**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.

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Tags: imagination, math, mathematics, maths, mind, phase space

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