Mathematical Impressions: Goldberg Polyhedra

0:06  0:15What do you see here? A soccer ball? I'll show you how to see it in a new way.

0:15  0:20This is what's called a Goldberg polyhedron. In the 1930's, the mathematician

0:20  0:24Michael Goldberg came up with a family of beautifully symmetrical forms made of

0:24  0:29pentagons and hexagons meeting three at each vertex. The number of hexagons

0:29  0:34varies, but a theorem tells us there must be exactly 12 pentagons. So the first

0:34  0:38thing to look for are these pentagons. Then we can characterize which Goldberg

0:38  0:43polyhedron this is by taking a kind of knight's move from any pentagon to its

0:43  0:47nearest neighbors. On this one, we go one, two, three steps out from the

0:47  0:51pentagon, then make a slight right and go one more step to land on another

0:51  0:57pentagon. So we call this the threeone Goldberg polyhedron. This is the twoone

0:57  1:02Goldberg polyhedron because a pentagon to pentagon path goes two steps out, then

1:02  1:07turns one to the side. The same type of path goes from any pentagon to its

1:07  1:11nearest neighbors. In a mirror image, we take a right turn instead of a left

1:11  1:18turn.We can make them arbitrarily complex. These tiny examples illustrate the

1:18  1:22eightthree Goldberg polyhedron. You have to look pretty carefully to find a

1:22  1:27pentagon and then count a knight's move that goes eight, then three to land on

1:27  1:33the nearest pentagon neighbor.Choose two numbers, say twoone, and make a line,

1:33  1:38on triangular graph paper,from one vertex to another, that is out two and over

1:38  1:44one. Then copy that motion,but rotated 120 degrees, and then again, rotated

1:44  1:50another 120 degrees, to make an equilateral triangle. Then take 20 copies of

1:50  1:54that equilateral triangle and assemble them like an icosahedron, five to a

1:54  1:59vertex. Finally, create what's called the dual by connecting the centers of

1:59  2:04adjacent triangles. This makes hexagons in most places but pentagons for just

2:05  2:09the 12 vertices of the icosahedron. You can imagine inflating it slightly to

2:09  2:14make it more spherical.Who would think that designs which were first presented

2:14  2:1880 years ago by a mathematician working on what's called the isoperimetric

2:18  2:23problem would end up being useful decades later in real life applications like

2:23  2:28geodesic domes,Pav\'{e} diamond jewelry, carbon nanostructures, and nuclear

2:28  2:32particle detectors?It's wonderful how abstract mathematics often finds

2:32  2:38unexpected applications.Here's a spherical jigsaw puzzle I made. It comes apart

2:38  2:42into pieces, and you have to figure out how to snap them together. When you join

2:42  2:47enough to find a pentagon to pentagon path, you discover it's the fivethree

2:47  2:52Goldberg polyhedron which guides you to complete it into a sphere.

2:52  2:56And I've always been impressed by these ivory balls of nested spheres which go

2:56  3:01back to the 1500's. A craftsman starts with a solid ball and drills holes into

3:01  3:05the center, cuts layers apart from each other on a lathe.

3:05  3:09All the layers have the same pattern of holes, but the traditional patterns

3:09  3:15aren't Goldberg polyhedra. So I designed this one with ten Goldberg layers. Each

3:15  3:19layer has a different pattern of holes, so it can't be made in the traditional

3:19  3:24drilling manner. I 3Dprinted it, and it's just amazing that all 10 layers can

3:24  3:29turn independently. What's also cool is that I can blow compressed air into it

3:29  3:38and get the inside spinning really fast!Here's another Goldberg variation. I

3:38  3:43used the pattern of faces from the sevenfour Goldberg polyhedron to make linked

3:43  3:47loopslike chain mail. But I didn't have to assemble anything. It's 3Dprinted

3:47  3:52all at once, just like you see it. There are over 3,000 links here. Do you see

3:52  3:59one of the twelve pentagon ones?Did you ever learn a new word and then suddenly

3:59  4:04start seeing it again and again?Math can be like that also. Once you understand

4:04  4:08a mathematical pattern, it becomes part of you, and you may find it anywhere,

4:08  4:14like this oneone Goldberg polyhedron. And once you learn to see in this way,

4:14  4:20you'll never confuse this twozero Goldberg polyhedron with a soccer ball.

4:20  4:24I'm drawn to forms which have a coherent underlying structure. So when I was

4:24  4:27designing this sculpture, it seemed natural to me to make the inner green

4:27  4:32skeleton a Goldberg polyhedron. Now that you're familiar with them, can you tell

4:32  4:36which one it is? It wouldn't be surprising if now you start to see Goldberg

4:36  4:37polyhedra everywhere.
 Title:
 Mathematical Impressions: Goldberg Polyhedra
 Description:

Because of their aesthetic appeal, organic feel and easily understood structure, Goldberg polyhedra have a surprising number of applications ranging from golfball dimple patterns to nuclearparticle detector arrays.
http://www.simonsfoundation.org/multimedia/mathematicalimpressionsgoldbergpolyhedra/
 Video Language:
 English
 Duration:
 04:46
media vision edited English subtitles for Mathematical Impressions: Goldberg Polyhedra 