 1-D is equivalent to a circle (seems like 2-D? but a topologist would say S¹ and in fact we don’t use the interior of the circle at all) is such a huge mental leap forward, I can’t express how much it amazes me or how much potential I think this metaphor has for everything else.

Think about the space of solutions of Rubik’s cube. Physically the cube is what it looks like:

but paths toward the solution are like a high-dimensional pyramid with “solved” at the top and entropy at the bottom.

Rather than being “just a plain-old `(ordered){red,blue,orange,yellow,green,white}⁹` with one particular configuration (starting point / solved) called the “centre”, all of that space is equivalence-classed by ,

• since some orientations can only be obtained by switching the stickers and not by legal moves,
• and since some members of `(ordered){red,blue,orange,yellow,green,white}⁹` are actually just setting the cube down on the table differently rather than twisting it. (and therefore equivalent-in-that-sense)

Anyway the Rubik’s Cube is a “knot” in phase space but nothing like a knot in right-in-front-of-you vision.

Here’s another example: if you have a Mac, you can invert the system’s colour scheme (presuming colours can be 2-inverted rather than in-in-in-verted or triverted but that’s another story) by pressing `Alt + Command + N`.

You can also 2-invert the colours of just one window (not the system) by pressing `Alt + Comand + M`. If you accidentally hit `M,N`, instead of `N`, your computer’s colour scheme would be 1-tangled. Or if you hit M on the wrong window, then hit `N`, then switched to another window and hit `M`, it would be even more tangled. But you can tell me that doesn’t make any sense! A Mac has a flat 2-D screen, with panes on it. Where do these “tangles” come from? It’s just some electronic signals zipping around and lighting up an LCD. Well, in phase space, in this particular mental representation which we can communicate about, it can be tangled.