Robert L. Read

@RobertLeeRead

Join A Free, All-online, Non-competitive Mathathon Organized Around Some Open Problems

The Boerdijk-Coxeter Helix, formed purely out of regular tetrahedra, not discovered until the 1950s.

On the weekend of Dec. 1st, we’re trying something that has never been done before. A hackathon is usually a weekend, often from Friday night to Sunday afternoon, of intense computer programming by teams. Often the artificial drama of competition and cash prizes are injected. Usually the teams choose their own project. Sometimes they are located in a large meeting space, sometimes they are completely virtual. Often they are organized around a theme, but rarely is that theme math.

We’re trying something different: a free, virtual, “math hackathon” — a mathathon. This is somewhat experimental, since, unlike hackathons, there is not an established tradition for mathathons. Although of course computers will be used and some of the problems are about algorithms, this is fundamentally about geometry. Our goals are ambitious. We want:

  • Participants to get real: be able to contribute to real problems whose solution is currently unknown, not simple exercises,
  • Participants to experience working on teams on real problems,
  • Amateur mathematicians and students to be able to attack real, if relatively easy, math problems whose answers are currently unknown,
  • The whole mathathon to cooperate in exploring a specific set of pre-defined problems in a specific area, in order to emphasize not individual achievement, but group progress, and
  • Others to be able to learn from our experience to host other mathathons in the future.

The spirit of the Public Invention Mathathon 2018 is cooperation, not competition.

In order to accomplish this, our mathathon is going to be “free and open.” We mean free as in “free speech”, but it also happens to be without cost as in “free pizza.” Participants will produce all of their work under the Creative Commons license that allows sharing, specifically, the Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) license.

Because we are not sure we could get enough people in any one town to participate and to allow as many people as possible, including internationally, the Public Invention Mathathon will be completely virtual. We will use a Zoom room for video chat, a public Slack channel for text communication, and “git”, a version control system, for sharing documents, hosted as a GitHub repository.

In math as well as science, sometimes asking the right question is as important as finding the right answer. We have, therefore, prepared a long list of “open” problems that form a framework for cooperation. All of these problems are in a branch of geometry, relating to how triangles or tetrahedra, (collectively simplices) can be placed together to form simplex chains. We believe the problems range in difficulty from things advanced High School students could address on up to publishable research problems.

As an example of the kind of problems we are addressing, here is a problem from the beginning of our work-in-progress article on the problems for the mathathon:

The term simplex means the simplest regular polyhedron. In two dimensions, a simplex is an equilateral triangle. In three dimensions, a simplex is a regular tetrahedron. In four dimensions, it is just called a simplex. Presumably one can define a simplex in higher- dimensioned Euclidean spaces as well.
Define a simplex chain to be a figure of many simplices adjoined face-to-face by a particular rule. The dimensionality of a face is always one less than the dimensionality of the space. In two dimensions, a simplex chain is a series of adjoined equilateral triangles joined edge-to-edge.
Let us number the simplices in a chain starting from 0. Then we can define a simplex chain via a rule that says which edge of the nth simplex to attach the n+1th simplex to. In two dimensions, we can label the edge of the last triangle in simplex chain as anti-clockwise or clockwise, or just left and right.
If is clear that the rules “always go left” “always go right” produced a pretty hexagon, but then starts putting the triangles right where one already is, creating a “self-collision.”
The rule “go left if n is even, right if it is odd” is a bit more interesting; it produces a “ladder” that goes off in a straight line. That is, the whole figure is contained with in two parallel lines separated by the height of the triangle.
A plane-filling spiral of triangles
Exercise: There is probably a simplex chain rule which tiles the entire plane with equilateral triangles. Can you define it?

In preparation for the mathathon, we’ve written an interactive webpage that let’s you try to solve this problem graphically.

We also deal with simplices in three dimensions, which are tetrahedra stacked together. In general, the harder problems are about 3D spaces, and the easier problems are about the 2D plane. To get a flavor for the kinds of 3D problems we are asking, watch the following video:

The current set of Open Problems is a work in progress, but contains more than 20 (more or less) crisp questions, to which we don’t know the answers. Often when you say “an open problem” it implies a difficult, well-recognized problem, the solution of which will bring great reputation to the solver. We do not mean that in these cases. We, like our participants, know only a tiny sliver of all that could be known. Here is a list of some things we don’t know:

  • The solutions to the problems we list,
  • Precisely how hard they are,
  • Whether they will be of interest to anyone, or if we will have more people than 100 that can fit in our Zoom room,
  • Whether solutions to these problems are worthy of publication,
  • Whether this mathathon will succeed in solving a single problem, or all of them.

But we believe it will be fun to spend a weekend exploring this!

A reader might ask: why should I, or anyone, care about how triangles and tetrahedra fit together? Why are simplex chains interesting?

To a “pure” mathematician the answer is:

They’re pretty.

To an “applied” mathematician the answer is:

Triangles and tetrahedra are inherently rigid structures, and therefore critical to structural and mechanical engineering. The strongest-for-their-weight two-dimensional structures, trusses, are formed of triangles, and the most efficient three-diemensional structures, or spaceframes, tend to be formed of tetrahedra…

Our evolving documentation of the open problems can be found here: https://github.com/PubInv/Mathathon-2018-Simplex-Chains/blob/master/SimplexChains.pdf. To suggest improvements in the problems or to make other suggestions, create a pull request or an issue, respond to this article, email <read.robert@gmail.com>, or join the Slack Team.

In order to create a safe, moderated, and helpful environment for everyone, we have enlisted several volunteer facilitators, (David Jeschke and myself) who will be on-line at all times except for a sleeping period late at night. We will attempt to answer questions, help teams form, and help people use git to share their work both publicly to the whole world, but more particularly, to the other members of the mathathon. We will encourage people to post their work even in a rough, incomplete, and possibly even wrong, state. The goal will be to spend this time developing an evolving body of work, not waiting until the end to reveal work to be judged. That is why we intend to make awards like “Best early contributor of shared work.”

When I was in graduate school I heard Prof. E. W. Dijkstra, one of the most famous Computer Scientists of all time, say:

If you are going to fail, fail spectacularly.

This all-virtual, cooperative mathathon organized around open problems is something that, as far as we know, has never been tried before. It may fail spectacularly. Maybe nobody will show up. Maybe not a single problem will be solved. Maybe our formulations of the problems are hopelessly unclear.

And for these reasons, it is a particularly exciting experiment. We don’t know what is going to happen. We hope you will join us. Registration is free, but limited to 100 participants: https://www.eventbrite.com/e/mathathon-a-cooperative-virtual-mathathon-tickets-50181898409

We intend to award superlatives, like “Best contribution by a student”, but are not offering any monetary prizes.

Public Invention is an educational non-profit being established by the author. Please share this article, or any of our other promotional material, to mathy persons, clubs, and communities.

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