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8
votes
1answer
265 views

computing average height-functions for lozenge tilings

Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to ...
3
votes
1answer
247 views

Radial tilings with variable area ratios

I was looking at this neat page on logarithmic spiral tilings when a question popped up: http://www.uwgb.edu/dutchs/symmetry/log-spir.htm It seems that in all of the tilings shown, the area of each ...
13
votes
1answer
415 views

Arctic regions in higher dimensional zonotopes

Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in this answer, one can consider higher dimensional zonotopes tiled by ...
14
votes
3answers
1k views

Rhombus tilings with more than three directions

The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...
9
votes
2answers
533 views

Name this periodic tiling

Hello MO, I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were ...
5
votes
1answer
325 views

Convex tilings of the plane

For convex polyhedra you have Steinitz's theorem characterizing them as the 3-connected planar graphs. My question is not about spheric tilings, but about periodic tilings of the euclidean plane. Is ...
2
votes
0answers
160 views

Are there any recommended texts that cover Turing Tilings?

I have read the original paper by Wang, as well as a paper by Boas [1996] entitled 'the Convenience of Tilings', but wanted to know if there were any other texts that people could recommend that ...
9
votes
1answer
508 views

Tiling survey that updates TIlings and Patterns?

Can anyone suggest a survey (or surveys) that provides an update to Tilings and Patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one. I am ...
13
votes
4answers
1k views

Decidability of tiling R^2

Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane? I know the ...
55
votes
5answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
13
votes
0answers
2k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
4
votes
1answer
311 views

A question about self-affine tiles

A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such ...
31
votes
1answer
7k views

Lecture notes by Thurston on tiling

I am looking for a copy of the following W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes. I see that a lot of papers in the tiling literature refer to it but I ...
21
votes
1answer
1k views

What can be tiled by T-tetrominoes?

The T-tetromino is a T-shaped figure made of four unit squares. An $m\times n$ rectangle can be tiled by T-tetrominoes if and only if both $m$ and $n$ are multiples of 4. This was proved in a 1965 ...
2
votes
1answer
307 views

Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?

The Gosper island tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosper-island' instead of a period parallelogram. In this case, I'm using ...