Polyhedra made from One Balloon

Some polyhedra can be made from one balloon! (For details, see this paper.) One example is the octahedron, pictured above. The instructions below are relatively simple, but the difficulty lies in getting the lengths right. This takes some practice!

You can also make the lengths smaller, leaving an extra long piece sticking off. Ta-da! You've made a magical octahedron wand, with the power of geometry! With practice, these are relatively quick to make, and are fun for kids (the balloon size 160 is better for this).

The above picture was taken at the Math Midway, where I made hundreds of magic octahedron wands for children of all ages, while of course explaining some of their mathematical properties!

Another simple one-balloon construction is the triangular dipyramid shown below.

More difficult is the cuboctahedron! By starting with a 160 balloon and twisting off small sections, it can be done.

The 4-simplex is a beautiful model, but presents a new difficulty: two different lengths of edges. Can you figure out how to twist it?

Many other polyhedra are theoretically possible to twist out of one balloon, but there are practical limitations! The picture below shows how far I got on an icosadodecahedron. If you can make a complete one, send me a photo!