WEBVTT 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. 00:00:14.629 --> 00:00:20.009 This is what's called a Goldberg polyhedron. In the 1930's, the mathematician 00:00:20.029 --> 00:00:24.070 Michael Goldberg came up with a family of beautifully symmetrical forms made of 00:00:24.089 --> 00:00:28.759 pentagons and hexagons meeting three at each vertex. The number of hexagons 00:00:28.779 --> 00:00:33.839 varies, but a theorem tells us there must be exactly 12 pentagons. So the first 00:00:33.859 --> 00:00:37.999 thing to look for are these pentagons. Then we can characterize which Goldberg 00:00:38.019 --> 00:00:42.539 polyhedron this is by taking a kind of knight's move from any pentagon to its 00:00:42.559 --> 00:00:46.599 nearest neighbors. On this one, we go one, two, three steps out from the 00:00:46.619 --> 00:00:50.669 pentagon, then make a slight right and go one more step to land on another 00:00:50.689 --> 00:00:57.209 pentagon. So we call this the three-one Goldberg polyhedron. This is the two-one 00:00:57.229 --> 00:01:02.109 Goldberg polyhedron because a pentagon to pentagon path goes two steps out, then 00:01:02.129 --> 00:01:06.560 turns one to the side. The same type of path goes from any pentagon to its 00:01:06.579 --> 00:01:10.879 nearest neighbors. In a mirror image, we take a right turn instead of a left 00:01:10.899 --> 00:01:18.000 turn.We can make them arbitrarily complex. These tiny examples illustrate the 00:01:18.019 --> 00:01:21.989 eight-three Goldberg polyhedron. You have to look pretty carefully to find a 00:01:22.009 --> 00:01:27.099 pentagon and then count a knight's move that goes eight, then three to land on 00:01:27.119 --> 00:01:32.900 the nearest pentagon neighbor.Choose two numbers, say two-one, and make a line, 00:01:32.920 --> 00:01:37.780 on triangular graph paper,from one vertex to another, that is out two and over 00:01:37.799 --> 00:01:44.159 one. Then copy that motion,but rotated 120 degrees, and then again, rotated 00:01:44.179 --> 00:01:49.590 another 120 degrees, to make an equilateral triangle. Then take 20 copies of 00:01:49.609 --> 00:01:53.670 that equilateral triangle and assemble them like an icosahedron, five to a 00:01:53.689 --> 00:01:59.010 vertex. Finally, create what's called the dual by connecting the centers of 00:01:59.030 --> 00:02:04.489 adjacent triangles. This makes hexagons in most places but pentagons for just 00:02:04.509 --> 00:02:09.269 the 12 vertices of the icosahedron. You can imagine inflating it slightly to 00:02:09.288 --> 00:02:13.609 make it more spherical.Who would think that designs which were first presented 00:02:13.629 --> 00:02:17.629 80 years ago by a mathematician working on what's called the isoperimetric 00:02:17.650 --> 00:02:22.659 problem would end up being useful decades later in real life applications like 00:02:22.680 --> 00:02:27.839 geodesic domes,Pav\'{e} diamond jewelry, carbon nanostructures, and nuclear 00:02:27.859 --> 00:02:32.309 particle detectors?It's wonderful how abstract mathematics often finds 00:02:32.329 --> 00:02:37.999 unexpected applications.Here's a spherical jigsaw puzzle I made. It comes apart 00:02:38.019 --> 00:02:42.309 into pieces, and you have to figure out how to snap them together. When you join 00:02:42.329 --> 00:02:46.589 enough to find a pentagon to pentagon path, you discover it's the five-three 00:02:46.609 --> 00:02:51.759 Goldberg polyhedron which guides you to complete it into a sphere. 00:02:51.780 --> 00:02:55.559 And I've always been impressed by these ivory balls of nested spheres which go 00:02:55.579 --> 00:03:01.109 back to the 1500's. A craftsman starts with a solid ball and drills holes into 00:03:01.129 --> 00:03:04.949 the center, cuts layers apart from each other on a lathe. 00:03:04.969 --> 00:03:08.909 All the layers have the same pattern of holes, but the traditional patterns 00:03:08.929 --> 00:03:15.079 aren't Goldberg polyhedra. So I designed this one with ten Goldberg layers. Each 00:03:15.099 --> 00:03:18.959 layer has a different pattern of holes, so it can't be made in the traditional 00:03:18.959 --> 00:03:24.219 drilling manner. I 3D-printed it, and it's just amazing that all 10 layers can 00:03:24.239 --> 00:03:28.979 turn independently. What's also cool is that I can blow compressed air into it 00:03:29.000 --> 00:03:37.539 and get the inside spinning really fast!Here's another Goldberg variation. I 00:03:37.560 --> 00:03:42.729 used the pattern of faces from the seven-four Goldberg polyhedron to make linked 00:03:42.750 --> 00:03:47.099 loops--like chain mail. But I didn't have to assemble anything. It's 3D-printed 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 00:03:52.449 --> 00:03:58.689 one of the twelve pentagon ones?Did you ever learn a new word and then suddenly 00:03:58.709 --> 00:04:03.739 start seeing it again and again?Math can be like that also. Once you understand 00:04:03.759 --> 00:04:07.849 a mathematical pattern, it becomes part of you, and you may find it anywhere, 00:04:07.869 --> 00:04:14.109 like this one-one Goldberg polyhedron. And once you learn to see in this way, 00:04:14.129 --> 00:04:19.660 you'll never confuse this two-zero Goldberg polyhedron with a soccer ball. 00:04:19.680 --> 00:04:24.040 I'm drawn to forms which have a coherent underlying structure. So when I was 00:04:24.060 --> 00:04:27.420 designing this sculpture, it seemed natural to me to make the inner green 00:04:27.439 --> 00:04:31.939 skeleton a Goldberg polyhedron. Now that you're familiar with them, can you tell 00:04:31.959 --> 00:04:35.990 which one it is? It wouldn't be surprising if now you start to see Goldberg 00:04:36.009 --> 00:04:37.009 polyhedra everywhere.