1 00:00:06,349 --> 00:00:14,609 What do you see here? A soccer ball? I'll show you how to see it in a new way. 2 00:00:14,629 --> 00:00:20,009 This is what's called a Goldberg polyhedron. In the 1930's, the mathematician 3 00:00:20,029 --> 00:00:24,070 Michael Goldberg came up with a family of beautifully symmetrical forms made of 4 00:00:24,089 --> 00:00:28,759 pentagons and hexagons meeting three at each vertex. The number of hexagons 5 00:00:28,779 --> 00:00:33,839 varies, but a theorem tells us there must be exactly 12 pentagons. So the first 6 00:00:33,859 --> 00:00:37,999 thing to look for are these pentagons. Then we can characterize which Goldberg 7 00:00:38,019 --> 00:00:42,539 polyhedron this is by taking a kind of knight's move from any pentagon to its 8 00:00:42,559 --> 00:00:46,599 nearest neighbors. On this one, we go one, two, three steps out from the 9 00:00:46,619 --> 00:00:50,669 pentagon, then make a slight right and go one more step to land on another 10 00:00:50,689 --> 00:00:57,209 pentagon. So we call this the three-one Goldberg polyhedron. This is the two-one 11 00:00:57,229 --> 00:01:02,109 Goldberg polyhedron because a pentagon to pentagon path goes two steps out, then 12 00:01:02,129 --> 00:01:06,560 turns one to the side. The same type of path goes from any pentagon to its 13 00:01:06,579 --> 00:01:10,879 nearest neighbors. In a mirror image, we take a right turn instead of a left 14 00:01:10,899 --> 00:01:18,000 turn.We can make them arbitrarily complex. These tiny examples illustrate the 15 00:01:18,019 --> 00:01:21,989 eight-three Goldberg polyhedron. You have to look pretty carefully to find a 16 00:01:22,009 --> 00:01:27,099 pentagon and then count a knight's move that goes eight, then three to land on 17 00:01:27,119 --> 00:01:32,900 the nearest pentagon neighbor.Choose two numbers, say two-one, and make a line, 18 00:01:32,920 --> 00:01:37,780 on triangular graph paper,from one vertex to another, that is out two and over 19 00:01:37,799 --> 00:01:44,159 one. Then copy that motion,but rotated 120 degrees, and then again, rotated 20 00:01:44,179 --> 00:01:49,590 another 120 degrees, to make an equilateral triangle. Then take 20 copies of 21 00:01:49,609 --> 00:01:53,670 that equilateral triangle and assemble them like an icosahedron, five to a 22 00:01:53,689 --> 00:01:59,010 vertex. Finally, create what's called the dual by connecting the centers of 23 00:01:59,030 --> 00:02:04,489 adjacent triangles. This makes hexagons in most places but pentagons for just 24 00:02:04,509 --> 00:02:09,269 the 12 vertices of the icosahedron. You can imagine inflating it slightly to 25 00:02:09,288 --> 00:02:13,609 make it more spherical.Who would think that designs which were first presented 26 00:02:13,629 --> 00:02:17,629 80 years ago by a mathematician working on what's called the isoperimetric 27 00:02:17,650 --> 00:02:22,659 problem would end up being useful decades later in real life applications like 28 00:02:22,680 --> 00:02:27,839 geodesic domes,Pav\'{e} diamond jewelry, carbon nanostructures, and nuclear 29 00:02:27,859 --> 00:02:32,309 particle detectors?It's wonderful how abstract mathematics often finds 30 00:02:32,329 --> 00:02:37,999 unexpected applications.Here's a spherical jigsaw puzzle I made. It comes apart 31 00:02:38,019 --> 00:02:42,309 into pieces, and you have to figure out how to snap them together. When you join 32 00:02:42,329 --> 00:02:46,589 enough to find a pentagon to pentagon path, you discover it's the five-three 33 00:02:46,609 --> 00:02:51,759 Goldberg polyhedron which guides you to complete it into a sphere. 34 00:02:51,780 --> 00:02:55,559 And I've always been impressed by these ivory balls of nested spheres which go 35 00:02:55,579 --> 00:03:01,109 back to the 1500's. A craftsman starts with a solid ball and drills holes into 36 00:03:01,129 --> 00:03:04,949 the center, cuts layers apart from each other on a lathe. 37 00:03:04,969 --> 00:03:08,909 All the layers have the same pattern of holes, but the traditional patterns 38 00:03:08,929 --> 00:03:15,079 aren't Goldberg polyhedra. So I designed this one with ten Goldberg layers. Each 39 00:03:15,099 --> 00:03:18,959 layer has a different pattern of holes, so it can't be made in the traditional 40 00:03:18,959 --> 00:03:24,219 drilling manner. I 3D-printed it, and it's just amazing that all 10 layers can 41 00:03:24,239 --> 00:03:28,979 turn independently. What's also cool is that I can blow compressed air into it 42 00:03:29,000 --> 00:03:37,539 and get the inside spinning really fast!Here's another Goldberg variation. I 43 00:03:37,560 --> 00:03:42,729 used the pattern of faces from the seven-four Goldberg polyhedron to make linked 44 00:03:42,750 --> 00:03:47,099 loops--like chain mail. But I didn't have to assemble anything. It's 3D-printed 45 00:03:47,119 --> 00:03:52,429 all at once, just like you see it. There are over 3,000 links here. Do you see 46 00:03:52,449 --> 00:03:58,689 one of the twelve pentagon ones?Did you ever learn a new word and then suddenly 47 00:03:58,709 --> 00:04:03,739 start seeing it again and again?Math can be like that also. Once you understand 48 00:04:03,759 --> 00:04:07,849 a mathematical pattern, it becomes part of you, and you may find it anywhere, 49 00:04:07,869 --> 00:04:14,109 like this one-one Goldberg polyhedron. And once you learn to see in this way, 50 00:04:14,129 --> 00:04:19,660 you'll never confuse this two-zero Goldberg polyhedron with a soccer ball. 51 00:04:19,680 --> 00:04:24,040 I'm drawn to forms which have a coherent underlying structure. So when I was 52 00:04:24,060 --> 00:04:27,420 designing this sculpture, it seemed natural to me to make the inner green 53 00:04:27,439 --> 00:04:31,939 skeleton a Goldberg polyhedron. Now that you're familiar with them, can you tell 54 00:04:31,959 --> 00:04:35,990 which one it is? It wouldn't be surprising if now you start to see Goldberg 55 00:04:36,009 --> 00:04:37,009 polyhedra everywhere.