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 are 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. (3-D.)
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 this sphere –
so that we can draw the mirror lines more easily.
Now this sphere has the same symmetry
as this 3-D 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 that will give me
something with the same symmetry
as a paper snowflake – except on this 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 this snowflake –
as far as symmetry is concerned.
Okay, so that's your regular old six-fold snowflake.
But I've seen pictures of twelve-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 twelve-fold symmetry.
But in 3-D, 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 six-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 you have twelve points
of rotational symmetry.
So I can fold, then slit it, so I can [indistinct]
around the rotation point.
And cut out each 'sphereflake'
with the same symmetry as this. Perfect.
And you can fold spheres
other ways to get others patterns.
Okay. 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
the 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 3-D symmetry groups,
such as the one water molecules follow
when they pack together into solid ice crystals.
And before you know it,
you'll be playing with multidimensional
quasi-crystallography,
or lie algebras or something.
So you should probably just stop now.