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]
