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Folding space-time

Music box, backwards Bach, orbifolds and wooden bowl. Created by Vi Hart.

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Video transcript

So in my sphere flakes video, I joked about folding and cutting space time, but then I thought, hey, why not? So, how do you do that? Well, when we wanted to fold and cut only space, we chose a medium that takes place in space. That is, static paper cut outs, or sphere sculpture. But to fold and cut time, we need a medium that happens over time. I choose music. Music has two easily recognizable dimensions. One is time, and the other is pitch space. Not quite the same as space-space, but it's one dimensional, which makes things easier. But let's not be confused with the notation. There's a few things to notice about written music. Firstly, that it's not music. You can't listen to this. Or, well, you can, but it'll be like-- [PAPER RUSTLING] It's not music, it's music notation. And you can only interpret it into the beautiful music it represents. Kind of like how a book is squiggles on a page that your brain interprets into a meaningful story. And maybe you don't understand it at all, or understand just a literal surface meaning of the action. Or maybe you can read deep and critically into a story that's simple on the surface, and get more from it than even the author intended. Math is like this, too. Secondly, written music represents a two-dimensional space of pitch and time, but only represents it. Like, there's the suggestion that this is the time axis, but it's not. This is exactly the same as this. Even though you're changing the values on the x-axis. At least, in standard music notation. Some more modern composers do make use of spatial notation, just like some poets do intentionally stretch out words or play with formatting. But in standard notation, a stretched out word only means your text editor is terrible at justified margins, and has nothing to do with the word itself. Pitch also doesn't entirely depend on the notes placed on the y-axis. So I'm going to use something a little closer to reality. I've got this music box that plays a paper tape. As you put the strip through the box, it plays the punched holes. Hold on, it's too quiet. This is why music boxes usually come attached to wooden boxes. I don't have a wooden box, maybe this nice wooden bowl. Um. Ah, there we go. Music bowl. Anyway, here distance along the strip does translate directly to time, assuming constant crank speed. [MUSIC PLAYING] The box also has a set C major scale on one staff, so pitch space is represented pretty directly. You don't have to worry about sharps and flats, or the space between staves, or the infinite possibilities on a continuous logarithmic frequency scale. The box magically ignores all notational elements, except for where and when the holes are. Each hole, each note, is a point on this strip of space time. So now, let's fold and cut it. It's easy to fold time so that it goes both forwards and backwards simultaneously. Then we can punch in some notes, and unfold it, into a symmetric melody that goes first forwards, and then backwards. Or first backwards and then forwards. Point is, it's reversible. It sounds the same whether I play it like this-- [MUSIC PLAYING] --or like this. [MUSIC PLAYING] But maybe you want to fold time into one finite chunk. Then you could put two mirror lines, which reflect themselves. Which means folding time back and forth infinitely, to get a little time chunk to cut up. And sometimes cutting through infinitely folded space time requires the use of power tools. But then you can unfold a repeating, back and forth musicy thing. [MUSIC PLAYING] Or you could leave time alone and fold space, if that's what you're into. And that sounds like this. [MUSIC PLAYING] Or you could do both. [MUSIC PLAYING] Maybe you could try folding at other angles, but now you're mixing up the space and time axes. And mixing up space and time is hazardous. Though you could flip both space and time, which is like rotating the music 180 degrees. This one is especially fun, because you can rotate any piece of sheet music, and try playing it upside down. And it's not just upside down, but backwards in time, too. [MUSIC PLAYING] Anyway, maybe you want patterns that aren't all mirror lines. Like, what about a simple repeating melody? Can we fold and cut that? Yes. Loop up the paper, cut, then unfold. Or, if you're feeling snazzy, do the melody once, then loop the paper through the box. [MUSIC PLAYING] This is probably the part where seasoned viewers will be yelling something about Mobius strips. In the first snowflakes video, I did fold up of a Mobius strip and cut it. OK, here's what happens. The strip unravels to give you your shapes. And then they repeat, but upside down. And then they repeat, but flipped again. It's a glide reflection. And an example of how glide reflections really are their own special sort of symmetry, different from just a combination of reflections. Which look like this. And translation, which looks like this. And which together, give you this. So if you fold space time into a Mobius strip, you get your melody, and then the inversion, the melody played upside down. And then right side up again. And so on. Or, rather than folding and cutting all of space time, just cut and tape a little loop of space time into a Mobius strip to be played over and over. [MUSIC PLAYING]