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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Which repeated shape covers a circle most efficiently

Imagine you have a ring and you want to fill as much of this ring as possible with one type of shape (triangle, square, hexagon, must be the entire shape repeated within the ring no overlaps). To be ...
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Find the number of ways of tiling a $3\times n$ rectangular grid with $2\times 1$ dominoes

I'm trying to find the number of ways $(a_n)$ of tiling a $3\times n$ rectangular grid with $2\times 1$ dominoes, where rotation is allowed. I want to find a recurrence relation for $(a_n)$ and an ...
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1 vote
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How to find the number of ways the $24$ unshaded squares in the grid can be tiled

Firstly, we see that some tiles can combine to $2 \times 3$ tiles lying around shaded square. and there are $2$ ways arranging $2 \times 3$ tiles around shaded square. In each case : in one $2 \times ...
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2 votes
1 answer
62 views

Prove that the number of ways of tiling a $1\times n$ rectangular grid with squares and dominoes is equal to $F_{n+1}$, where $n\geq1$ [closed]

Here $F_n$ refers to the Fibonacci numbers. I know the definition of the Fibonacci numbers and various recurrence relations that they satisfy but I'm not sure how to prove this statement.
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Tiling curved 3D space

A flat plane can be tiled, for example, with regular hexagons. If you try to tile a sphere with hexagons, however, it doesn't work--you have to introduce 12 pentagons to complete the tiling. Try to ...
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2 votes
0 answers
103 views

Is there a particular Turing Machine which halting is undecidable in all formal systems?

Hanf and Myers showed in 1974 that there exists a single set of tiles that will tile the plane only in a non-computable way. How are we to interpret this? Does it imply that there is a particular ...
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1 vote
1 answer
155 views

How many squares of each colour are in a generalized checkerboard $C$-coloured $m \times n$ rectangle?

How many squares of each colour are in a generalized checkerboard $C$-coloured $m \times n$ rectangle? Assume an $m\times n$ rectangle has been been divided into a grid of $mn$ unit squares, and the ...
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Covering the plane with compact sets, II

A previous question Covering the plane with compact sets received a (to me) unexpected, simple, and elegant solution, which, moreover, possessed the merit of continuity. This time the constraints are ...
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2 votes
1 answer
81 views

Covering the plane with compact sets

Let $X$ be a non-empty compact subset of the plane and $\{Y_{\alpha}\}$ a family of pairwise disjoint compact subsets of the plane each of which intersects $X$ in a singleton. Is it possible to choose ...
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A rectangular board was covered without overlapping by $1×4$ and $2×2$ polyminos . Then the polyminos were removed from the board but one $2×2$ was

A rectangular board was covered without overlapping by $1×4$ and $2×2$ polyminos . Then the polyminos were removed from the board but one $2×2$ was lost . Instead, another $1×4$ polymino was provided ....
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Definition of aperiodic tiling

I think I got confused with the definition of aperiodic tiling. Look at the following example: First, try to find a "1-dimensional aperiodic tiling". Start with the string 0, then make the ...
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Mapping "valid" tile sequences to number series

I'm interested in understanding the maths behind a satisfying time-waster game called Noodles!. In one of the game varieties there's a grid of tessellating hexagons with pipes drawn on them, and you ...
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recurrences for tilings part 2

The following problems are from chapter 9 of Arthur Engel's Problem solving strategies (questions 63-65) respectively. Let $a_n$ denote the number for a $k\times n$ rectangle (for instance, for ...
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1 answer
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recurrences for tilings part 1

The following problems are from chapter 9 of Arthur Engel's Problem solving strategies (questions 60, 61, 62) respectively. For each question, let $a_n$ denote the number of ways for a $k\times n$ ...
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What is 12-fold Stampfli-inflation tiling and where/how can I recognize it in this analysis of dodecagonal 30° twisted bilayer graphene?

Dodecagonal 30° twisted bilayer graphene is just two graphene honeycomb nets with an exact 30° rotation with respect to the other. If you treat it as a 2D pattern rather than a 3D stacked structure, ...
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How many patterns could $n$ connected bricks form?

As shown below is a floor pattern with identical bricks, each with width of 1 unit and length of 2 units. Picking $n$ bricks from the floor , how many kinds of patterns could they form, if all of ...
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3 votes
1 answer
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Minimality of aperiodic SFTs for $\mathbb{Z}^2$

Let $A$ be a finite set, a subshift of $A^{\mathbb{Z}^2}$ is a closed subset $X\subseteq A^{\mathbb{Z}^2}$ (with respect to the product topology, $A$ as a finite set is of course considered to be ...
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2 votes
0 answers
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Can we do regular geodesic tiling on a quadric surface, such as hyperboloid and paraboloid?

Can we do regular tiling on a quadric surface, such as hyperboloid and paraboloid? Assume that the straight lines are the geodesics. Can you show the pictures of the tiling?
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Can we enforce a P3 Penrose tiling using unmarked rhombi with a small set of local matching rules?

Suppose I've got some rhombi that I want my friend to construct a P3 Penrose tiling out of: However, the edge markings on my tiles have worn off, so I need to give my friend instructions about how to ...
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1 vote
1 answer
52 views

Beatrix's board and holes

This is the last question from UKMT's JMC 2016: Beatrix places dominoes on a 5 × 5 board, either horizontally or vertically, so that each domino covers two small squares. She stops when she cannot ...
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5 votes
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How much of a Penrose tiling must be specified to uniquely determine the tiling?

Every Penrose tiling contains every valid finite patch of tiles, as shown e.g. in Theorem 8 here. So in order to figure out exactly which of the uncountably many Penrose tilings one is looking at, we ...
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-2 votes
1 answer
97 views

Number of rectangles on a Rubik's cube

Beside the standard $3\times3\times3$ Rubik’s cube, there are other cubes where each side is divided into $n$ pieces ($n = 3$ for the standard Rubik’s cube). Ignoring the colors, the small squares on ...
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1 answer
35 views

Combinations to arrange sub areas in an area

Assuming a $3\times3$ area is given. Furthermore there are sub areas given, e.g.: O O O O O O O O O X X X X X X X X X The aim is to arrange these sub areas so that they are not overlapping ...
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2 votes
0 answers
59 views

On which 2D surfaces in 3D can a Kagome lattice pattern be drawn using three sets of parallel lines"?

In the background/introduction to my Space SE question Have Kagome lattice patterns been used as structural reinforcement in spacecraft in non-Iranian spacecraft? Can we help Scott Manley "unsee&...
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3 votes
2 answers
116 views

Tiling of a deficient $7\times7$ chessboard with L trominoes

Prove that a $7\times7$ chessboard with one square removed can always be tiled by $L$ trominoes. I'm looking for a reasonably simple proof. I was able to prove some specific cases, For instance, when ...
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1 vote
1 answer
53 views

Tiling with sticks

An $n$-stick is a $1×n$ or $n×1$ tile. Let $P$ be a lattice polygon (whose edges are parellel to either the $x$-axis or the $y$-axis) which can be completely tiled only with $n$-sticks for a given ...
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2 votes
1 answer
44 views

Can a scaled $L$-tromino be cut into two congruent polyominoes?

The $L$-tromino can trivially be cut into two congruent trapezoidal pieces: It can also be trivially cut into three squares, and into four other $L$-trominoes of half the side length. I am curious ...
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1 vote
1 answer
60 views

Proving Tiling Problem with 1x3 Vertical & Horizontal Tetrominoes [duplicate]

Suppose I have a 6x6 board that I wish to tile with 1x3 vertical or horizontal trominoes. I want to prove the number of horizontal trominoes in any tiling must be divisible by 3. Intuitively, it makes ...
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1 vote
1 answer
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Battleship placement proving that the number of battleships is divisible by 3

We have a grid of 6 columns and $x$ number of rows. All battleships are three units long and can be placed like this: 1 or like this: 2 Where the entire grid is filled with ships with no square units ...
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0 votes
1 answer
80 views

Clarifying the meaning of the tiling/tessellation notations?

I would like to come up with a final list of "tilings", but am having hard determining what the name or even a standard representation of the tiling is. Sidenote, it appears that the terms &...
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Why are some of the tilings in this matrix left with question marks?

Why are these left with question marks, for k-uniform tilings where $k > 6$? What is preventing us from figuring it out? What are the latest techniques people are trying to apply to figure them out?...
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1 vote
1 answer
41 views

How do points on a grid satisfy the grid equation?

Currently, I'm working on a presentation regarding Penrose tilings. During my research, I've become interested in the Pentagrid method of construction, that was introduced by N.G. de Bruijn in 1981 (...
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0 votes
1 answer
50 views

Help with Colouring Proof for tiling a 3x9 board with L-Dominos?

I define an L-Domino as a 2x2 square with one piece removed. I want to prove that it is not possible to tile a 3x9 board with L-Dominos. My initial thought is to tile the board with 3 colours such ...
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2 votes
0 answers
113 views

How to divide a sphere into many equally sized triangular tiles?

I am inspired by spherical tilings: How do you take a sphere, such as one roughly the size of earth, which has a radius (assuming perfect sphere) of 6,356,000 meters, and divide it into triangular ...
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5 votes
1 answer
80 views

Tiling a square with 3:1 rectangles

It is known that a square can be tiled with $n$ rectangles whose length is double their width for any $n > 4$. In particular, no two rectangles can overlap and no part of any rectangle is outside ...
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0 votes
0 answers
27 views

periodic Euclidean tiling functions with counting, and ordinal to cartesian functions. (here floret pentagonal tiling)

The floret pentagonal tiling, also known as the snub trihexagonal tiling, is pictured below. Given a specific horizontal/vertical range (the grey box is an example (9,14) in the image) starting at the ...
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0 votes
2 answers
53 views

How to tranform a square-ish tile into an arbitrary quadrilateral

Consider the spiral tiling shown here. Two notable features of this tiling are that (1) the tiles increase then decrease in size radially, and (2) the tiles are unique, there is no similarity. This ...
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2 votes
1 answer
47 views

Transform point in the Poincaré disc to point in tile

I have a set of image files that I want to use to use to texture a set of tiles with (one image per tile), in order to render a textured version of the Poincaré disc with a specific tiling by using ...
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0 votes
0 answers
17 views

Clebsch Graph Surface Tiling and the Hurwitz Automorphism Theorem

The Hurwitz Automorphism Theorem defines the upper bound for automorphisms of an oriented compact Riemann surface of genus $g=5$ to be $|Aut(S)| \leq 84(g-1) = 336$. Known largest $|Aut(S)|$ for genus ...
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0 answers
28 views

how to determine that there are 21 vertex types of tilings by regular polygons?

How to determine the 21 vertex types of tilings by regular polygons? ive been searching all through internet to find its exact process to determine the sets of polygons and how they comeup on that ...
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Penrose tilings, infinite overlap?

After watching this episode about Penrose tilings: https://www.youtube.com/watch?v=48sCx-wBs34 This got me wondering, are there any built-in rules as for how much overlap there can be between two ...
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3 votes
2 answers
134 views

Edgematching tiles

Consider a 3×3 grid. Now, look at the patterns which generate 1 to 7 dots around the edges, taking into account rotations and reflections. Turns out there are 49 patterns, as seen in the set below ...
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2 votes
0 answers
74 views

What are non-trivial dissections of a $45^\circ-60^\circ-75^\circ$ triangle into smaller $45^\circ-60^\circ-75^\circ$ triangles?

At this link, a square is divided into $45^\circ-60^\circ-75^\circ$ triangles in various ways. To solve that problem, several people built databases of shapes that could be built with that triangle. ...
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1 vote
1 answer
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Are there polyominoes which tile half-plane but tile no strip with any width?

In Golomb's hierarchy: If a polyomino tiles strip then tiles half-plane. (Ok, it's trivial.) But what is with other direction? Is there an example which tiles half-plane but doesn't tile any strip?
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-2 votes
2 answers
107 views

How many ways are there to form a 5 𝑐𝑚 × 3 𝑐𝑚 rectangle from squares of side lengths 1𝑐𝑚, 2 𝑐𝑚 and 3 𝑐𝑚? [closed]

"How many ways are there to form a 5 𝑐𝑚 × 3 𝑐𝑚 rectangle from squares of side lengths 1𝑐𝑚, 2 𝑐𝑚 and 3 𝑐𝑚 ?" Above is a question from SEAMO(South East Asian Math Olympiad). I tried ...
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1 vote
0 answers
18 views

Partioning rectangles into rectangles and valid sub-rectangle extension rules.

Given a rectangle $A$ composed of unit squares, we then fuse grid squares into sub-rectangles $B_i$ in a way that the $B_i$ partition $A$. Example: XXO YZO YWW ...
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0 votes
1 answer
30 views

What's the min# $x$ colors of squares spanning an $n×m$ rectangle satisfying adjacency of $\:0$ same-color orthogonally & $≤2$ of ea color diagonally?

An $n×m$ rectangle consisting of $nm$ total squares is to be arranged such that no orthogonally-adjacent squares have same color and no more than one pair from each color are diagonally-adjacent. What ...
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2 votes
0 answers
73 views

Tiling a Rectangle by Squares - alternative proof

This is a known problem about tiling a rectangle of $ 1 \times x $,$x$ is irrational with $n$ squares with length of their sides $s_1,s_2,\ldots , s_n$. You have to prov this is not possible. I want ...
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  • 331
4 votes
3 answers
103 views

Cutting a polygon into 2 or 3 smaller, rationally-scaled copies of itself?

I've noticed that many 2D geometric figures can be tiled using four smaller copies of themselves. For example, here's how to subdivide a rectangle, equilateral triangle, and right triomino into four ...
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1 vote
1 answer
61 views

Linear function applied to sides of rectangle and its tiles

I'm trying to follow miniature 12 in Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra where Matousek proves that it isn't possible to tile a rectangle the ratio of ...
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