All Questions
Tagged with tiling reference-request
24 questions
47
votes
1
answer
13k
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 ...
- 1,795
21
votes
0
answers
453
views
Does every 5-celled animal tile the plane?
An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) One may ...
- 3,068
17
votes
1
answer
457
views
The sparsest planar net that captures every unit segment
Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...
- 151k
15
votes
2
answers
737
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 ...
- 881
15
votes
0
answers
573
views
Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane
This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE.
There exists a rather remarkable ...
- 715
11
votes
1
answer
406
views
Thinnest 2-fold coverings of the plane by congruent convex shapes
It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...
- 151k
11
votes
1
answer
475
views
Examples of games developed purposely to analyze players' strategies for mathematics research
Background
This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
- 4,829
10
votes
2
answers
803
views
Reference for Wang tile
I am working on projects in solving ground state of generalized Ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results:
...
- 867
10
votes
2
answers
678
views
Name this periodic tiling
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 previously ...
- 23.2k
9
votes
4
answers
902
views
quasicrystal and penrose tiling, mathematical introduction
Starting to research on quasicrystal from material science, I want to know more about how to understand quasicrystal from a purely mathematical (especially tiling) perspective (probably start from ...
- 867
9
votes
2
answers
2k
views
How to divide a square into three similar rectangles
Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles?
With a bit of algebra it can ...
- 99
6
votes
2
answers
424
views
A class of tilings with amazing visual qualities
For more examples please see my related question on MSE:
Interesting tiling with a lot of symmetrical shapes
This is achieved by rotation of square grid over itself by atan(3/4).
Resulting ...
- 161
5
votes
2
answers
382
views
What is known about tiling a rectangle in an irreducible way by smaller rectangles?
Given two naturals $s<t$. Is there always a square (or at least a bigger rectangle) that can be tiled with $s\times t$ rectangles in an irreducible way (i.e. any grid line splitting it cuts at ...
- 13.4k
5
votes
0
answers
108
views
Non-monotileable amenable groups
This is crossposted from MSE.
We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$.
In his article Monotileable Amenable Groups, B. Weiss ...
- 10.6k
5
votes
0
answers
150
views
monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
- 43.1k
4
votes
1
answer
549
views
Tiling with restricted overlap
Non-overlapping tilings of regions is a well-studied topic.
I wonder if the following variant has been considered:
A tile can be partitioned into several regions, where such regions from different ...
- 15.8k
4
votes
2
answers
207
views
Classification of symmetries of tilings in surfaces?
Is there a general study of the symmetries of tilings on surfaces?
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
- 561
4
votes
0
answers
304
views
References and upper bounds for the SONNAT tiling game?
Introduction
In a video released about a month ago, Pembesita describes1 a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling.
In the single-player game2, the player may employ ...
- 4,829
3
votes
2
answers
397
views
An "incomplete" tiling?
Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them?
When each square of the board is covered by a domino this ...
- 380
2
votes
2
answers
131
views
Reference request: Cut-and-project method gives rise to a fiber bundle over the torus
I apologize in advance for how vague this request is.
A few weeks ago, I came upon a paper that (if I recall correctly) proves that the hull of a cut-and-project tiling is a fiber bundle over a torus. ...
- 243
2
votes
1
answer
129
views
Reference on relation between SFTs and Wang-tiles
I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a ...
- 633
1
vote
2
answers
103
views
A variation of domino tiling problem with fusions
I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to ...
- 633
1
vote
2
answers
232
views
What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 13.6k
0
votes
0
answers
140
views
Graph theory: Closed neighourhoods and generalized clustering coefficients
The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$.
The number of edges between neighbours divided by the number of pairs of neighbours is ...
- 9,720