All Questions
Tagged with co.combinatorics tiling
94 questions
95
votes
5
answers
4k
views
Can a row of five equilateral triangles tile a big equilateral triangle?
Can rotations and translations of this shape
perfectly tile some equilateral triangle?
I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
- 3,137
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
42
votes
2
answers
2k
views
Does any set of dominoes tile some common figure?
Let $D_1,\dots,D_n \subset \mathbb{Z}^2$ be two-point sets, i.e. 'dominoes' (unlike common dominoes, these are not necessarily connected, but I couldn't come up with a better name).
Does there always ...
- 1,209
33
votes
1
answer
7k
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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
25
votes
1
answer
2k
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Polyomino that can tile itself
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ ...
- 1,462
24
votes
1
answer
3k
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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
23
votes
1
answer
1k
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Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$
When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...
- 508
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 ...
21
votes
1
answer
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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 ...
21
votes
1
answer
1k
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Tiling rectangle with trominoes — an invariant
There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes.
EDIT: we do not admit ALL ...
- 15.8k
20
votes
4
answers
2k
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Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, ...
- 24.2k
20
votes
2
answers
741
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Can every tromino (including those with gaps) tile the plane?
I've generalized trominos to include "gaps", i.e. they are formed by removing all but $3$ squares from an $n$-omino where $n$ is finite.
The generalized trominos pictured above can tile the plane ...
- 440
19
votes
1
answer
616
views
How hard is it to tell when a finite set tiles the integers?
Given a nonempty set $B$ of integers between 1 and $n$, we wish to determine whether or not $\mathbb{Z}$ can be tiled with translates of $B$ (that is, covered by disjoint translates of $B$). I know an ...
- 19.7k
18
votes
2
answers
2k
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♢ ⧫ ⬠: the fourth kind of Penrose tiling?
It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There ...
- 437
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votes
2
answers
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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
15
votes
4
answers
887
views
Tiling a rectangle with all simply connected polyominoes of fixed size
For which values of $n$ can we tile some rectangle with one copy of each free simply-connected $n$-omino (that is, each polyomino with $n$ squares that has no holes)?
It appears that it is possible ...
- 421
15
votes
3
answers
384
views
Bicoloring of $\mathbb{N}^2$, avoiding set of patterns, is the maximal limit density rational?
Consider a bi-coloring of $\mathbb{N}^2$, (black and white), where we wish to maximize the limit (limsup) of the density of black squares in $[n] \times [n]$ as $n \to \infty$. Here, we identify each ...
- 15.8k
14
votes
3
answers
1k
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What exact number of domino tilings cannot be realizable?
Inspired by some other questions, (this and this),
I wonder what numbers $n$ there are that satisfy
$$p(n)=\text{there is no region that admits exactly } n \text{ domino tilings}.$$
If this is true, $...
- 15.8k
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 ...
- 85.6k
14
votes
1
answer
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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
14
votes
0
answers
4k
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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
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, ...
- 85.6k
12
votes
2
answers
454
views
Random Walk on Pentagonal Tiling
I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
12
votes
0
answers
168
views
Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
- 1,209
11
votes
1
answer
807
views
Soft question: mathematics about truchet tiles
It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...
- 191
10
votes
1
answer
401
views
How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
- 13.4k
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
9
votes
0
answers
292
views
Tilings in finite (not necessarily Abelian) groups
Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that
$$ G = \bigsqcup_{b\in B} bA.$$
...
- 1,354
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 ...
- 85.6k
8
votes
1
answer
248
views
For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?
I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture).
I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...
- 4,014
8
votes
0
answers
139
views
Inequality among domino tilings of large triomino shapes
Inspired by this question, which asks for what shape maximizes the number of domino tilings, I want to ask the following seemingly simpler question, which I have been thinking about for a while:
...
- 15.8k
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
7
votes
3
answers
981
views
Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N
For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
- 3,707
7
votes
3
answers
2k
views
Partitioning a rectangle into different isosceles triangles
After all the discussion raised by this old question, I am wondering about a somewhat complementary one:
For any given rectangle, does there exist a finite set of pairwise different isosceles ...
- 13.4k
7
votes
1
answer
354
views
Changing tiles by swapping squares
In an $n\times n$ table, initially there is a $1\times n$ tile in each row. A swap is an operation that involves choosing two tiles, move one square from the first to the second tile, and ...
- 1,209
6
votes
5
answers
542
views
Tiling with ten-fold symmetry and (unoriented) Penrose tiles?
Consider tilings of the plane made out of rhombi of side 1 and either angles $\pi/10$ and $2\pi/5$ or angles $\pi/5$ and $3\pi/10$. If we give a certain orientation to the edges and respect that ...
- 20.2k
6
votes
2
answers
148
views
Decidability of (restricted) periodicity of Wang tilings
Consider a Wang tiling (given a subset of $C^4$ for a finite set $C$ of colours, e.g.). It's well-known to be undecidable whether there exists a tiling, and also whether there exists a periodic tiling....
- 2,519
6
votes
0
answers
657
views
Unique domino tiling
Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset $S$ of the $xy$-plane is star-convex if there ...
- 1,090
5
votes
3
answers
748
views
Aperiodic graphs
The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph $G$...
- 329
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votes
1
answer
213
views
Aperiodic set of corner Wang Tile [closed]
There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...
- 867
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2
answers
382
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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
1
answer
397
views
How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 20.2k
5
votes
0
answers
157
views
Tiling rectangles using all squares of sides 1, 2, 3, ..., n
Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle?
- 3,707
5
votes
0
answers
105
views
Hooks, monomers, dimers and Young diagrams: Part II
As promised, I've upgraded my last question.
Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...
- 43.2k
5
votes
0
answers
150
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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 $...
- 43.2k
5
votes
0
answers
145
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Complexity of $\mathbb{Z}^n$ tilings
Let $\mathcal{T} \subset \mathbb{Z}^n$ be a finite set. Let $\Lambda \subset \mathbb{Z}^n$ be a full rank lattice.
We say that $\mathcal{T}$ is a $\Lambda$-tile for $\mathbb{Z}^n$ if the following ...
- 800
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votes
0
answers
131
views
For which sidelengths are there polyominos composed of three squares that tile the plane?
Given three naturals $a<b<c$. We consider polyominos, connected or not, which are composed of three squares of sides $a,b,c$.
How can one characterize all triples $a,b,c$ for which such a ...
- 13.4k
4
votes
1
answer
340
views
Triangling the triangle
Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case ...
- 613
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
4
votes
0
answers
138
views
Hyponontiling Wang tiles
Call a finite collection of tiles that can tile the plane if we have to use each tile at least once tiling.
Is there a collection of at least 3 tiles that is not tiling, but such that after removing ...
- 18.7k