Mathematical Impressions: Goldberg Polyhedra

Because of their aesthetic appeal, organic feel and easily understood structure, Goldberg polyhedra have a surprising number of applications ranging from golf-ball dimple patterns to nuclear-particle detector arrays.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

polyhedra everywhere.