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.