My favorite balloons to twist
Posts tagged Math
Orderly Tangle
Jan 26th
Orderly tangle / regular polylink
This is called an “orderly tangle” (or also “regular polylink.”) This particular tangle is made up of four triangles, which when woven together in a certain symmetric fashion form a stable structure. You may have seen tangles in the form of puzzles before. My college roommate was really into puzzles and had a bunch of wooden ones. I first saw tangles that were made from balloons on Vi Hart’s web site.
To make this balloon, I used four 260 balloons. I used my neon colors for a cool color combination. I left around a 3-4 inch tail (instead of inflating the entire balloon) as I didn’t want the final structure to be too big/loose. There are no tricky twists – it’s just four triangles. But it is a geometric exercise to figure out how to put it together. You’ll have to make each triangle one at a time as you figure out how to weave them. I added a 5″ round balloon in the middle for fun.
To learn more about these cool geometric structures, check out this video below by George Hart, who has lots of photos and videos of really cool mathematical sculptures at his web site and YouTube channel.
Fin RegPolylinks from Simons Foundation on Vimeo.
Balloon ball (icosahedron)
Oct 1st
Here’s a balloon ball. Sure, you could just inflate a round balloon, but this is cooler. 
Balloon ball – icosahedron
In mathematical terms, it’s actually an icosahedron – a polyhedron with 20 triangular sides. It may look complicated, but because of its symmetry and basic units, it’s actually quite easy to put together. I first saw this on Vi Hart’s web site, where she has posted great instructions for this icosahedron, as well as many other mathematical shapes, such as fractals, tangles, and other polyhedra! Check it out!
To make this balloon, I took three 160 balloons and cut each in half. Each section was then used to make one of the six units. (I wanted to make an icosahedron that wasn’t too big.)
If you’d like to read a mathematical paper written about balloon twisting, check out: Computational Balloon Twisting Theory: The Theory of Twisting Polyhedra, co-authored by Hart, Martin Dermaine, and Erik Dermaine (who was one of my college professors!)
Balloon design and graph theory
Apr 13th
When figuring out balloon designs, it’s optimal to minimize the number of balloons you need and the number of times you break a balloon. This way, you save on the number of balloons you use up, you save time (because breaking balloons and tying knots adds time,) and you save your fingers a little. (Try tying 50 knots and see how your fingers feel!)
Balloon twisting can be used to illustrate some ideas in graph theory. A balloon design can be represented as a graph. The places where the balloon is twisted represent vertices in the graph. The edges in the graph indicate which vertices are connected. For example, the picture below shows a graph of the top of a cake (the pink “frosting.”)
Is it possible to do the frosting with one single balloon without having to break the balloon? (That is the equivalent of asking if this graph has a Euler walk – can all edges in the graph can be visited exactly once?) Take out a pencil and paper and give it a try! See if you can trace over every edge exactly once without picking up your pencil.
Graph of frosting on top of the cake
Actually, as you’ve probably figured out, it’s not possible. (More than two vertices have an odd degree.) So then, what is the least number of balloon segments necessary to do the frosting, going over every edge exactly once? Can you figure it out?
The next time you are figuring out a balloon design, you might also be solving some equivalent problem of determining the best route for a road trip, package delivery, etc.
