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So in my last video,

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I joked about folding and cutting

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spheres instead of paper.

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But then I thought, why not?

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I mean, finite symmetry groups

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on the Euclidean plane are fun and all.

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But there are really only two types:

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some amount of mirror lines around a point,

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and some amount of rotations around a point.

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Spherical patterns are much more fun.

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And I happen to be a huge fan of some of

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these symmetry groups – maybe just a little bit.

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Although snowflakes are actually

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three dimensional. (3-D.)

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This snowflake doesn't just have

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lines of mirror symmetry –

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but planes of mirror symmetry.

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And there's one more mirror plane –

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the one going flat through the snowflake –

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because one side of the paper mirrors the other.

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And you can imagine that snowflake

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suspended in this sphere –

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so that we can draw the mirror lines more easily.

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Now this sphere has the same symmetry

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as this 3-D paper snowflake.

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If you're studying group theory,

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you could label this with group theory stuff.

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But whatever.

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I'm going to fold this sphere on these lines,

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and then cut it – and that will give me

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something with the same symmetry

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as a paper snowflake – except on this sphere.

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And it's a mess.

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So let's glue it to another sphere.

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And now it's perfect and beautiful in every way.

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But the point is it's equivalent to this snowflake –

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as far as symmetry is concerned.

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Okay, so that's your regular old six-fold snowflake.

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But I've seen pictures of twelve-fold snowflakes.

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How do they work?

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Sometimes stuff goes a little oddly

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at the very beginning of snowflake formation,

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and two snowflakes sprout, basically,

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on top of each other – but turned 30 degrees.

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If you think of them as one flat thing,

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it has twelve-fold symmetry.

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But in 3-D, it's not really true.

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The layers make it so there's

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not a plane of symmetry here.

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See. The branch on the left is on top.

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While in the mirror image,

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the branch on the right is on top.

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So, is it just the same symmetry

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as a normal six-fold snowflake?

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What about that seventh plane of symmetry?

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But no, through this plane,

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one side doesn't mirror the other.

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There's no extra plane of symmetry.

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But there's something cooler –

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rotational symmetry.

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If you rotate this around this line,

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you get the same thing –

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the branch on the left is still on top.

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If you imagine it floating in a sphere,

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you can draw the mirror lines,

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and then you have twelve points

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of rotational symmetry.

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So I can fold, then slit it, so I can [indistinct]

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around the rotation point.

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And cut out each 'sphereflake'

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with the same symmetry as this. Perfect.

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And you can fold spheres

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other ways to get others patterns.

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Okay. What about fancier stuff like this?

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Well, all I need to do is figure out

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the symmetry to fold it.

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So, say we have a cube.

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What are the planes of symmetry?

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It's symmetric around this way,

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and this way, and this way.

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Anything else?

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How about diagonally across this way?

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But in the end, we have all the fold lines.

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And now, we just need to fold

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the sphere along those lines

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to get just one little triangle thing.

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And once we do,

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we can unfold it to get something

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with the same symmetry as a cube.

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And, of course, you have to do something

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with tetrahedral symmetry,

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as long as you're there.

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And, of course, you really

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want to do icosahedral.

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But the plastic is thick and imperfect,

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and a complete mess.

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So who knows what's going on.

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But at least you could try some other ones

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with rotational symmetry and other stuff –

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and make a mess.

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And soon, you're going to want to fold and cut

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the very fabric of space itself to get awesome

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infinite 3-D symmetry groups,

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such as the one water molecules follow

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when they pack together into solid ice crystals.

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And before you know it,

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you'll be playing with multidimensional

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quasi-crystallography,

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or lie algebras or something.

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So you should probably just stop now.