# Questions tagged [tiling]

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### Polyomino that can cover an arbitrarily large square but not the entire plane

https://userpages.monmouth.com/~colonel/nrectcover/index.html For a polyomino with no holes that cannot tile the plane, we may ask what are the maximal rectangles and infinite strips that it can ...
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### Tiling space with supertile of hypercube unfoldings

Two students in my class asked and answered what might be a novel question. It is well known that the cube has exactly $11$ edge-unfoldings (or "nets"), as shown below:         (Image from ...
86 views

### Automorphism group of a normal tiling of the plane

A normal tiling of the plane is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a connected set or the empty set, and all tiles are ...
272 views

### Family of shapes that can be tiled into one another

Okay, I'm trying to ask a question which hasn't been asked before, it may be futile, but let's see. So let's take a square, this will be our shape A. We can tile a 2x1 rectangle by using shapes ...
1 vote
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### Which pentagon gives least packing density?

We extend Which convex pentagon gives least packing density? by going from convex pentagons to general ones. Question: Which pentagon gives the least packing density on the Euclidean plane? Note: All ...
109 views

### On sets of rectangles that can all together form at least one big rectangle

Let us say a set of $n$ rectangles is rectifiable if all $n$ rectangles together form a big rectangle without gaps or overlaps. Question: How hard computationally is the question of deciding whether a ...
30 views

### Thinnest 3-fold and n-fold coverings of the plane by congruent convex shapes

This post is totally based on Thinnest 2-fold coverings of the plane by congruent convex shapes - indeed, am only recording a couple of very natural further questions here rather than add them as ...
83 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 ...
72 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 ...
67 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
148 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 ...
1 vote
380 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 ...
140 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 ...
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### 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 ...
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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, ...
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### 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
234 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 ...
255 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 ...
223 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 ...
135 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 ...
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### 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 ...
222 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 ...
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### 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/...
945 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 ...
68 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 ...
164 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 ...
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### 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 ...
305 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 ...
155 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 ...
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### 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
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### 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 ...
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### 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. ...
302 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 ...
143 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 ...
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### 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 ...
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### 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 ...
240 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 ...
167 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 ...
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### 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 ...
76 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 ...
211 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 ...
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### 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? ...
159 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 ...
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### 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
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### 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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### 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
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### 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. ...
708 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
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\...