This week we began studying rectangles. Rectangles were our polygon-of-choice because they easily form a tiling that is amenable to a C artesian coordinate system. Additionally, since squares can be seen as a particular case of the rectangle, the work we did on rectangles could easily be applied to squares as well. Later in the week we worked on 45-45-90 triangles and over the subsequent weekend, we wrote down the conjectures we formulated for rectangles, squares and 45-45-90 triangles. Surprisingly, many of the results for total orbits of a certain period and non-iterated orbits of a certain period in these shapes retained similar properties to the equilateral triangle's. As we go onward, one of the largest questions will be the level of similarity between different shapes - what factors lead some shapes to share properties and what leads them to diverge?

This week began with little hope of making progress in the 30-60-90 triangle. But, with much diligence, a large excel document and -of course- colored pencils, we were able to count the orbits of this triangle with relative ease. Although the actual orbits are not directly related to other triangles (in the way that 60-60-60 was related to 45-45-90), the method of counting orbits turned out to be strikingly (!) similar. The connection between partitions of a number "n" into specific sized groups has proven invaluable. Does an underlying reason exist as to why partitions are such an effective way to count periodic orbits in triangles? Can the conclusions be extrapolated to other shapes? The rest of the shapes we are dealing with have at least one angle greater than or equal to 120 degrees which makes the analysis of their tessellation much more difficult to consider. Hopefully the setbacks of these shapes will not be insurmountable!

So far this week we have begun classifying the orbits of the 30-30-120 triangle. This triangle was a reasonable next step in our project, because, it is one of the unique shapes which is able to edge-tessellate the plane. Using techniques that we implemented earlier, Over the week I hope to finish a program I was writing in Java to predict a billiard word for a periodic orbit given an ending coordinate (i.e. bw(1,1)=acabacab...).

Last week was a week of work with little (to no) results. At the beginning of this week though, after indefinitely putting aside the billiard words problem, we began to finalize our approach to 30-30-120 triangles. The non-acuteness of this triangle makes this shape unique; the fact that it contains an angle (120 degrees) whose degrees whose degrees divide 360 an odd number of times makes the 30-30-120 triangle special. Specifically, the 120 angle leads to a phenomenon in this triangle where, for a given angle, different positions along the starting side of the triangle will lead to different length periods. However, despite irreconcilable differences from our previous studies, the orbits in this shape were, too, reducible to spaces of orbits generated from two basis orbits. The 120 degree angle in this triangle does differentiate it from other triangles but after accounting for this distinction, the method for identifying and classifying orbits in this triangle are strikingly similar to previous cases.

On Thursday of this week was my final presentation. After, to some extent, breaking through the problem of edge-tessellators containing angles of 120 degrees, the innovating element of the summer's research is largely over, and what remains is the synthesis of my work thus far into a final presentation on Thursday, a general summary for the administration of DIMACS and -hopefully- the writing up of our work for the math research world for publication.

This eighth week is the last week of the REU program. A significant part of the program has flown to Prague and my office has been completely vacated (except for myself). With the final presentations presented, most of this week has been spent writing up results we found over the summer and making a rough draft of a writeup for the DIMACS program, detailing my work and experience this summer. Thank you very much for the opportunity to share my activities on this blog and I look forward to sharing more successes in the future!