0:00:00.075,0:00:01.264 So in my last video, 0:00:01.264,0:00:02.646 I joked about folding and cutting 0:00:02.646,0:00:03.914 spheres instead of paper. 0:00:03.914,0:00:05.375 But then I thought, why not? 0:00:05.375,0:00:06.720 I mean, finite symmetry groups 0:00:06.720,0:00:08.797 on the Euclidean plane are fun and all. 0:00:08.804,0:00:10.563 But there are really only two types: 0:00:10.563,0:00:12.305 some amount of mirror lines around a point, 0:00:12.305,0:00:14.563 and some amount of rotations around a point. 0:00:14.563,0:00:16.554 Spherical patterns are much more fun. 0:00:16.554,0:00:18.321 And I happen to be a huge fan of some of 0:00:18.321,0:00:22.443 these symmetry groups – maybe just a little bit. 0:00:22.443,0:00:23.865 Although snowflakes are actually 0:00:23.865,0:00:25.002 three dimensional. (3-D.) 0:00:25.002,0:00:26.250 This snowflake doesn't just have 0:00:26.250,0:00:27.442 lines of mirror symmetry – 0:00:27.442,0:00:29.389 but planes of mirror symmetry. 0:00:29.389,0:00:31.529 And there's one more mirror plane – 0:00:31.529,0:00:33.998 the one going flat through the snowflake – 0:00:33.998,0:00:36.389 because one side of the paper mirrors the other. 0:00:36.389,0:00:37.750 And you can imagine that snowflake 0:00:37.750,0:00:38.827 suspended in this sphere – 0:00:38.827,0:00:41.021 so that we can draw the mirror lines more easily. 0:00:41.021,0:00:43.114 Now this sphere has the same symmetry 0:00:43.114,0:00:45.183 as this 3-D paper snowflake. 0:00:45.183,0:00:46.497 If you're studying group theory, 0:00:46.497,0:00:48.446 you could label this with group theory stuff. 0:00:48.446,0:00:48.918 But whatever. 0:00:48.918,0:00:50.619 I'm going to fold this sphere on these lines, 0:00:50.619,0:00:52.092 and then cut it – and that will give me 0:00:52.092,0:00:53.437 something with the same symmetry 0:00:53.437,0:00:55.075 as a paper snowflake – except on this sphere. 0:00:55.075,0:00:55.789 And it's a mess. 0:00:55.789,0:00:57.137 So let's glue it to another sphere. 0:00:57.137,0:01:00.110 And now it's perfect and beautiful in every way. 0:01:00.110,0:01:02.580 But the point is it's equivalent to this snowflake – 0:01:02.580,0:01:04.431 as far as symmetry is concerned. 0:01:04.431,0:01:06.963 Okay, so that's your regular old six-fold snowflake. 0:01:06.963,0:01:09.208 But I've seen pictures of twelve-fold snowflakes. 0:01:09.208,0:01:10.399 How do they work? 0:01:10.399,0:01:11.785 Sometimes stuff goes a little oddly 0:01:11.785,0:01:13.842 at the very beginning of snowflake formation, 0:01:13.842,0:01:15.553 and two snowflakes sprout, basically, 0:01:15.553,0:01:17.949 on top of each other – but turned 30 degrees. 0:01:17.949,0:01:19.518 If you think of them as one flat thing, 0:01:19.518,0:01:20.864 it has twelve-fold symmetry. 0:01:20.864,0:01:23.040 But in 3-D, it's not really true. 0:01:23.040,0:01:24.569 The layers make it so there's 0:01:24.569,0:01:26.192 not a plane of symmetry here. 0:01:26.192,0:01:27.968 See. The branch on the left is on top. 0:01:27.968,0:01:29.037 While in the mirror image, 0:01:29.037,0:01:30.846 the branch on the right is on top. 0:01:30.846,0:01:31.981 So, is it just the same symmetry 0:01:31.981,0:01:33.651 as a normal six-fold snowflake? 0:01:33.651,0:01:35.879 What about that seventh plane of symmetry? 0:01:35.879,0:01:37.271 But no, through this plane, 0:01:37.271,0:01:39.156 one side doesn't mirror the other. 0:01:39.156,0:01:41.117 There's no extra plane of symmetry. 0:01:41.117,0:01:42.612 But there's something cooler – 0:01:42.612,0:01:44.422 rotational symmetry. 0:01:44.422,0:01:45.786 If you rotate this around this line, 0:01:45.786,0:01:47.209 you get the same thing – 0:01:47.209,0:01:48.894 the branch on the left is still on top. 0:01:48.894,0:01:50.645 If you imagine it floating in a sphere, 0:01:50.645,0:01:52.329 you can draw the mirror lines, 0:01:52.329,0:01:53.937 and then you have twelve points 0:01:53.937,0:01:55.313 of rotational symmetry. 0:01:55.313,0:01:57.673 So I can fold, then slit it, so I can [indistinct] 0:01:57.673,0:01:59.411 around the rotation point. 0:01:59.411,0:02:00.764 And cut out each 'sphereflake' 0:02:00.764,0:02:04.209 with the same symmetry as this. Perfect. 0:02:04.209,0:02:04.993 And you can fold spheres 0:02:04.993,0:02:06.816 other ways to get others patterns. 0:02:06.816,0:02:08.993 Okay. What about fancier stuff like this? 0:02:08.993,0:02:10.446 Well, all I need to do is figure out 0:02:10.446,0:02:11.700 the symmetry to fold it. 0:02:11.700,0:02:13.061 So, say we have a cube. 0:02:13.061,0:02:14.669 What are the planes of symmetry? 0:02:14.669,0:02:16.307 It's symmetric around this way, 0:02:16.307,0:02:18.169 and this way, and this way. 0:02:18.169,0:02:19.499 Anything else? 0:02:19.499,0:02:21.668 How about diagonally across this way? 0:02:21.668,0:02:23.867 But in the end, we have all the fold lines. 0:02:23.867,0:02:24.953 And now, we just need to fold 0:02:24.953,0:02:26.494 the sphere along those lines 0:02:26.494,0:02:28.671 to get just one little triangle thing. 0:02:28.671,0:02:29.525 And once we do, 0:02:29.525,0:02:31.169 we can unfold it to get something 0:02:31.169,0:02:32.545 with the same symmetry as a cube. 0:02:32.545,0:02:34.026 And, of course, you have to do something 0:02:34.026,0:02:35.061 with tetrahedral symmetry, 0:02:35.061,0:02:36.530 as long as you're there. 0:02:36.530,0:02:37.551 And, of course, you really 0:02:37.551,0:02:38.575 want to do icosahedral. 0:02:38.575,0:02:39.988 But the plastic is thick and imperfect, 0:02:39.988,0:02:41.005 and a complete mess. 0:02:41.005,0:02:41.851 So who knows what's going on. 0:02:41.851,0:02:43.366 But at least you could try some other ones 0:02:43.366,0:02:45.704 with rotational symmetry and other stuff – 0:02:45.704,0:02:46.573 and make a mess. 0:02:46.573,0:02:47.750 And soon, you're going to want to fold and cut 0:02:47.750,0:02:49.742 the very fabric of space itself to get awesome 0:02:49.742,0:02:51.067 infinite 3-D symmetry groups, 0:02:51.067,0:02:52.839 such as the one water molecules follow 0:02:52.839,0:02:55.784 when they pack together into solid ice crystals. 0:02:55.784,0:02:56.586 And before you know it, 0:02:56.586,0:02:58.006 you'll be playing with multidimensional 0:02:58.006,0:02:58.998 quasi-crystallography, 0:02:58.998,0:03:00.420 or lie algebras or something. 0:03:00.420,0:03:02.925 So you should probably just stop now.