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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.