Math Monday: Hula Hoop Geometry

Science
Math Monday: Hula Hoop Geometry

By Glen Whitney for the Museum of Mathematics

hoop

Math Mondays have so far featured a wide array of different items from which one can make a tremendous variety of geometric constructions, but there has not yet been one on hula hoops. This week and next we’ll remedy that oversight. Also, the postings so far have almost entirely shown the constructions as fait accompli, so this series will also try to give a bit of insight into the process of devising a new creation.

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:

hoop
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:

hoop

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.

hoop

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

hoop

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:

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

Continued in Math Monday: Hula Hoop Geometry, Part II

14 thoughts on “Math Monday: Hula Hoop Geometry

  1. Colecoman1982 says:

    At first glance, it looks like an issue with their method of attaching hoops to each other. The pictures make it look like they’re using zip-ties or Velcro straps. Much like with lashing based construction, this allows some shifting. I have a feeling that’s what caused the final collapse. If they adopt a more ridged form of connection, I think they’d find it much easier.

  2. danny says:

    Is there a world record for this? I’m tempted to get my 7th grade class to attempt the record….or a public free standing hula hoop structure? :)

    1. Glen Whitney says:

      Well, if you do, please be sure to send a photo to mondays@momath.org — good luck!

  3. Hula Hoop Geometry | Hooping.org says:

    […] constructions, but there has not yet been one on hula hoops – until now. Why hula hoops? Glen Whitney explains, “They’re a pretty cheap source of large, pre-made circles, generally decently symmetric […]

  4. MAKE | Math Monday: Hula Hoop Geometry, Part II says:

    […] Hula Hoop Geometry, Part I See all of our <a href=”http://blog.makezine.com/tag/MathMonday”>Math Monday</a> columns Share this: Pin ItLike this:LikeBe the first to like this post. […]

  5. MAKE | Math Monday: Those Circles Are GREAT! says:

    […] few weeks ago, I challenged readers to arrange six great circles so that every intersection of a pair of circles […]

  6. Math Monday: Those Circles Are GREAT! says:

    […] few weeks ago, I challenged readers to arrange six great circles so that every intersection of a pair of circles […]

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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. His free weekly-ish maker tips newsletter can be found at garstipsandtools.com.

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