Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
2
votes
0
answers
222
views
Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size
Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares
of length $l_1,\dots,l_k$?...
- 17.9k
6
votes
3
answers
1k
views
Consecutive Integer Squared Square
Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.
- 61
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
10
votes
2
answers
1k
views
Decidability of periodic tilings of the plane
I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (...
- 2,519
2
votes
1
answer
1k
views
Minimizing the Perimeter of a polyomino
Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...
- 23
1
vote
1
answer
392
views
Frustrating the number of possible common edges between two connected components composed of square Penrose tiles
Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...
- 13
3
votes
1
answer
520
views
tiling a rectangle with squares: how unique are the minimal solutions?
This is a follow-up of my recent thread about tiling a $m\times n$ rectangle with squares:
I'm wondering to what extent a minimal tiling is essentially unique, that is, up to reflections of the whole ...
- 13.4k
33
votes
1
answer
7k
views
tiling a rectangle with the smallest number of squares
This is based on another thread. For $m,n\in \mathbb N$, let $f(m,n)$ be the minimum number of squares with integer sides needed to tile a $m\times n$ rectangle. Recently, a table of values for $n\le ...
- 13.4k
2
votes
1
answer
445
views
What are some properties of Delone sets that come from Barlow packings of spheres?
Given a Barlow packing of $\mathbb{R}^n$ by balls with at most a finite number of different radii, the centers of the balls will form a Delone set in $\mathbb{R}^n.$
For a highest density sphere ...
- 1,010
2
votes
1
answer
252
views
Cut and Project Sets Using Hyperbolic Space
One strategy for creating aperiodic sets in $\mathbf{R}$ is to take a line $L$ of irrational slope in $\mathbf{R}^2$ along with a compact window $W \subset \mathbf{R}$ which is thought of as a subset ...
- 1,010
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
4
votes
1
answer
878
views
"Aztec Diamond" analogue for Square-Octagon graph
I have been reading David Speyer's Perfect Matchings and the Octahedron Recurrence, trying to carry out his "cross-wrenches" generalization of the Aztec diamond. In what follows, I'm asking for a ...
- 22.8k
1
vote
1
answer
202
views
Overlaying two domino-like constructions such that all individual pairs of domino-like cells in the overlay have matching symbols
Imagine I have two $n$ x $m$ assemblies of $P = (n*m)$ unit square cells on the plane, $(c_{(a,1)}, ..., c_{(a,P)}) \in A$ and $(c_{(b,1)}, ..., c_{(b,P)}) \in B$, where every cell, $c_k$, in a ...
- 13
8
votes
3
answers
3k
views
What are Penrose Tilings, and how do they relate to Quasicrystals?
The question is in the title, but let me elaborate a little.
Background
Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...
- 15.5k
21
votes
1
answer
1k
views
Monomer-Dimer tatami tilings need better relationships with other math. Summary of results
A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the tatami condition if no four tiles meet at any point. (Or you can think of it as the removal of a matching from ...
8
votes
1
answer
403
views
one-dimensional (sort of) tilings
Consider the following one-dimensional tiling problem. Each "tile" is a sequence of nonnegative integers. A "region" is also such a sequence. I can shift the "tiles", or reverse them. A tiling is ...
- 423
8
votes
0
answers
239
views
Possible structures for minimal tiling sets
Inspired by Col. Sicherman's results here, my speculations have so far outrun my expertise that I thought I might pass my question along to others who might find it equally intriguing, but perhaps ...
- 475
18
votes
7
answers
4k
views
Mathematics of quasicrystals
I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.
- 1,099
9
votes
2
answers
1k
views
Fractal Tiling of Rhombic Dodecahedra
Hello, this is my first question on Math Overflow...
Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres.
Consider a "nucleus" ...
user24228
4
votes
1
answer
315
views
Representing groups with two generators as graph automorphisms
Suppose we have a group $G$ which can be generated by two elements $x$, $y$. Call $H$, $K$, $L$ the subgroups of $G$ generated by $x$, $y$ and $y^{-1}x^{-1}$, respectively.
With these data, we can ...
- 2,527
12
votes
2
answers
664
views
Detecting tilings by toric geometry
This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
- 85.5k
8
votes
1
answer
394
views
Does every polycube tiling imply a regular polycube tiling?
Let's define d-polycubes to be a union of unit hypercubes from the $\mathbb Z^d$ tiling of d-dimensional Euclidean space which has connected interior. Given a tiling of $\mathbb R^d$ by identical ...
- 85.5k
1
vote
1
answer
2k
views
Tetromino tiling.
There is a rectangle grid given with some of the tiles already filled with tetrominoes. I want to find out the minimum number of tetrominoes required to fill the remaining tiles such the filling is ...
- 111
14
votes
1
answer
1k
views
slick-proof-of-trick-for-counting-domino-tilings
The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
- 19.7k
4
votes
0
answers
117
views
symmetric difference of temperate zone and inscribed disk
For random domino tilings of the Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference ...
- 19.7k
9
votes
1
answer
394
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 ...
- 19.7k
3
votes
1
answer
373
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 ...
- 93
12
votes
1
answer
872
views
Tiling by regular simplices
The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional ...
- 123
14
votes
1
answer
543
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 ...
- 85.5k
21
votes
4
answers
2k
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 ...
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
5
votes
1
answer
394
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 ...
- 563
2
votes
0
answers
169
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 ...
- 83
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
17
votes
3
answers
2k
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 ...
- 453
79
votes
6
answers
4k
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 ...
- 82.6k
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
- 4,922
14
votes
0
answers
4k
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?
- 1,015
4
votes
1
answer
461
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 ...
- 1,795
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
4
votes
1
answer
1k
views
What Islamic tiling patterns are constructible?
Eric Broug in his book Islamic Geometric Patterns gives
straightedge and compass construction of some simpler patterns.
It is clear his techniques will provide constructions for many
Islamic patterns.
...
24
votes
1
answer
3k
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 ...
- 32.4k
2
votes
1
answer
341
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 '...
- 975
19
votes
5
answers
21k
views
Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
- 383
11
votes
4
answers
608
views
What is the right way to think about / represent general tilings?
For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
