# Questions tagged [tiling]

The tiling tag has no usage guidance.

166
questions

**3**

votes

**1**answer

39 views

### Tilings of lattice polytopes by transformations of lattice polytopes

A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...

**1**

vote

**1**answer

59 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 [......

**4**

votes

**2**answers

260 views

### Domino tiling obtained from space-filling curves, is possible to predict basic properties?

Periodic and aperiodic domino tiling systems can be obtained by the following construction rules:
Draw a regular square grid n×n of n2 cells.
Select a space-filling curve that is consistent with ...

**1**

vote

**1**answer

78 views

### Is there a rectangular tiling based on the Padovan sequence? [closed]

I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. ...

**7**

votes

**1**answer

233 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 ...

**1**

vote

**1**answer

82 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\...

**5**

votes

**3**answers

295 views

### Can local flip moves connect dimer matchings on 'quadrangulated' planar bipartite graphs? (perfect matching reconfiguration problem)

I'm interested in the structure of dimer matchings on planar graphs with a bipartite structure. In particular, I'm interested in whether any two perfect matchings can be connected, i.e. transformed ...

**1**

vote

**0**answers

77 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 $n=1$, ...

**2**

votes

**1**answer

203 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 ...

**2**

votes

**0**answers

131 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 ...

**3**

votes

**1**answer

82 views

### Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...

**7**

votes

**1**answer

210 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 ...

**3**

votes

**0**answers

105 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 ...

**12**

votes

**2**answers

273 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 ...

**2**

votes

**0**answers

72 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 ...

**3**

votes

**0**answers

41 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 (...

**5**

votes

**0**answers

146 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?

**11**

votes

**1**answer

355 views

### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...

**4**

votes

**1**answer

82 views

### Tiling the surface of a hypersphere with regular simplices

Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...

**5**

votes

**0**answers

85 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, ...

**4**

votes

**0**answers

130 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 ...

**5**

votes

**0**answers

125 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 $...

**6**

votes

**3**answers

744 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?

**4**

votes

**0**answers

134 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

**0**answers

126 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 ...

**5**

votes

**0**answers

94 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 ...

**9**

votes

**1**answer

298 views

### Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...

**3**

votes

**0**answers

84 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 ...

**10**

votes

**1**answer

311 views

### Is there a triangle which makes dense set of angles by drawing medians?

This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...

**1**

vote

**0**answers

74 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 ...

**4**

votes

**0**answers

155 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 $(...

**1**

vote

**0**answers

185 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?

**1**

vote

**0**answers

90 views

### Minimal period for a bounded Langton's ant moving on a tessellation

We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...

**1**

vote

**1**answer

120 views

### Order question about pentagonal tiling type 9 and type 10

People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see Pentagonal tiling.
These 15 types of pentagonal was named by finding date except type 9 and type ...

**5**

votes

**2**answers

318 views

### Smallest tile to *isohedrally* tessellate the hyperbolic plane

Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...

**19**

votes

**2**answers

2k views

### Smallest tile to tessellate the hyperbolic plane

Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...

**3**

votes

**1**answer

135 views

### Aperiodic tiling of compact space by small number of basic tiles

Suppose we have compact space, like sphere or torus in particular dimension $d$.
Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just ...

**3**

votes

**0**answers

92 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 ...

**1**

vote

**1**answer

108 views

### Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?

My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...

**9**

votes

**1**answer

257 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 ...

**9**

votes

**1**answer

206 views

### Thinnest covering of the plane by regular pentagons

Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...

**5**

votes

**2**answers

501 views

### What rectangles can a set of rectangles tile?

(I asked this question first on math.stackexchange, but did not get any responses so I thought I would try here.)
If we have a set of $p_i \times q_i$ rectangles ($p_i, q_i \in \mathbf{N}$), which $m ...

**3**

votes

**1**answer

228 views

### Are there unique additive decompositions of the reals?

Given $b\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+bU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?
Here is ...

**2**

votes

**0**answers

80 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 ...

**15**

votes

**2**answers

1k views

### ♢ ⧫ ⬠: 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 ...

**6**

votes

**3**answers

645 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 ...

**7**

votes

**1**answer

197 views

### Covering the plane with line segments with local hexagonal constraints

Can we characterize the following kinds of plane coverings? (Open-ended, but provide some description more "useful" than the constraints given.) For a more answerable question, is there an effective ...

**4**

votes

**1**answer

160 views

### Game theoretic aspects of Wang tiles?

Wang tiles are interesting in that they can simulate Turing machines. My question is whether anyone has studied their game theoretic properties?
In particular, we could imagine a game in which you ...

**2**

votes

**2**answers

161 views

### Does one pieces of every kind of connected polyominoes P in $\mathbb{R}^2$ which has no hole cover a plane?

Or polyominoes with no hollow in $\mathbb{R}^3$?
I created this conjecture and tried to make counterexample, but it doesn't work well. Thank you for any answer or correcting question.

**18**

votes

**1**answer

533 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 ...