# Questions tagged [tiling]

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202
questions

**3**

votes

**1**answer

70 views

### Is there a formula for a number of one-sided N-ominoes?

As we all know, Polyominoes are shapes which consist of certain number of squares connected together. A famous videogame - Tetris - has a gameplay based around tetraminoes - polyominoes with 4 squares ...

**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 ...

**3**

votes

**0**answers

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 ...

**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\...

**2**

votes

**1**answer

63 views

### What is the average component size of a coloring?

Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...

**1**

vote

**1**answer

137 views

### Tiling a square with similar non-congruent rectangles. What is the aspect ratio of the rectangles as n grows large?

I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a ...

**3**

votes

**2**answers

228 views

### For what n and t can a square be partitioned into n similar rectangles in t congruence classes?

It is known that a square can be partitioned into three similar rectangles, all mutually non-congruent. I don't think it's possible with four. With what numbers of rectangles can this be achieved? And ...

**1**

vote

**0**answers

161 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 ...

**4**

votes

**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 ...

**2**

votes

**1**answer

108 views

### Packing densities of non-centrally symmetric planar convex regions

Reference: https://en.wikipedia.org/wiki/Smoothed_octagon
Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...

**0**

votes

**1**answer

133 views

### maximum number of colors for an optimal tiling which guaranties infinite paths

This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...

**5**

votes

**1**answer

793 views

### How to characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk from a purely analytic point of view?

I wonder how can one describe the generators of the triangle group for the tesselation of Poincare unit disk by triangles with angles $\pi/4, \pi/4 , \pi/4 $ in terms of Mobius transformations. I also ...

**20**

votes

**4**answers

1k views

### 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, ...

**1**

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 ...

**2**

votes

**0**answers

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/...

**3**

votes

**1**answer

126 views

### Construct by compactness (Pentagonal tiling – Rao paper)

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that:
"... We keep a connected component $H_d'$ of $H_{d}$ such ...

**7**

votes

**1**answer

196 views

### Are there any convex pentagonal rep-tiles?

A rep-tile is a shape that can tile larger copies of the same shape.
Question 1: Are there any convex pentagons that are also rep-tiles?
Remarks: 15 convex pentagonal tiles of the plane are known and ...

**5**

votes

**2**answers

207 views

### Distribution over Penrose Tilings?

The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow ...

**25**

votes

**1**answer

1k views

### Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?

A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...

**2**

votes

**1**answer

199 views

### Objects in bijection with integer partitions (and lattices)

A partition of $n$ is a non-increasing sequence of positive integers of sum $n$. Several lattices are defined over integer partitions, in particular the dominance order and the Young lattice.
Several ...

**26**

votes

**3**answers

10k 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 ...

**20**

votes

**0**answers

286 views

### 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 ...

**29**

votes

**5**answers

927 views

### Fair cutting of the plane with lines

An infinite countable family $\cal{L}$ of lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:
$\bullet$ No circle intersects infinitely many ...

**19**

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 ...

**0**

votes

**0**answers

59 views

### On Covering a Planar Region with Copies of a Tile of Different Shape

Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...

**5**

votes

**1**answer

146 views

### Random domino tilings: Is this distribution well-defined, and how can it be sampled from?

I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means.
My first instinct was to do ...

**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 ...

**15**

votes

**3**answers

726 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 ...

**1**

vote

**0**answers

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 ...

**74**

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 ...

**14**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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**

votes

**1**answer

141 views

### Absolute and relative tilings of the hyperbolic plane

In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it).
The ...

**12**

votes

**1**answer

771 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 \...

**0**

votes

**0**answers

118 views

### Graph theory: Closed neighourhoods and generalized clustering coefficients

The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$.
The number of edges between neighbours divided by the number of pairs of neighbours is ...

**0**

votes

**0**answers

28 views

### Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...

**5**

votes

**1**answer

163 views

### A claim on partitioning a convex planar region into congruent pieces

Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...

**3**

votes

**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 ...

**4**

votes

**1**answer

202 views

### Structures for random graphs with structure

Background[You may skip this and go immediately to the Definitions.]
Crucial features of a (random) graph or network are:
the degree distribution $p(d)$ (exponential, Poisson, or power law)
the mean ...

**7**

votes

**6**answers

466 views

### Tiling with similar tiles

Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...

**3**

votes

**1**answer

155 views

### Monotile that tiles when you apply a rubber band

My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...

**3**

votes

**1**answer

111 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 ...

**89**

votes

**5**answers

3k 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**

votes

**1**answer

57 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

89 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 [......

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vote

**1**answer

194 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 ...

**4**

votes

**2**answers

378 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 ...

**8**

votes

**2**answers

683 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 (...

**1**

vote

**1**answer

96 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. ...