Each year I visit Mathcamp
week to lead some fun hands-on activities.
Here's the complete graph on eleven vertices, one of two ribbon
constructions we made.
We also made a lovely expanded
model from Zometool.
And we had a bagel-cutting workshop to make linked bagel halves and
And here is a very tricky geometric construction from twenty identical
pieces of paper.
The four activities shown above are detailed in the four sections below.
1. Paper Construction
This is a paper model of a sculpture I call Snarl
I brought the cut paper parts, so the Mathcampers had "only" to
assemble and tape them.
It is really hard, trust me! Getting the first cycle of five is
the key step. Then add parts one at a time.
I made the paper model to help make the structure clear, but the "overs
and unders" are subtle.
It is easier to do if one has an appropriate thinking cap.
I don't recommend this technique of trying to add the last five parts
as a single module.
It is quite an accomplishment when done! Only a dozen or so
in the history of the universe have ever
succeeded in building this.
If you want to try assembling your own copy, this 3-fold axis view may
But I warn you, it is hard. The template for the parts is here
2. Bagel Cutting
In this workshop, I showed how to cut a bagel into two congruent linked
Anyone can do it in ten seconds once they learn the steps.
An optional extra activity is to do the integral that gives the surface
area of the twisted cut.
Here, the bagel itself was a convenient manifold for writing on, to
work it out.
Despite having only tough non-New-York bagels, these came out quite
I have the instructions for cutting bagels into two linked halves
And the instructions for cutting a trefoil knot bagel are online here
3. Zometool Workshops
I always do an assortment of 3D and 4D construction workshops at
These are dodecahedral modules for making the expanded 120-cell, an
attractive 4D polytope.
With a lot of teamwork, the modules gradually come together over about
And here is the finished polytope, with me looking through it.
You can read about the fifteen Zome-constructable H4 polytopes here
4a. Math Mob Complete Graph
The idea of a math mob
propose a concept for a mathematical event
that is scalable so many people can come together and do something cool.
Here we will make K11
from pink "marking ribbon" used by
After spacing ourselves into a circle, one person ties the end of the
ribbon to his left wrist.
The ribbon is passed around and around. Initially it is tied to
every person's left wrist.
On the second and third cycle, it is tied to every second person's left
wrist. And so on.
Since eleven is prime, the algorithm is particularly simple in this
The chords get closer and closer to the center as the number of skips
When complete, every person is connected to each of the other people in
a large piece of
The view looking up from underneath is quite cool.
Anyone walking by is drawn in with an irresistible urge to stand in the
We walked inside while still wearing it. It pops nicely back into
shape after going
4b. Math Mob Parabola
As a second math-mob ribbon construction, we made a giant parabola.
The participants line up in two rays making 90 degrees, with the vertex
at the top.
Passing the ribbon along the line, wrapping around the wrists, we
construct a series of lines.
The amphitheater steps are helpful for equal spacing. The
envelope of the lines is a parabola.
Above is the standard axes orientation. Can you prove these lines
are tangent to a parabola?
Here is the final parabola.
(More math mob ideas are sketched here
somewhere on a future occasion.)