Adri's REACH Log

  
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REACH stands for "Research Experiences in Algebraic Combinatorics at Harvard." This log is a record of my homework, research, and progress.

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Wednesday, January 29, 2003

 
Spent Tuesday's meeting discussing the new direction of our problem. We have decided to switch from hex tiling to triangular tiling, using Sheffield's ribbon tiles (of sorts) to successfully determine tilability. This sounds a lot more promising. Jen explained to me the new coloring sheme and new height functions for triangular regions. Also, we met the two new REACH participants, and Dave explained to them what we worked on for the past few months.
Adri 12:07 PM

Tuesday, January 07, 2003

 
Small turnout today- but I was able to catch up on REACH sessions missed during MIT finals week. Currently reviewed Pak's article. Obvioulsly since Pak/Shefield disregard a key disjoint tile, making up for this tiling will be the tricky part. I spent time today researching where the original translations of our five hex tiles into five rectangular tiles came from. Conway and Lagarias bring it up, but do not give a formal translation. If the fourth disjoint tiling translation can be "altered," then we can possibly apply Pak/Shefield's work to our problem...

HOURS THRU JANUARY 15th = 61
Adri 2:59 PM

Tuesday, December 03, 2002

 
Started reading Igor Pak's article on ribbon tiling. Currently trying to find a relationship between his paper and our original algorithm that transformed hex tiles into rectangular tiles. Found a possible relationship for our two moves (two triangles to two lines and the triangle/line transformation).

HOURS THRU DECEMBER 3rd = 53.
Adri 8:43 PM

Tuesday, November 26, 2002

 
On Thursday, the group tested David's breakthrough conjecture -- "Make the shortest possible cut. If both areas are tilable, repeat with the next shortest cut. If eventually we are left with two tilable regions, the entire region is tilable. If we are left with at least one region not-color balanced (and therefore not tilable), then the original region was not tilable." So far, we cannot find a counterexample. And of course, intutition leads us to believe that this is true. The hard part now is proving this. The upper bound on number of cuts necessary is 8, and this will occur only when we need to "free" a completely impacted triangle. We can use this as a starting point.
Adri 11:31 AM

Friday, November 15, 2002

 
Spent time yesterday during REACH determining all untilable, color balanced paths (path = 0) for two of each color (hence 6 tiles). Disregarding rotation, reflection, and translation, I was able to come up with a definitive five unique paths. Two seems to appear in pairs, and 1 is symmetric and therefore does not have a pair. I am going to do this for 3 of each color (i.e. paths of 9 tiles) this weekend. The only concern with this method is that it will blow up quickly. However, danl suggested writing a program to check my hand written work, so this might be very helpful. Also Jim suggested looking into the bottleneck problem that we saw for domino tilings. However, it seems almost trivial to do this with hexes, since the features making a region "untilable" often lie on the outside of the regions, rather than in the inner, connecting regions between two larger tilable regions. But using the two pieces danl determined previously in conjunction with the nontilable regions might provide insight to a bottleneck problem posed by hexes.
Adri 10:02 AM

Thursday, November 14, 2002

 
Working on hex-tiling with group. I have put work on the word problem on hold, basically because 1. I don't think it will be so easily solvable 2. the headway I was making was interesting, but seemed circuitous (i.e. determining conjugates of paths/words) and 3. Jim figured it might not necessarily be the most efficient path to solving the word problem. So I have spent the past few REACH sessions focusing on the height function. I started with 3 dimensional height functions to gain familiarity with the function, and have been testing regions with the 1-D function. On Tuesday I spent time determining the number of untiliable regions with heights of 0, since this will be our biggest challange in forming an algorithm. (Obviously for tilable regions the path = 0 and for non-tilable regions MOST paths are not equal to 0). Therefore focusing on untilable regions with path = 0 is key. I am currently looking at patterns where paths are untilable for all regions with 2 of each color, 3 of each color, etc.
Adri 8:18 AM

Tuesday, November 05, 2002

 
Spent 4 hrs this weekend reading about Conway's "word problem." Spent time at REACH today trying to get conjugates/cycles of 1 tile. Made progress with the help of other team members to see how groups of tiles (of the same kind) can be reduced down to the same word for a single tile of that type. A lot of this was "assumed to be obvious" in Conway's paper but was not until today. Hopefully mastering 1 tile will lead us to an insight for all 5 tiles. I will be working on other arrangements of 1 tile and multiple tile types for Thursday.
Adri 8:47 PM