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

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This is what's called a Goldberg polyhedron. In the 1930's, the mathematician

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Michael Goldberg came up with a family of beautifully symmetrical forms made of

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pentagons and hexagons meeting three at each vertex. The number of hexagons

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

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thing to look for are these pentagons. Then we can characterize which Goldberg

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

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nearest neighbors. On this one, we go one, two, three steps out from the

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

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pentagon. So we call this the three-one Goldberg polyhedron. This is the two-one

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Goldberg polyhedron because a pentagon to pentagon path goes two steps out, then

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

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nearest neighbors. In a mirror image, we take a right turn instead of a left

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turn.We can make them arbitrarily complex. These tiny examples illustrate the

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eight-three Goldberg polyhedron. You have to look pretty carefully to find a

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

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the nearest pentagon neighbor.Choose two numbers, say two-one, and make a line,

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

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one. Then copy that motion,but rotated 120 degrees, and then again, rotated

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another 120 degrees, to make an equilateral triangle. Then take 20 copies of

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that equilateral triangle and assemble them like an icosahedron, five to a

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vertex. Finally, create what's called the dual by connecting the centers of

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adjacent triangles. This makes hexagons in most places but pentagons for just

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the 12 vertices of the icosahedron. You can imagine inflating it slightly to

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make it more spherical.Who would think that designs which were first presented

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80 years ago by a mathematician working on what's called the isoperimetric

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problem would end up being useful decades later in real life applications like

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geodesic domes,Pav\'{e} diamond jewelry, carbon nanostructures, and nuclear

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particle detectors?It's wonderful how abstract mathematics often finds

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unexpected applications.Here's a spherical jigsaw puzzle I made. It comes apart

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

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enough to find a pentagon to pentagon path, you discover it's the five-three

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Goldberg polyhedron which guides you to complete it into a sphere.

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

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back to the 1500's. A craftsman starts with a solid ball and drills holes into

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the center, cuts layers apart from each other on a lathe.

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All the layers have the same pattern of holes, but the traditional patterns

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aren't Goldberg polyhedra. So I designed this one with ten Goldberg layers. Each

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

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drilling manner. I 3D-printed it, and it's just amazing that all 10 layers can

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

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

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used the pattern of faces from the seven-four Goldberg polyhedron to make linked

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loops--like chain mail. But I didn't have to assemble anything. It's 3D-printed

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

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

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start seeing it again and again?Math can be like that also. Once you understand

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a mathematical pattern, it becomes part of you, and you may find it anywhere,

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like this one-one Goldberg polyhedron. And once you learn to see in this way,

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you'll never confuse this two-zero Goldberg polyhedron with a soccer ball.

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

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designing this sculpture, it seemed natural to me to make the inner green

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

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

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polyhedra everywhere.