# Questions tagged [tiling]

The tiling tag has no usage guidance.

56
questions with no upvoted or accepted answers

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### 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 ...

**16**

votes

**1**answer

669 views

### 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 ...

**14**

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297 views

### Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...

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500 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 ...

**13**

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**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?

**8**

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129 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:
...

**8**

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**0**answers

226 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 ...

**6**

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**0**answers

167 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 ...

**5**

votes

**0**answers

90 views

### If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula ...

**5**

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151 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?

**5**

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96 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, ...

**5**

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130 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 $...

**5**

votes

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102 views

### What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?

Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...

**5**

votes

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103 views

### Minimal covers instead of tilings in Maxwell Allman's problem

The question I'm going to ask is inspired by this thread. I wonder what happens if instead of tilings we consider minimal covers, i.e., families of convex closed polygons that cover the square and ...

**5**

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140 views

### 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 ...

**5**

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**0**answers

124 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 ...

**4**

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**0**answers

135 views

### Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!

Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...

**4**

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74 views

### Possible cardinalities of spherical tiling

Suppose that we have a tiling of $n$-dimensional (I want to get answer for $n = 4$, but general result would be nice!) sphere by isometric tiles strictly contained inside the right-angled simplex. ...

**4**

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**0**answers

59 views

### Unbalanced colourings of Penrose tiles

It is known that both the rhombus and kite-and-dart Penrose tilings are three-colourable, from
Babilon, Robert. "3-colourability of Penrose kite-and-dart tilings." Discrete Mathematics 235, no. 1-3 (...

**4**

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142 views

### Hooks, monomers, dimers and Young diagrams: Part I

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it.
Consider the one-line partition $\lambda_n=(n)$ and its ...

**4**

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**0**answers

140 views

### Tiling squares with oblongs

An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so ...

**4**

votes

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150 views

### Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it.
The following picture illustrates the request:
In this example, the starting tile is a "thin rhombus" (the pink ...

**4**

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**0**answers

157 views

### Tileability and computabilty

Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(...

**4**

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137 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 ...

**4**

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113 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 ...

**3**

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57 views

### Chromatic number of rectangle tilings

Suppose we have a region of the plane tiled by finitely many
rectangles. We want to color the rectangles so that two
rectangles have different colors if they share a part of an
edge or if they share ...

**3**

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**0**answers

73 views

### Distance spectra of uniform tilings

Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that ...

**3**

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120 views

### Aperiodic tile with rational area

Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...

**3**

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92 views

### chromatic number of plane using Cairo pentagonal tiling

Scale the Cairo pentagonal tiling so the short side is of length 1. Then it is easy to colour the tiling with 8 colours, two parallel ribbons of four colours each, to establish that the chromatic ...

**3**

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100 views

### How many positions of a tile can occur in a periodic tiling?

In my recent question about polygonal tilings where tiles can occur in infinitely many positions, both constructions given as solutions are of self-similar nature. This means in particular that there ...

**3**

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106 views

### Tilings of the plane and meromorphic functions on the plane

This question has three up-votes on m.s.e. but isn't getting any answers.
Every textbook says every doubly-periodic meromorphic function on $\mathbb C$ has a fundamental domain that is a ...

**3**

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405 views

### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...

**3**

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134 views

### Is there a Fourier transform for principal r-discrete groupoids?

I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...

**2**

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97 views

### Cutting polygons into mutually similar and non-congruent pieces

It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) - for example, see https://en.wikipedia.org/wiki/...

**2**

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137 views

### Tradeoffs in translation-invariant tilings of $\mathbb{R}^3$

Suppose I tile $\mathbb{R}^3$ in a ($\mathbb{Z}^3$-)translation-invariant manner. If we insist on the tiling being regular, then we are left with only the cubic tiling. However, suppose that we ...

**2**

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108 views

### Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...

**2**

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83 views

### Tiling of polygons in $\mathbb{R}^2$ by squares

Let $X\subset \mathbb{R}^2$ be a polygon (possibly nonconvex, but not intersecting itself) with all the sides parallel to one of the axes.
I am interested on whether $X$ can be tiled by (finitely ...

**2**

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**0**answers

57 views

### Number of labelings of symmetric hexagonal tilings P(a,b,c) with j descents

I am searching for the Number of labelings of symmetric hexagonal tilings
If the hexagon is of the form P(n,n,n) then the coefficients can be found here
A217311
I am looking for the coefficients of ...

**2**

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**0**answers

136 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 $\mathcal{H}...

**2**

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49 views

### What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?

I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of Weaire–...

**2**

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209 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 $l_1,\dots,l_k$?...

**2**

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168 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 ...

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159 views

### How to fill a rectangle with smaller rectangles of given sizes?

I have a problem. I try to find an algorithm to fill up a given rectangle with smaller ones. Something like in this picture:
I know the size of the big rectangle, the size of all the little ...

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vote

**0**answers

52 views

### Rigid monohedral tilers

Say that a tile $T$ that alone can tile the plane—a monohedral tile—is rigid
if it is not the case that $T$ can be slightly deformed to $T'$ so that:
$T'$ can also tile the plane
$T'$ is arbitrarily ...

**1**

vote

**0**answers

220 views

### Which polygons tessellate the hyperbolic plane?

The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
It is well known that in Euclidean geometry, all triangles and all ...

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41 views

### How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?

When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...

**1**

vote

**1**answer

251 views

### Looking for an efficient way of maximising 'pair scores' for subset of 30 selected from 50 to 10 000 objects

Context: I have a tiling program that uses a directed breadth first search algorithm. It is 'directed' by what I call 'pair scoring'. There are $N$ polyforms (pieces) used in the tiling. I have an $N\...

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239 views

### Generalizations of classical tiling problem

A classic problem using an inductive construction is to show that the $2^n \times 2^n$-square, with a missing corner, can be tiled with L-triominoes.
The proof goes like this:
It is true for $n=1$, ...

**1**

vote

**0**answers

77 views

### dividing a square into unique rectangles with the same perimeter

There's a solution for dividing a square into unique rectangles with the same area which is the blanche dissection.
There's also a solution for dividing a square into unique rectangles with the same ...

**1**

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190 views

### Squares as sum of squares

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?