Questions tagged [tiling]

Use this tag for questions about (not necessarily periodic) tilings of metric spaces, their combinatorial, topological and dynamical properties, as well as basic definitions and concepts.

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Transform point in the Poincaré disc to point in tile

Given a tiling of the hyperbolic plane with a finite set of tiles, how can I transform a point in the Poincaré disc model to the tile that occupies that point and get the corresponding point in that ...
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1answer
56 views

Can you tile a heart with dominoes?

For a positive integer $n$, let $R_n$ be the set of integer lattice points $(x, y)$ such that $0 \leq x < 2n$ $0 \leq y < 4n$ $x \leq y$ $y \leq 5n - x$ $y \leq x + 3n$, and let $L_n = \{(-x, ...
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Can similar convex heptagons tile the plane?

It is a fairly straightforward matter to apply Euler's formula $V-E+F=2$ for planar graphs to see that congruent convex heptagons cannot tile the plane. The graph associated to the heptagons within ...
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1answer
19 views

Assigning Tiles on a Hyperbolic Grid with Unique Coordinates

I am creating a location in an RPG campaign that deals with non-euclidean space, and I'm currently toying with the idea of a forest with a finite border that takes up infinite space. The idea is that ...
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Smallest number of $45^\circ-60^\circ-75^\circ$ triangles that tile a rectangle

In this wonderful question we learned that a square can be divided into forty six $45^\circ-60^\circ-75^\circ$ triangles. Now I am wondering what is the smallest number of $45^\circ-60^\circ-75^\circ$ ...
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1answer
86 views

Tilings of a 5×5 square with smaller squares.

There is a $5 × 5$ array of lights, such that at each step, we may toggle all the lights in any $2 × 2, 3 × 3, 4 × 4$ or $5 × 5$ sub-square. Initially all the lights are switched off. After a certain ...
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Covering a square with crosses

I am trying to find the smallest number of "crosses" needed to cover an n by n square with overlap. A "cross" is basically the "X" pentomino, the following figure: The ...
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106 views

Filling a Board with Tiles that can be overlapped

Let's say I want to fill an $n \times n$ board with tiles that have the shape of a $3 \times 3$ square with the $4$ corners cut out (the tile makes a plus sign) such that the sides of each tile are ...
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Why are there exactly eight convex polygons that can appear in a Penrose tiling?

Problem Suppose that we have a P$3$ Penrose tiling, using rhombi of smallest angles $36^\circ$ and $72^\circ$:                                              Which convex shapes can be found as a union ...
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3answers
109 views

How to prove that a hexagon is the regular polygon with the most sides that can tile a plane [duplicate]

I need to know the answer to this question to find out why bees use hexagonal cells in hives. I know that a circle takes up the most area using the least perimeter, so bees would try to make shapes as ...
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Tiling problem : tiling square allowing space.

Given we have a 4 by 4 square and we place 2 by 1 tiles until we can not place it anymore without overlapping. For example, if $1$ denotes the tile and $0$ means the empty space, the tiling below is ...
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If a subset of the square grid can be tiled by $1\times n$ rectangles for every $n$, can it be tiled by infinite rays?

Suppose that we have some set $S$ of grid-aligned squares in the plane; equivalently, we can think of our set as $S\subset \mathbb{Z}^2$. Suppose that for every positive integer $n$, $S$ can be tiled ...
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Construct a set of local rules in plane such that all permitted tilings are non periodic.(there exists at least one tiling)

This is the problem given to me on a practice. I imagine the problem is understandable but if there is anything unexplained, please tell me. I have tried various things to solve this problem but all ...
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How many “prime” rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, ...
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1answer
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How many combinatorially distinct ways are there to tile an equilateral triangle with $k$ $60^\circ-120^\circ$ trapezoids?

I believe there is exactly one way (up to combinatorial equivalence) to arrange 3 trapezoids with angles of $60^\circ$ and $120^\circ$ into an equilateral triangle: With $4$ trapezoids, I see two ...
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Can a monohedral tiling have more than $21$ corner-adjacencies at every tile?

Suppose we tile the plane with a simply-connected tile $T$, and we wish to ensure that every tile touches at least $k$ other tiles. If we restrict "touching" to require sharing a positive-...
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If an animal tiles the plane via translation, can it do so in a lattice configuration?

It is known that if a polyomino tiles the plane using only translated copies, then it has at least one such tiling where the centroids of each tile form a lattice; see for instance the paper Arbitrary ...
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1answer
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Tiling $3k\times (3k-2)$ board with $L$- Trominoes

Consider a $3k\times (3k-2)$ board. For which values of $k$ can we cover the board with $L$- trominoes? For this, It was clear that for $k$ even, we are done, because then we can divide the board ...
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How few $(42^\circ,60^\circ,78^\circ)$ triangles can a regular hexagon be divided into?

While working on an answer to this excellent post by Edward H. on dissections of an equilateral triangle into similar triangles with angles of $42^\circ, 60^\circ,$ and $78^\circ$, I wondered about a ...
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1answer
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Given dihedral angles, find a set of edges

In the paper Space Vectors Forming Rational Angles a special set of tetrahedra is mentioned. "The remaining three are in the R-orbit of the tetrahedron with dihedral angles (π/7, 3π/7, π/3, π/3, ...
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1answer
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What's difference between Penrose Arc Decoration and Ammann Line Decoration for tiling kite and dart?

Penrose Arc for edge tiling rule Ammann line for edge tiling rule The result seemingly the same. It could be use to join the tile to be penrose tiling. So what is the difference between these two ...
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115 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 a polyomino tiles the plane, is there necessarily a larger tiling polyomino formed by two copies of it?

Say that we have a polyomino $P$ which tiles the plane. In may cases, it can do so by forming a two-tile "patch" which tiles the plane. For instance, with the T pentomino: Is there always a ...
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1answer
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How quickly can you mess up a domino tiling in 3D?

Suppose that we are trying to tile $\mathbb{Z}^3$ with dominoes, i.e., two face-adjacent cubes. We start going about this haphazardly, laying dominoes in random spots. How long can we keep this up ...
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209 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 ...
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Does there exist an infinite path of one colored sub-tiles?

This question came to my mind when I was contemplating on the tilling of bathroom's floor: What is the maximum number of colors we can use for an optimum coloring of $N^2$ square sub-tiles of $N×N$ ...
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1answer
115 views

Number of ways to partition $2 \times N.$ Tile into $m$ parts

Given a $2 \times N$ Tile , how to find the number of ways to partition it into $M$ parts ? Meaning of a part : Cells having same number which are adjacent to each other form a part . Lets take $N = ...
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3answers
1k views

How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?

This is the parallel question to this other post with many answers already, in the sense that the $(42^\circ,60^\circ,78^\circ)$-similar triangles form the only non-trivial rational-angle tiling of ...
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1answer
49 views

Existence of a special pentagonal tiling

Is there a pentagonal tiling composed of only one shape of pentagon so that each pentagon touches exactly 5 other pentagons? Two pentagons are in touch if they share at least one common point. Few ...
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What is the asymptotic behavior of large polyominoes? How many of them tile the plane?

The free polyominoes on $n$ cells can be classified into three categories: those with holes, those that tile the plane, and those without holes that do not tile the plane. (No polyomino with holes ...
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Simplest Discrete 3D Model of a Regular 2D Hyperbolic Tiling

I only have a beginners level understanding of hyperbolic geometry, and I am afraid that the following question might be too vague, but here goes. I know one can make real 3D models of regular tilings ...
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1answer
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How to prove a tiling of a hexagon must form a 3D cubic stack?

How to prove that a tiling of a big hexagon consisting of triangles, using only $2$-triangle tiles (three possible orientations), must resemble a continuous, convex (for each small cube), manifold, $3$...
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1answer
49 views

Does a tiling being monohedral and monogonal imply that it is isohedral or isogonal?

I’m working through Grünbaum and Shepard’s Tilings & Patterns and I have been unable to make much progress on this problem. Does there exist a tiling that is monohedral and monogonal, but is ...
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1answer
54 views

Neighbours of white squares on a chess board and Hall's theorem.

Consider an $m \times n$ chess board with $mn$ even, $m,n \geq 2$, and one black and one white square removed. Label the white squares $1,\dots, \ell$ where $l = mn/2 - 1$, and for each $i \in \{1,\...
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219 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.) In this ...
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1answer
74 views

Which equilateral triangles does the P-hexiamond (the “sphinx”) tile?

There's been lots of work investigating the polyominoes which can tile a square (equivalently, a rectangle). However, as far as I can tell there's been less investigation into the polyiamonds which ...
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1answer
56 views

System of distinct representatives and chessboards

I encountered the following problem, which was presented in the context of the topic of SDRs (system of distinct representatives) - I am able to solve the problem, but I make no use of a SDR, and I am ...
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1answer
83 views

Covering an 8x8 board with L and O Tetromino [duplicate]

I solved a puzzle about proving that if a rectangular board can be covered by L-Tetrominoes then the number of squares must be a multiple of 8. I based the solution on a colored board (like a ...
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1answer
93 views

Is there a “nice” rep-tile of order $6$?

A planar set is said to be a rep-tile if it can be tiled by congruent shapes, each similar to the original. If there are $k$ such shapes, each scaled down by a factor of $\sqrt{k}$, it is said to be ...
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1answer
34 views

Covering a rectangular board with Tetrominoes

I am reading about a puzzle question that is about Tetrominoes and proving that if a rectangular board can be covered with T-Tetrominoes the board's number of squares has to be a multiple of 8. The ...
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1answer
71 views

Geometric tiling puzzle [closed]

yesterday my sister send me a YouTube link about a tiling puzzle game: https://youtu.be/A9KU_gPOaGU. Can one deduce from the geometry of the pieces alone whether there is something special about this ...
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How many isohedral ways are there to stack $1\times 1\times 2$ blocks?

If I have blocks of size $1\times 1\times 2$ cubes, how many ways there are to stack them isohedrally in 3D space? I now have pretty robust system for solving this kind of problem in 2D, but 3D ...
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What is the math behind the “IQ Blox” puzzle?

One of my younger relatives showed me this curious puzzle (called "IQ Blox") while I was staying at their place. The premise is as follows: there's a leaflet included with the puzzle listing ...
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1answer
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Wang Tiling problem - If you can tile the first quadrant then you can tile the whole plane

I have seen in some articles (see here for instance) - the following claim without a proof regarding Wang tiles: The first quadrant can be tiled iff the whole plane can be tiled Can anyone explain ...
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1answer
37 views

Proof that a $2^n . 2^n$ Chessboard with a Missing Square can be covered by L-Shaped Tiles - Missing Step

I understand the recursive solution to tiling a $2^n. 2^n$ chessboard with a missing square using L-shaped tiles such as these: using the following method to divide the original problem into ...
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1answer
63 views

How to generate vertex-transitive tilings?

It is trivial to construct vertex-transitive polytopes: choose any finite matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and some point $v\in\Bbb R^d$. Take the convex hull of the orbit $\Gamma v$, ...
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1answer
175 views

Can an equilateral triangle be dissected into 5 congruent convex pieces?

There is a rather surprising dissection of an equilateral triangle into 5 congruent pieces:                                                     However, these pieces aren't very "nice", ...
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1answer
208 views

Is a finite tiling of 45-45-90 triangles uniquely determined from the resulting union (up to trivial flips)?

Suppose I take a union of nonoverlapping $1-1-\sqrt{2}$ triangles in the plane: The same shape can be tiled in another way, by flipping two triangles joined together into a square: In general, are ...
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131 views

Tiling an equilateral triangle using trapezoid by a divide and conquer algorithm.

An equilateral triangle is partitioned into a smaller equilateral triangles by parallel lines dividing each of its sides into n > 1 equal segments. The topmost equilateral triangle is chopped off ...
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2answers
138 views

Recurrence relation to find number of ways to cover $2×n$ chessboard with red and blue $2 \times 1$ tiles.

Let an be the number of ways to cover a 2 x n chessboard using 2 x 1 red tiles and 2 x 1 blue tiles. Find a formula for an. Find a recurrence relation for an, with initial conditions. Solution: given ...

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