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Yeah, those are both pretty big words in the title today. “Prodigious” means “extraordinary” or “remarkably impressive,” while a polyhedron is a three-dimensional figure made of multiple flat-plane faces, like a perfect cube or pyramid. If you’ve ever played Dungeons and Dragons, you’re prodigiously familiar with lots of different polyhedrons: dice.
Get your brain ready for some serious learnin’, cause this week’s Saturday Science will take us on a wild ride involving sports, geometry, chemistry, and even architecture. The polyhedron we’re talking about is so prodigious because it shows up all over the place, from the tiny world of atoms and molecules to the not-so-tiny world of major architectural design. We are going to be talking about something called a…drumroll please…TRUNCATED ICOSAHEDRON! BOOM!
Okay, yeah, that probably means nothing to you. But I bet you’ve played with them many times without realizing it because the most common everyday use of a truncated icosahedron is in the humble soccer ball. Today we’re going to construct ourselves a truncated icosahedron/soccer ball out of cardboard so you can see just what makes it go together so well.
Materials
- Old cereal boxes or tissue boxes or any boxes that are made out of that same thin but strong cardboard
- Scissors
- Tape
- A soccer ball (optional)
- This PDF, printed out for the pentagon (the five-sided one) and hexagon (the six-sided one):
Tip: Make sure you use these patterns. Not just any size of shapes will work. Each has to be perfectly regular (all sides and angles are exactly the same) and the length of the sides on the two shapes has to be the same. That’s why the hexagon is slightly bigger than the pentagon: same side lengths, but one more side.
Process
- Print out these instructions (be sure to use the correct PDF file) and cut out the two patterns very carefully.
- Using those patterns and your cardboard, cut out 12 pentagons and 20 hexagons.
- Get out your tape, because it’s time to begin assembly. The trick to making a truncated icosahedron is to make sure every pentagon is connected to a hexagon on all five sides, so…
- Tape together one pentagon and five hexagons, with the hexagons each sharing one edge with the pentagon.
- Right now you have a sort of snowflake-looking thing. Time to start forming it into a ball. See those places where two hexagons are almost touching? Bring those edges together and tape them. When you’re done with this step, you should have a little mini dome.
- Since we know that the pattern is one pentagon surrounded by five hexagons, constructing the rest of it is as simple as following that pattern. Now, it’s simple, but that doesn’t mean it’ll be easy. You’ll have to tape your shapes carefully and work on the correct angles, so everything comes together at the end. Start by putting a pentagon in every angle between two hexagons, and go from there.
- Admire your homemade cardboard soccer ball! You can’t really play too much with it, but would you want to after all that work? I say decorate it and show it off.
Summary
Okay, it’s time to define some more of our terms here. Specifically the big one: icosahedron. A standard icosahedron is a regular polygon with 20 sides. That means that all of the sides are the same shape and size; a cube is one of the simplest regular polyhedron, with six sides that are all equal squares. A standard icosahedron has 20 triangular sides, and those sides come together in 12 vertices, the points where edges meet. Again, back to Dungeons and Dragons: the 20-sided die is a standard icosahedron.
If you go back and count the faces on your soccer ball, you’ll realize that a truncated icosahedron has 32 faces. So if an icosahedron has 20 faces, how do we get away with calling this 32 face thing an icosahedron? It’s weird, but it makes sense if you think about it. See, a truncated icosahedron is created if you slice off those 12 vertices from a standard icosahedron. Each vertex (that’s the singular) is where five triangles connect, so when you slice it off, you get a five-sided polygon: a pentagon. And since there are 12 vertices, you get 12 pentagons. The other effect of these slices is to create a bunch of hexagons where once there were triangles.
So basically, you start with a standard icosahedron, do some fancy slicing (or truncating, which just means to shorten something by cutting the top off), and you get a truncated icosahedron.
So there’s the geometry. But what about the rest of it? Easy! The truncated icosahedron shows up all over the place. For example, in the 20th century, there was a famous architect named Buckminster Fuller. Fuller popularized the building of something called a geodesic dome. Think the big ball at Epcot Center: that’s a really complex geodesic dome, a ball made of many much smaller polygons. But smaller, simpler geodesic domes can be made with fewer polygons than Epcot’s. Like, for example, a truncated icosahedron. If you need a bigger and more complex dome, you start truncating the truncated icosahedron, which smooths things out, and you go from there.
Because of Buckminster Fuller’s affinity for the geodesic dome, when chemists discovered a series of carbon molecules shaped like them, they named the whole category of them “Fullerenes.” They start at 20 carbon atoms (C20) and go up from there. But the most common one is C60, or Buckminsterfullerene, which has its atoms arranged to form…A truncated icosahedron.
WOW! That was quite the experiment. Now it's time to take this black and white beauty to the soccer field! Head out to the Riley Children's Health Sports Legends Experience® and make your way to the Soccer Experience to show off your fancy footwork.
Want more Saturday Science? See all of our at-home activities on the blog or on Pinterest.
