Zachary Stier's REU 2018 Web Page

About Me

Name: Zachary Stier
Email: zachary.stier (at)
zstier (at)
zjstier (at)
Office: CoRE 434
Hill 323
but I'm usually in Hill 705
Home Institution: Princeton University
Project: Sphere Packings and Number Theory
Mentor: Professor Alex Kontorovich

About My Project

Debra Chait, Alisa Cui, Gabriel Eiseman and I are all working with Professor Kontorovich on various aspects of Apollonian circle packings. My work will focus on Kontorovich and Nakamura's notion of crystallographic packings, which are a type of Apollonian packing which conveniently generalize to higher dimensions, in contrast to general-position configurations which in dimensions 4 and higher are not guaranteed to generate valid packings.

My abstract for the work I will be presenting at the Young Mathematicians Conference: The Apollonian circle packing, generated from four mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. More recently, this has yielded a connection between $n$-dimensional packings and configurations of planes in $\mathbb{H}^{n+1}$ for various quadratic forms in $n+2$ variables. In particular, Vinberg's algorithm, in conjunction with Kontorovich and Nakamura's Structure Theorem, allows us to ask questions about whether certain Coxeter diagrams in $\mathbb{H}^{n+1}$ for a given quadratic form admit a packing at all. Further, Kontorovich and Nakamura's Finiteness Theorem shows that there only exist finitely many classes of these packings, none of which are in dimension 21 or above. In this work, we systematically determine and enumerate all known examples of higher-dimensional sphere packings arising in this way.

The abstract for the paper Alisa, Devora and I are writing:

Research Log

Week 1: May 29 - June 1

This week, we spent time familiarizing ourselves with the fundamental notions of these packings through lectures and discussions with Professor Kontorovich. This included: Apollonian circle packings; Descartes' kissing circles theorem; circle inversions; inversive coordinates; hyperbolic geometry as motivation for some inversive notions; Coxeter diagrams; and crystallographic packings.

Week 2: June 4 - 8

This week we primarily worked on implementing ideas from the Koebe-Andreev-Thurston Theorem in Mathematica. We managed by Friday afternoon to complete a method that, given appropriately-ordered and indexed polyhedron data, is able to reproduce the associated strip packing and Gram matrix (along with inversive coordinates).

Week 3: June 11 - 15

This week, we debugged some cases of the K-A-T implementation and learned more about the theory of circle packings (bend matrices, Vinberg's algorithm). We also started on putting together the website, which included implementing Bend, Orbit, and Strip functions. I then began focusing on my individual aspect of the project, which entailed translating Vinberg's coordinates for higher dimensions into inversive coordinates.

Week 4: June 18 - 22

This week we primarily worked on uploading more data onto the website, and understanding the theory of doubling. We spoke with Kei Nakamura and Alice Mark about various confusions with the theory (including doubling and vinberg's algorithm). On the non-mathematical side, we had a field trip on Wednesday to IBM Headquarters in Yorktown, NY.

Week 5: June 25 - 29

This week I computed doublies of Coxeter diagrams that lacked isolated clusters, in part in order to identify which ones will require more advanced methods to turn into packings. This occurs in parallel with computing data for the website. Later in the week, I pivoted to understanding Kontorovich and Nakamura's method for transforming the transformation of the correct Coxeter diagram for the Eisenstein Bianchi group Bi(3) [the 4-1-1 diagram] into one with two isolated clusters [infinity-4-1-infinity]. I found this extremely difficult, and was unable to find the correct series of transformations.

Week 6: July 2 - 6

A main idea that I had for the 4-1-1 transformation, exploiting the common 4-1 subdiagram shared between it and the known solution, turned out to be a red herring. Professor Kontorovich managed to reconstruct the solution he and Kei had come up wtih last year. I was then tasked with a handful of ideas to start thinking about: doing the same for 4-1-2 (which was not known to necessarily have a solution), work on a robust method for testing if a configuration excends, Poincare-ly, to a hyperbolic polytope of finite volume, and a conjecture about integral quadratic forms. I set out on the 4-1-2 diagram first. After finding an almost-solution (it generated a packing that was missing some circles but was otherwise complete) I sent it to Professor Kontorovich, who managed to find a subdiagram lurking in a layout of the orbit of 4-1-2 on itself. I spent the week working through a paper by John Milnor on hyperbolic volume and working to implement the Lobachevsky function in Mathematica. Issues arose when it became apparent how nontrivial the problem of hyperbolic volume really is.

Week 7: July 9 - 13

The workaround to hyperbolic volume is just to test if the interiors in the real boundary of hyperbolic space are of dimension 0. Mathematica has a built-in method for this that is terribly slow in high dimensions (even as low as 5). However, it is also just an inequalities problem, and it turns out that the inequalities are really easy to analyze by the eye test. Using this, I managed to find packings arising from the n=6,7,10,11,13 (dimension=5,6,9,10,12) cases of the d=3 quadratic form! The main step is to note that all of the fundamental polyhedra in those spaces feature 4-1-1 subdiagrams, which proved to be very exploitable.

Week 8: July 16 - 20

This week is largely dedicated to preparing the final paper. There might be some research occuring in the latter half of next week, depending on how much revision the writeup requires.

Week 9: July 23 - 27

This weeks was spent on paper writing, poster writing (Alisa and I will be at the Young Mathematicians Conference at Ohio State in two weeks!), and error correcting.

References & Links

Click here to see our website of crystallographic packings.

Click here to read our paper.

Here are the presentations that I gave:
  1. Initial presentation (June 4)
  2. Final presentation (July 13) (joint with Alisa and Devora)
Here are the papers I have read for my project: