Public Invention Mathathon 2018

This repo holds information defining open problems, and hopefully solutions to them, for the Nov. 30th through Dec. 2nd, free of charge, free-libre all-virtual mathathon/hackathon hosted by Public Invention.

Introduction and What to Expect

We’ve prepared an introductory slide deck that will be presented on Friday night to introduce the weekend’s cooperation.

When you come to the zoom room on Friday at 7:00 Central, David and I will go over the slide deck introduction. After that, we will be ready to answer any questions you may, and also to help people form teams.

I can’t be sure, but I suspect people will want help finding problems appropriate to their interest and skill levels.

Throughout the weekend, I and others will be in the Zoom Room at all times (except sleeping and short breaks) to answer questions and help with any work. If your interent connectiong doesn’t support the Audio/Video in Zoom or you just prefer chat, please join the slack team devoted to this purpose. David and I will be monitoring the slack channell all weekend.

It is our deepest hope that some teams will form and people will have fun, and maybe solve some interesting problem. We’ll try to make that happen, if you are willing to share with us what you are most interested in.

Math is not easy; but it is fun. We have some very hard problems, but we have identified some problems that we think an advanced High School student can work on.

Usage of the Software

We have written a 2D web page for testing interactively testing the easiest class of problems: building chains of equilateral triangles. We call instructions for building such a chain is called a 2D simplex generator.

The page allows you to dynamically load a JavaScript function which can generate a simplex chain made of equilateral triangles joined edge-to-edge.

The JavaScript function takes a natural number (starting at 0) and tells which edge of the last triangle in the chain the next triangle should be joined to. These edges are simply named “L” and “R” for left and right. The string “S” is returned when it is time to stop. Every chain starts with one triangle.

As an example, consider this JavaScript function which generates a beam or ladder.

(n) => { return ((n < 10) ? (((n % 2) == 0) ? "L" : "R" ): "S"); }

By turning left and then right, a straight “ladder” is constructed. This is one of the simpler generators.

If you create a generator which causes a simplex chain to self-collide, the colliding triangles will be rendered as black.

A number of sample generators can be selected, but the goal is to write your own! Play around and see what you can create. For example, can you write a generator that fills all space? If you find a good generator, make a pull request or just place it in an issue and we will add it to our library of sample generators.

Draft of 3D Playground

We are currently attempting to develop a similar piece of code for 3D regular tetrahedra.

Motivation and More Math

To understand why problems like this, and in particular 3D versions of these problems are valuable to mechanical engineers and robotocists, you may want to read our technical article that also lists many of the open problems that will be addressed by the mathathon.

Improvements

The purpose of the mathathon is to foster cooperation, not competition. We welcome cooperation in making this repo more fun, and in getting ready for the Mathathon. All sorts of suggestions are welcome.

With respect to the current experimentation page, we would like to:

• Allow the grid size to be controlled (in particular, increased.)
• The starting point and direction could be better specified.
• A system for evaluating how closely a simplex chain matches and geometric curve would be very useful.
• A system for grading specific problems, such as space-filling or creating a logarithmic spiral, would be fun.
• Expanding the library of generators is always valuable.
• Is there a more elegant and powerful way to program generators? (For example, could we create subroutines for turns and straight lines?)
• Can we build a set of evalutions, such as if a generator is periodic, or closed, or unbounded, or minimal?

Open Problems

In an attempt to make that Mathathon serious and seriously fun we have defined a large set of open problems tagged with particular numbers. You can find these in our pdf. Precisely defining interesting problems is a major part of this effort; if you can suggest a problem, please send it to us in an issue.

Code of Conduct

We have published a code of conduct.

License

All code is this repo is licensced under the GNU General Public License (v3). Non-code assets are licensed under the Creative Commons Share Alike-By Attribution (v4) license.