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**5**

votes

**1**answer

240 views

### Generating function for number of different tessellation checkered rectangle

Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$.
Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$.
$\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ ...

**6**

votes

**1**answer

127 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 ...

**6**

votes

**0**answers

95 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:
...

**19**

votes

**1**answer

584 views

### 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 ...

**29**

votes

**5**answers

1k views

### Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
...

**14**

votes

**3**answers

275 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 ...

**9**

votes

**4**answers

554 views

### Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...

**9**

votes

**1**answer

247 views

### What can we learn from the newly discovered monohedral convex pentagonal tiling?

Wikipedia: https://en.wikipedia.org/wiki/Pentagonal_tiling#Stein_.281985.29_and_Mann.2FMcLoud.2FVon_Derau_.282015.29
Media coverage: ...

**16**

votes

**3**answers

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 ...

**2**

votes

**0**answers

124 views

### Lights Out game over GF(p)

On Jaap's Puzzle Page
http:// www.jaapsch.net/puzzles/lomath.htm#domtilings
Theorem 7 says:
If standard Lights Out is played on a m x n grid-like board, ...

**5**

votes

**1**answer

93 views

### Penrose tiling substitution is bijective

Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles.
It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...

**59**

votes

**6**answers

3k 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 ...

**9**

votes

**3**answers

11k 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.

**5**

votes

**0**answers

105 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 ...

**15**

votes

**2**answers

410 views

### 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 ...

**3**

votes

**3**answers

352 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 ...

**14**

votes

**1**answer

997 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 ...

**5**

votes

**2**answers

164 views

### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...

**4**

votes

**1**answer

187 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 ...

**10**

votes

**1**answer

324 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 ...

**6**

votes

**1**answer

136 views

### Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...

**16**

votes

**2**answers

1k views

### “a shape that … lies halfway between a square and a circle”

An article in the
Notices of the AMS, Volume 61, Issue 10, 2014
(PDF download link),
on Khot's Unique Games Conjecture, says this:
Another group ... found a
shape that in a certain sense lies ...

**4**

votes

**3**answers

614 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.

**4**

votes

**0**answers

123 views

### Optimal planar net for catching convex shapes

Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...

**6**

votes

**1**answer

242 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 ...

**4**

votes

**2**answers

147 views

### (non-)existence of the aperiodic monotile

The aperiodic monotile problem asks whether there exists a single tile that every tiling made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? ...

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votes

**4**answers

451 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 ...

**16**

votes

**1**answer

295 views

### Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., ...

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votes

**1**answer

100 views

### What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]

I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...

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votes

**0**answers

94 views

### Mixing time for dimers on the square-octagon graph

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...

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votes

**1**answer

740 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 \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...

**5**

votes

**1**answer

146 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 ...

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votes

**0**answers

714 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 ...

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votes

**3**answers

809 views

### 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, ...

**22**

votes

**3**answers

2k views

### Can a unit square be cut into rectangles that tile a rectangle with irrational sides?

For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...

**5**

votes

**2**answers

210 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 ...

**9**

votes

**1**answer

425 views

### Are there irregular tilings by L-polyominoes?

I wonder if one can tile the plane with an order-$n$ L-polyomino
in a fundamentally irregular manner.
I seek help in defining what should constitute "irregular."
An L-polyomino of order $n \ge 2$ is ...

**3**

votes

**1**answer

121 views

### Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space?

In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose ...

**8**

votes

**1**answer

141 views

### Exact enumerations from two-dimensional stat mech models

Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge ...

**4**

votes

**0**answers

322 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:
...

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votes

**1**answer

252 views

### Random walk on a Penrose tiling

Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the
origin, or, equivalently, returns to the origin infinitely often.
It was subsequently established that in $\mathbb{Z}^3$, ...

**9**

votes

**3**answers

401 views

### Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...

**1**

vote

**0**answers

109 views

### Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function ...

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votes

**4**answers

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 ...

**29**

votes

**1**answer

631 views

### Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...

**4**

votes

**1**answer

228 views

### Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...

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vote

**0**answers

78 views

### Periodic Tiling of Wang tile [duplicate]

Without considering any aperiodic tiling, is there any established (or better efficient) algorithms that tries to determine whether a set of tile can tile Wang tile periodically (or better determine ...

**10**

votes

**1**answer

311 views

### A question about tiling Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...

**3**

votes

**1**answer

131 views

### Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...

**0**

votes

**1**answer

160 views

### Can aperiodic tilings be non-hierarchical? and confusion over domino problem

Anyone experienced with the undecidability of aperiodic tiling?
It's related to the halting problem which Turing proved was undecidable in the 30's and basically superimposes tiles onto other tiles ...