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Current time:0:00Total duration:3:05

So in my last video I joked
about folding and cutting spheres instead of paper. But then I thought, why not? I mean, finite symmetry
groups on the Euclidean plane are fun and all, but there's
really only two types. Some amount of mirror
lines around a point, and some amount of
rotations around a point. Spherical patterns
are much more fun. And I happen to be a huge
fan of some of these symmetry groups, maybe just a little bit. Although snowflakes are
actually three dimensional, this snowflake doesn't just
have lines of mirror symmetry, but planes of mirror symmetry. And there's one
more mirror plane. The one going flat
through the snowflake, because one side of the
paper mirrors the other. And you can imagine
that snowflake suspended in a sphere, so that we can draw
the mirror lines more easily. Now this sphere has
the same symmetry as this 3D paper snowflake. If you're studying
group theory, you could label this with group
theory stuff, but whatever. I'm going to fold this sphere
on these lines, and then cut it, and it will give me something
with the same symmetry as a paper snowflake. Except on a sphere, and
it's a mess, so let's glue it to another sphere. And now it's perfect and
beautiful in every way. But the point is it's
equivalent to the snowflake as far as symmetry is concerned. OK, so that's a regular,
old 6-fold snowflake, but I've seen pictures
of 12-fold snowflakes. How do they work? Sometimes stuff
goes a little oddly at the very beginning
of snowflake formation and two
snowflakes sprout. Basically on top of each
other, but turned 30 degrees. If you think of them as one flat
thing, it has 12-fold symmetry, but in 3D it's not really true. The layers make it so there's
not a plane of symmetry here. See the branch on the left is on
top, while in the mirror image, the branch on the
right is on top. So is it just the same symmetry
as a normal 6-fold snowflake? What about that seventh
plane of symmetry? But no, through this plane one
side doesn't mirror the other. There's no extra
plane of symmetry. But there's something cooler. Rotational symmetry. If you rotate this around this
line, you get the same thing. The branch on the
left is still on top. If you imagine it
floating in a sphere you can draw the
mirror lines, and then 12 points of
rotational symmetry. So I can fold,
then slit it so it can swirl around
the rotation point. And cut out a sphereflake with
the same symmetry as this. Perfect. And you can fold spheres other
ways to get other patterns. OK what about fancier
stuff like this? Well, all I need to do is figure
out the symmetry to fold it. So, say we have a cube. What are the planes of symmetry? It's symmetric around this way,
and this way, and this way. Anything else? How about diagonally
across this way? But in the end, we have
all the fold lines. And now we just need to fold
a sphere along those lines to get just one
little triangle thing. And once we do, we can
unfold it to get something with the same
symmetry as a cube. And of course, you
have to do something with tetrahedral symmetry
as long as you're there. And of course, you really
want to do icosahedral, but the plastic is
thick and imperfect, and a complete mess, so
who knows what's going on. But at least you could
try some other ones with rotational symmetry. And other stuff and make a mess. And soon you're going
to want to fold and cut the very fabric of space
itself to get awesome, infinite 3D symmetry groups,
such as the one water molecules follow when
they pack in together into solid ice crystals. And before you
know it, you'll be playing with multidimensional,
quasi crystallography, early algebra's, or something. So you should probably
just stop now.