Craft & Design Science
Math Monday: An Octahedron in a Balloon

By George Hart for the Museum of Mathematics


You may know how to make a ship in a bottle, but how would you make this octahedron in a balloon? The twelve edges of the octahedron are made of strips of blue balloon rubber, glued to the inside of the clear balloon at the six vertices. Think of how you might make this before reading my solution below.

I did it by first making a spherical octahedron on the outside of the balloon. It is glued with rubber cement at the six cardinal points: top, bottom, left, right, front, back.

Then I untied the slip knot to let the air out of the balloon, carefully turned it inside out, and reinflated it.

If you master this technique, the next step is to try to make a nice cube. And be sure to send me a photo if you can make the compound of five tetrahedra in five different colors.

See all of George Hart’s Math Monday columns

10 thoughts on “Math Monday: An Octahedron in a Balloon

  1. Ok, here’s an idea I’ve been bouncing around: Vacuum-filled balloons.There’s no way they can be anything but lighter than air. You just need a vessel that can withstand atmospheric pressure, which is what? 15 psi? Make some kind of lightweight carbon/kevlar chamber about the size of a weather balloon and instead of filling it up with something, you suck all the air out. It has to float like a helium balloon.

    Maybe you can’t achieve perfect vacuum, but 75% would be fine. Heck, 50% would probably still float. I’m thinking there’s got to be a way to get that done.

    1. This stirred up an old memory, so I did a quick search of the Usenet archives.  I found this thread from 1997:

      The idea comes up once in a while, but as this thread mentions, the density of hydrogen or helium is so low that you don’t gain much from enclosing a vacuum, but any rigid structure is likely to be much heavier than an equivalent elastic structure.

      If it wasn’t NaNoWriMo, and if I wasn’t 8000 words behind, it would be fun to work out the theoretical limits …

      1. Well I’m not one to be so easily dissuaded. We’ve come a long way since 1997. There’s materials that are incredibly strong and lightweight like aerogels. Maybe a large sponge of such a material wrapped in an airtight coating could achieve the effect.

        There’s no reason it has to be a single, huge vessel. Maybe by nesting vessels containing successively smaller pressure, you could achieve a combined strength and buoyancy. Maybe you could create a foam of bubbles containing vacuum. Maybe complete neutral buoyancy is the wrong goal. Maybe there is an advantage in simply being lighter.

        Instead of putting so much effort into proving why it won’t work, why not put some thought into thinking of a way it could work? When people present ideas, you could use it as a source of ideation for yourself instead of an ego-gratifying endzone dance where you’ve done nothing but shut down a conversation that might have gone somewhere fascinating?

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Gareth Branwyn is a freelance writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor to Boing Boing, Wink Books, and Wink Fun. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.

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