Grand total hours as of 1/1:
100 hours

12/15--1/12
I have been away from my computer and not documenting my hours.
I have spent a few hours here and there thinking about Sheffield's
paper, in particular his bijection between tilings and graphs.
I have been trying to adapt this idea to our tiles, thus far without
success.

12/10
2 hours at harvard

12/12
2 hours at harvard

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12/3-12/9
I read some of Pak's Paper.  I read Sheffield's paper probably 4 times
byu now, each time understanding a little more.
I now have a pretty good grasp of his ideas, and they seem like they
should help us.
Unfortunately, his results use some specific properties of ribbon
tiles, whereas we have a tile which is not a ribbon tile.
So we're going to have to see if we can make it work without these
properties.

~12 hours
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12/3
2 hours at harvard

12/4
1 hour updating web page

12/5
2 hours reading sheffield's paper, taking another look at height
functions and local moves

12/6
2 hours continuing yesterday's work

12/7
2 hours reading sheffield's paper

12/8
3 hours reading sheffield's paper, trying to adapt his methods to our
problem

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11/26-12/2
So last week's problem turned out to be a biggie.  We found a region in
particular, which admitted only one tiling, and had a unique shortest
cut which did not follow the tiling.
We tried to revise/expand our algorithm in a few ways to fix or
categorize & avoid such problems but we have not been able to do this in a
satisfactory way as of yet.
Jim mentioned to us that he had talked to Igor Pak who mentioned that
he thought our problem could be solved by using some ideas he had
previously developed.
So I emailed pak and scott sheffield and they referred me to a couple
of papers which treat similar problems.
I'm going to try to get my head around these papers next week and maybe
resume progress on this problem (!).
~3 hours

11/26
2 hours at harvard

11/27
1 hour emailing pak and sheffield, downloading & printing papers,
reading abstracts, emailing group

11/28
Gobble, gobble

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11/19-11/25
We discussed some width conjectures, all of which proved to be wrong,
but which gave us some interesting (i.e. smooth) untileable regions.
We continued to try to use the idea that untileability comes from local
color imbalance.
We discussed various iterative algorithms for dividing a region, such
that if the region were tileable, the division would follow the bounday
of tiles.
The most successful of these ideas was the "shortest cut" algorithm,
where one would make the shortest cut in a region which preserved color
balance.
The idea was that if this algorithm finished, you had a tiling, if not,
the tiling was impossible.  So we set ourselves the task of testing
this algorithm.
We were able to put an upper bound (of 7-ish) on the length of a cut,
then tested cases to see if all of the ways of "crossing the cut" could
be transformed by local moves into tilings which "followed the cut."
All cases where the cut was 1, 2, or 3 worked, but there was a case
where the cut was 4 which did not work.
Our task then became to see how big a problem this was.
~7.5 hours
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11/19
2 hours at harvard

11/21
2 hours at harvard

11/22
3.5 hours validating and then disproving "shortest cut" algorithm

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11/12-11/18
So my original height function isn't yielding many fruits. Dean and
jessica each came up with height functions which have some desirable
characteristics
that my function doesn't have.  We spent some time playing with those,
again looking for algorithms which yield discontinuities.
We went back to this idea on thursday, trying it with a 3-D height
function, which generates (another!) beautiful picture, which we think is
the same as our old manhattan/hoboken picture.
I spent some time saturday thinking about jim's bottleneck suggestion
and trying to sketch out a framework for how we coulod use local
conditions (which I am calling with functions) of tiles to inductive eat away
at any region in polynomial time.
I have a couple of very weak conjectures, but I'll share them with the
group on tuesday,and see if we think they're worth pursuing.
~8 hours
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11/12
2 hours at harvard
1 hour exploring dean's height function

11/14
2 hours at harvard

11/16
2 hours thinking about bottleneck ideas, developed the idea of a
"width" function.

11/17
1 hour writing summaries, updating web page

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11/5-11/11
I spent a bunch more time this week playing with height functions.
At first, I thought the hoboken/manhattan approach might be the extra
we needed to get a satisfactory criteria, but I soon convinced myself
that that was wrong.
I was pretty discouraged with the whole enterprise until jim suggested
that our old abandoned 1-d height function might actually be useful.
So I spent a bunch of time trying to develop an algorithm for extending
this height function into the interior of a region which would give a
criteria for tileability.
My basic idea is still to follow thurston:  If the tiles are lifted
one-by-one to the overlying graph, their boundaries will be continuous.
Thus the idea is to find an algorithm which , when extended to the
interior, will give discontinuities on the interior of the tiles, or show
an impossible pattern if the region is untileable.
Also toyed with the idea of trying to turn my 3-d graph into a 3-d
manifold, but I"m not really sure what I"m doing there.
~11 hours
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11/5
2 hours at harvard
3 hours playing with signed area (manhattan/hoboken) and other models

11/6
4 hours playing with original height function

11/7
2 hours at harvard

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10/29-11/4
This week I did (and am still doing) a bunch of exploration with height
functions on the hexagonal grid.
My instinct is that I'm going to need to use a little more
sophisticated group theory to get a satisfactory solution to this problem.
I have a few idle ideas at this point, which I will hopefully explore
over the weekend.
~10 hours
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10/29
2 hours at harvard
joined the polyhex group
explored some more with my height function

10/30
talked to jim about my height function.  he suggested he thought it
would need to be multi-dimensional in order to give a sufficient
condition.
So I tried a different height function, and ran into some of the same
difficulties (i.e. it is nec., but not sufficient).
Have to explore a bit more with this one.

The height function is as follows:  three-color the hexagonal grid,
R-G-B, with R on top of B on top of G.
In a consistent way:
Alternately label the edges of R's with b's and c's
Alternately label the edges of G's with a's and c's
Alternately label the edges of B's with b's and a's
Direct the edges so that:
As you walk around any given hexagon, you alternate between walking
with the direction and against it.
Each edge labelled a is oriented counterclockwise around the B hexagons
Now as you walk around the edge of the region, keep track of the a's
b's and c's as you go (a 3-d height function).

10/31
2 hours at harvard
shared some results with polyhex group.

11/1
updated web page
read about automatic groups and solving the word problem.

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10/22-10/28
Read the Thurston article anout tiling.  It was really well-written and
gave me a bunch of ideas for the tri-hex tiling problem.  Spent the
remainder of the week in sunny florida.
Developed a height function for a hexagonal grid which is really pretty
geometrically but may not be sufficient for our purposes.
~10 hrs
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10/28
Developed a height function for trihexes by analogy with Thurston's for
dominos, which gives at least a necessary (but not sufficient)
condition for a region to be tiled by tri-hexes. I played around with a bunch
of examples of tileable and non-tileable regions to try to see if I
could extend my criteria to make it sufficient, just as Thurston had done,
by extending the height function into the interior of the region.
My height functionis as follows:  Three-color the hexagon grid, R, G,
B.  As you walk around the grid, if you traverse an edge with a green on
your left, add 1 to your height, if you traverse an edge with a blue on
your left, subtract 1.
These assignments yield a graph where above the green hexagons are
right-handed helices, above the blue hexagons are left-handed helices, and
above the red hexagos are "sheets" which alternate
up-down-up-down-up-down.
The nice thing about this height function is that the boundary of any
tri-hex is closed in this graph (i.e. the height comes back to zero
after you walk around it).
Thus a necessary condition for any region to be tiled by trihexes is
that the boundary be closed in this graph.
Unfortunately, it is not hard to find regions which meet this condidion
which are not tileable.  Thus we need to strengthen the condition.

10/23
1 hour reading thurston article

10/22
2 hours at Harvard
1 hour reading Thurston article

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10/15-10/21
Read all of the one-pagers.  The tiling problems seem interesting, so I
tracked down some of the relevant articles and got a start on reading
them.  If I can find a couple of other people who are interested in this
project, I'd like to pursue it.  By the next meeting, I'm hoping to
have as good a grasp as possible of exactly what other people have done
with these problems.
~10 hrs.
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10/15
2 hours reading:
integrality
symmetry
letting algebra do it
grant proposal
one-page introductions
minutes
(all from reach web page)
2 hours at harvard
1/2 Hour more reading a pedestrian approach...

10/17
1 hour finding and xeroxing articles
1 hour reading conway and lagarias article
2 hours at harvard

10/18
updated web page
sent email to DONALD west
took another stab at conway article 1 hr.

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10/8-10/14
Learned a lot more about recurrences, octahedral and otherwise.  Spent
a bunch of time trying to formulate combinatorial interpretations for
recurrences generated by matchings.
~7.5 hrs.
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10/8
2 hours at harvard
1.5 hours working on matching recurrence, combinatorial interpretation

10/10
2 hours reading frieze/condensation paper
2 hours at harvard

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10/1-10/6
Learned some more techniques, like using Cayley-Hamilton to solve 1-d
recurrences.  Learned about laurentness property and why we care about
it.
~6 hrs.
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10/1
2 hours at harvard

10/3
read domino tilings paper
2 hours at harvard

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9/24-9/30
This first week I spent trying to gain exposure to different techniques
which I expected we might use.
~10 hrs.
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9/24:
2 hours at REACH-introduction, resources
15 min. on matching recurrence problem

9/25
started REACH diary

9/26
2 hours at REACH
1.5 hours deriving generating function (2 ways) from matching
recurrence
to do: figure out how to get recurrence relation from g.f.
30 minutes finishing and proofreading minutes

9/27
made web page
posted minutes
started generatingfunctionology

9/29
Read first two chapters of gfology

9/30
worked on homework problems

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pre-history:
Read and did most of the problems in Wilf's East-Side, West-Side
Learned basics of MAPLE