# 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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### Show you can tile 3D space with this structure: center cube with one cube attached to each face

This is problem 2.1 from THE BLACK BOOK OF PROBLEM SOLVING: Define a Czech cube to be a center cube with one cube attached to each face. Prove that all of R3 can be tiled by Czech cubes. Just '...
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### mapping an aperiodic tiling into a periodic tiling

I came across an aperiodic tiling yesterday, and I was wondering, being just a stupid engineer and not a mathematician: Is it possible to map this into a periodic tiling using a continuous function? ...
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### Why does this application of Jacobsthal numbers defined by the recurrence relation: $a_n$ = $a_{n-1}$ + 2$a_{n-2}$ work in 2D tiles / grids? [duplicate]

Problem Statement: Find the Recurrence Relation for $a_n$, where $a_n$ is the number of ways to tile a (2xn) rectangular board with (1x2) or (2x2) pieces. . . Note: A (1x2) piece can be placed either ...
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### When is a 3D polyhedron stacking aperiodic. [closed]

Suppose you find a 3D "einstein". For me, its stacking has no dominant continuous planes*. How do you determine whether you can or can't make a periodic stacking (3D tiling) with it? (*) In ...
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### Is it possible to uniformly subdivide a sphere into arbitrarily small cells?

I am not a geometer, so I might be misusing some terms. So let me try to be more explicit regarding what I mean. "Subdivide a sphere into cells" means to partition the set of all points on a ...
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### Need Help Proving The Impossibility of A Prisoner Problem

Imagine a prison consisting of 64 cells arranged like the squares of an 8-by-8 chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he will be ...
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### Group of Oriented Edges of a Tiling

This is an idea I'm sure exists already, but is quite complicated, so it's hard to find without appropriate terminology. We consider a tiling of the plane by regular $n$-gons, containing an edge $e_0$ ...
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### Minimum number of tiles that are needed.

This is a coloring proof problem from Arthur Engel: Consider an $m \times n$ rectangle, what is the minimum number of $1 \times 1$ tiles that are needed to be colored such that the remaining portion ...
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### Aperiodic Tilings and Squarefree Words [closed]

Here is the definition of aperiodic tiling on Wikipedia. "A tiling is called aperiodic if its hull contains only non-periodic tilings. The hull of a tiling $T\subseteq\mathbb{R}^d$ contains all ...
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### How can we find sizes in a uniform tiling of the hyperbolic plane?

Given a certain uniform tiling of the hyperbolic plane (for example, one given by its vertex configuration, if that specifies a tiling unambiguously, or a tiling specified by some other means, ...
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### Explanation of using wallpaper groups with vertex figure for k-uniform tilings

In my project I need to implement various uniform tiling of a 2D-plane, so some time ago I started to dig a little bit into sources related to subject. From what I understand, any k-uniform ...
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### Covering a $kn+1\times kn+1$ region on a $(k+1)n-1\times (k+1)n-1$ square grid

We are given a $\left((k+1)n - 1\right)\times \left((k+1)n-1\right)$ square grid and tiles of size $1\times n$. We can place the tiles anywhere on the board, provided that they never cover the same ...
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### How to infinitely tile a cartesian plane with the least "mine-coverage" for various "8-proximity" values?

thanks for opening my question! So I started playing Minesweeper again recently and started asking myself this question. Firstly, I refer to an infinite square grid with squares of arbitrary side ...
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### What is special about the "pentagrid" and "heptagrid" in Margenstern's work on Cellular Automata in Hyperbolic Spaces?

In his work he mainly focuses on the pentagrid {5,4} and heptagrid {7,3}: In what ways are these tilings special? How do they compare to hyperbolic tilings in general? I am wondering what insights ...
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### Possible number of L-tetrominoes in covering an $8 \times 8$ chessboard

Assuming that we do not have T-tetrominoes in our play, what are the possible numbers of L-tetrominoes that can appear in a covering with tetrominoes? I have proven that the number can be any even ...
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### How many tiling?

Recently, I study tiling of the plane with regular polygon which is edge-to edge. If we restrict to the two triangles and two hexagons, then we can slide rows of tiles so that in adjacent rows, we ...
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### Classification of monohedral plane tilings with non-convex hexagons

Is a classification of monohedral plane tilings with non-convex hexagons known or unknown? I prepare a seminar for talented basic school pupils and I would like to talk about monohedral tiling problem....
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### Projection of a Pentagonally-tiled Sphere

I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any ...
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### Is there a database of all domino tilings of an n by m rectangle (for small n, m)?

I would like to see the actual dominos in all domino tilings of any small rectangle (or similar). I'm not sure what an efficient algorithm is for constructively enumerating them by computer. Yes, ...
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### What pentagonal tiling is this?

I'm a little baffled by the following pentagonal tiling: It clearly has an 8-tile primitive unit, and thus should be one of types 7-8 (Richard Kershner) or else one of types 9, 11, 12 or 13 (Marjorie ...
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### 3D Ising model and hexagonal domino tillings

I'm preparing an exam and in a preparation sheet there is an exercise that I just don't know how to deal with. Could someone please explain it to me? a) Explain why the 3D Ising model on the cubic box ...
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### How to obtain translations vector in periodic tiling

A tiling of the plane, $\mathcal{T}$, is a family of sets- called tiles- that cover the plane without gaps or overlaps. Assume that tiles are regular polygons and tiling is edge-to-edge. A tiling is ...
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### Which isometries allow us to tile aperiodically?

A recent Quanta article  discusses a new result of Greenfeld & Tao : There is some dimension $d$ with a $d$-dimensional tile which aperiodically fills $\mathbb{R}^d$ but cannot do so ...
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### Tiling 2 by N rectangle with Tetris shapes

There are similar questions here, but not exactly what I'm looking for. I'm a physics students and we are asked to prove that using the following 6 shapes, The number of ways to build a 2 by N tiling ...
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### Square Tetris: covering a $4 \times 4$ square with (any) tetrominoes - Why no solutions with exactly $1$ $T$-piece?

(Given a certain application of this problem, I'm surprised I couldn't find any discussion about it specifically. I'm probably just searching for the wrong terms. In any case, this one's been ...
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### Tile homotopy and T-tetromino packing of rectangles

From my old question (Which rectangles can be tiled with L-trominos, when only two orientations are allowed?), I learned a very interesting way to deal with tiling problems. I was wondering about T-...
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### Which rectangles can be tiled with triangles $T_n$, when only two orientations are allowed?

This question is a generalization of another question asked here: Which rectangles can be tiled with L-trominos, when only two orientations are allowed? A triangle $T_n$ is a polyomino with columns on ...
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### Smallest remainder using linear combination of two numbers $a$ and $b$.

Given two numbers $a$ and $b$. We need to find all the linear combinations i.e $$ax+by \le N$$ such that $x \geq 0$ and $y \geq 0$. Please notice the non-negative constraint. What is the smallest ...
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### Penrose tiling "nesting number"

I discovered a simple way to generate assign a "nesting number" for each tile in a Penrose "kites and darts" tiling, which results in a really nice way to visualize the tiling (see ...
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### Recurrence for filling a $2\times 2\times n$ box with $1\times 1\times 2$ blocks.

I am asking about how to derive a particular recurrence for this combinatorial problem: In how many ways can a $2\times 2\times n$ box be filled with $1\times 1\times 2$ blocks? Letting $F(n)$ be ...
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### A recursive relation for the number of ways to tile a 2 x n grid with 2x1, 1x2, 1x1 and 2x2 dominos

I'm trying to solve this problem: In how many ways can you cover a 2xn grid with 1x1, 1x2, 2x1, 2x2 dominos? And here is my attempt: Let a(n) be the number of ways we can cover the grid. Then if we ...
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### How many ways there are to cover an $n \times n$ tiling with $2 \times 1$ dominoes?

I came across the famous dimer problem in statistical physics and I'm struggling to come up with a mathematical formula for covering an $n \times n$ tiling with $2\times1$ dominoes? How does a ...
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### Tiling of a grid using triominoes

I want to tile a $12 \times 12$ grid using L-shaped triominoes. There must be no overlaps or missing spaces, and I know that it is possible to do so. Now, I want to know about a new condition: each ... 48 views

### What is an elegant algorithm for tiling a rectangle with identical squares based on 2 constraints (ratio of rectangle and minimizing uncovered area)?

Squares should all have the same dimension. The ratio constraint is that the length of the rectangle should be equal to its width (a square) or no more than two times its width (2:1 ratio). The ...
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### Squarefree parts of integers of the form $xy(x+2y)(y+2x)$

The motivation for this question comes from Theorem 3.3 of the 1995 paper Tilings of Triangles by M. Laczkovich, which states: Let $x$ and $y$ be non-zero integers such that $x+2y\neq 0\neq y+2x$. ...
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### Which rectangles can be tiled with L-trominos, when only two orientations are allowed?

This is a question that I got after reading this: https://www.cut-the-knot.org/Curriculum/Games/LminoRect.shtml. (This link already gave me the same result as theorem 1.1 of the article https://www....
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### What does it mean for the hexagon to be efficiently packing space?

The general claim goes something like this the best regular polygon that tiles the 2D (Euclidean?) plane with equal size units and leaves no wasted space is the hexagon I have seen similar claims ...
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### Hexagons are best for tiling 2D space in terms of perimeter vs area. What's best for 3D space?

If you think of the bee-hive problem, you want to make 2D cells that divide the plane of honey into chunks of area while expending the least perimeter (since the perimeter of the cells is what takes ...
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### Finding all 15-ominoes that tile the plane and have distinct internal adjacencies

Problem Description: This problem oddly came up in Minecraft with some friends. Not sure what the best terms are; but that's partly why I'm here. So a polyomino is built up from squares. This problem ...
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### Algorithm to get the maximum size of n circles that will fit within a given rectangle with a given width and height, packed hexagonally like honeycomb

I found an algorithm for determining the maximum size that squares can be packed in a given rectangle. This is working well, but actually it's equally-sized circles that I have to pack, and since the ...
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