General Information

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Project Description

Crystallographic sphere packings, first introduced by Kontorovich and Nakamura in 2017, are collections of oriented n-spheres which have disjoint interiors, fill space densely, and are generated by the action of a geometrically finite hyperbolic reflection group. Theorems by Koebe-Andreev-Thurston and Kontorovich-Nakamura show that every polyhedron has an associated circle packing. By examining the bend (1/radius) of each circle in the packing, we can determine whether a packing is integral—that is, if its circles have exclusively integer bends. This question is of fundamental interest, particularly because packings associated with polyhedra are not well documented. This work classifies circle packings associated with polyhedra on up to 7 vertices, identifying a set of fundamental integral polyhedra that cannot be decomposed by slicing along faces or edges.
This work was done as one part of a larger project by Zachary Stier, Debra Chait, and myself.

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Weekly Log

Week 1:

This week has consisted mainly of learning background information to help contextualize and understand the problems we will be working with. We are beginning to focus on our specific projects - I will be studying the Koebe-Andreev-Thurston theorem - and understand the background and tools needed to study them. I am reading this paper by Kontorovich and Nakamura to get background.

Week 2:

After learning more about the Koebe-Andreev Thurston theorem, we followed Ziegler and wrote a function in Mathematica that takes in information about the faces of a polyhedron, calculates the radii of each circle in the associated circle packing, finds their positions and renders an image, and calculates the associated Gram matrix.

Week 3:

This week we learned about Bianchi groups and Vinberg's algorithm while working on adding to and adjusting code so that we could start setting up a website. This website will catalogue various crystallographic sphere packings associated with polyhedra and Bianchi groups, as well as higher dimensional sphere packings.

Week 4:

Using the framework we set up last week, we began adding as much information as possible to the website. This data will help us begin to verify or disprove conjectures about the integrality of circle packings. A version of the website can be seen here.

Week 5:

A program called plantri was used to generate data in an efficient and methodical way, leading to discovery of potentially 3 new distinct integral polyhedra. Integrality has not yet been proven, but empirically seems to hold in each case.

Week 6:

At the end of last week I found potential candidates for new integral polyhedra; upon further inspection, this only seems to be true for one of them. However, the other two are rational but not integral, which is interesting in its own right, and I have spent time this week trying to prove nonintegrality despite being rational. I also added more to the website and made it more user friendly.

Week 7:

I spent a lot of time towards the beginning of the week trying to refine our Mathematica code for finding the exact layout of the circles in a packing from just the polyhedron. I have a more robust version for part of it, but not all. The latter half of the week was spent focused on our final presentation, which we did as a group.

Weeks 8 & 9:

The last weeks of the program were spent writing a final paper with Zachary and Debra. I also attempted to write a version of the Mathematica program that combines two approaches, but reached a roadblock with Mathematica's precision errors. In addition to the final paper, I have created a poster specific to my own portion of our project to present at the Young Mathematician's Conference in August.

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Presentations

• First Presentation
• Second Presentation

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Additional Information

• My Mentors
  • Dr. Alex Kontorovich