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By Glen Whitney for the Museum of Mathematics



First, why hula hoops? They’re a pretty cheap source of large, pre-made circles, generally decently symmetric and strong. So they’re a candidate for any large-scale building project that can be based on the geometry of a circle. What are some examples? Well, you can envision each circle as a great circle on a sphere, and ask: is there a way to arrange four of these so that every intersection point between great circles is equidistant from its nearest neighbors? That leads to a pleasant construction something like this:

Assignment: can you do the same thing with six hula hoops?

For a recent event, MoMath wanted a large-scale public construction activity, so based on our success with hula hoops to date, designer Tim Nissen envisioned a gigantic pyramid of hoops — here’s the initial conception:

Now that’s a lot of hoops, so we decided to try a Sierpinski tetrahedron instead of a solid pyramid, which is at least as mathematically cool and requires significantly less material. (It’s interesting to think about just how much less…) All good building events require a rehearsal, so a bunch of folks got together to try tying hula hoops together on a Sunday afternoon.

The initial tie of four hoops into a sort of truncated tetrahedron went well, as well as combining four of these into an order-1 Sierpinski tetrahedron, as you can see from the following photo. It’s interesting to note that when you attach four solid tetrahedra at the vertices to create an order-one Sierpinski tetrahedron, the cavity remaining is a different shape (what shape?) — whereas in this construction based on circles, the central void is identical to the four units that were combined.

We even managed to combine four of the order-1 units into an order-2 tetrahedron pretty nicely:

Note at the next stage, the order-2 tetrahedra were too tall for us to put one atop three directly, so we planned to put this order-2 atop three order-1 tetrahedra, one at each corner, and then lift that entire structure atop three “bases”, each formed of three order-1 tetrahedra. However, we never got that far: when we fastened the order-2 tetrahedron atop the three order-1s, here’s what happened:

Total structural collapse, leading to hula chaos! What to do?

To be continued…

See all of our Math Monday columns

Gareth Branwyn

Gareth Branwyn
