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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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Can we fill the plane with a certain operation?

Background: Paint the origin $(0,0)$ black in $\mathbb{R}^2$. Let $S$ be a set $\{ (x,y) \in \mathbb{R}^2 ~|~ x^2 + y^2 =1 \}$. Paint $S$ black. Paint $(u,v) +S$ black for all $(u,v) \in S$. (...
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Nonperiodic space-filling polyhedra

For periodic space-filling polyhedra, the maximum number of faces seems to be 38, according to On Space Groups and Dirichlet-Voronoi Stereohedra. For non-periodic space-filling polyhedra, the ...
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1answer
50 views

Generating all possible Domino tilings on a $4 \times 4$ grid

I have a task to write a program which generates all possible combinations of tiling domino on a $4 \times 4$ grid. I have found many articles about tilings, but it is for me quite difficult and I ...
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Fill a surface with cuttable tiles

I have a problem that I have a surface. The surface can be of random shape. There can be holes at random places at the surface. I need to fill the surface with rectangular tiles of a fixed size (...
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1answer
62 views

Cover a chessboard

Let $2n\times 2n $ board. I cover it with dominoes $1\times 2$ s.t. every cell is adjacent exactly one cell coverd by a domino. I have to find the maximal number of dominoes that can be placed in ...
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1answer
62 views

Is there some way of proving that this simple pattern tiles the plane? Is a formal proof even necessary?

I’m thinking about the well known pattern generated by constructing a series of squares with side lengths following the Fibonacci sequence. Each time we add in a new square, we choose a side of the ...
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Are periodic tilings stable against defects outside of some region away from the defect?

Suppose that I have a set of Wang tiles on a 2D infinite grid, and that normally the tiling pattern is periodic. Assume it has period $p$ in both vertical and horizontal direction. Then at fixed ...
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Algorithms to generate random fault-free rectangulation? [closed]

I want to generate random rectangular partition of a given $m*n$ rectangle under the constraint that it must be fault-free partition. Basically, no two adjacent rectangles share a common side and at ...
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How can I generate tilings using a computer? [closed]

I was reading this wikiperdia article on polyminos. The pictures look very nice. Example I want to learn how to generate these figures myself in an interactive way. Basically I want to start with a ...
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Counting approximations of a flat shape by subsets of square tiling

A closed topological disk $K$ is approximated by the maximal subset of faces of the square tiling that are contained in the interior of $K$. As $K$ is translated and/or rotated in the plane, the ...
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1answer
29 views

Does the determinant of a substitution matrix need to be $\pm 1$?

The substitutional rule for the Fibonacci sequence is $\sigma: L \rightarrow LS, S \rightarrow L$, is: $$ \sigma : \left ( \begin{array}{c} L \\ S\\ \end{array} \right ) \rightarrow \underbrace{\left ...
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Neighboring solids in tetrahedral-octahedral honeycomb

In the tetrahedral-octahedral honeycomb, each vertex seems to be incident to 6 octahedra and 8 tetrahedra: Such simple combinatorial fact is probably well-known, or perhaps even obvious. However, ...
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2answers
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Dudeney’s solutions to haberdasher's problem exact measures of sections

What is the IG length if the side of the square is 1? I wonder if it is half of the square side. The triangle below represents the haberdasher's problem. version 2 version 1 (added after edit, here ...
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2answers
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Number of different fault-free $2 \times 1$ domino tilings on a $6 \times 5$ rectangle

Fifteen $2 \times 1$ dominoes can be used to tile a $6 \times 5$ rectangle. In tiling the rectangle we might generate what are known as fault-lines. A fault-line is any horizontal or vertical line ...
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1answer
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Substitution matrix for the Ammann-Beenker tiling

On the wikipedia page for the Ammann-Beenker tiling, it says the following: I am trying to derive the substitution matrix for and show that its eigenvalues are $(1\pm\sqrt{2})^1$... so far the things ...
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1answer
93 views

Is there a simple perfect squaring of a 1366 by 768 rectangle?

So, a simple perfect squaring of a rectangle is a tiling of that rectangle by squares whose side lengths are all distinct integers. Additionally, not subset of the squares must form a smaller ...
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1answer
117 views

tiles covering a $7\times 7$ square

A $7 \times 7$ board is divided into $49$ unit squares. Tiles, like the one shown below, are placed onto this board. The tiles can be rotated and each tile neatly covers two squares. Note that each ...
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39 views

Finding the Proportions of Topological Disks

I am currently in the process of writing an internal software package that will be used for computational geometry research. I am interested in being able to programatically generate isotoxal ...
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2answers
58 views

How to find lengths and corner coordinates of an irregular pentagon

How do I find the side lengths and therefore corner coordinates of a pentagon with the following internal angles: A = 140°, B = 60°, C = 160°, D = 80°, E = 100° ?...
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Heesch numbers in 3D

At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10. In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found ...
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1answer
115 views

How do I solve this tile-covering problem?

Consider an $n\times n$ chessboard whose top-left corner is colored white. But Alice likes darkness, so she wants you to cover those white cells for her. The only tool you have are black L-shaped ...
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How to accurately tile a Conch tiling with no knowledge of any of it.

I am trying to use this tiling but I can't reconstruct it accurately and really have no knowledge of this type of math. I'm seriously really ignorant to how this can be produced. If there is anyway ...
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1answer
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What is the smallest number of $45^\circ$–$60^\circ$–$75^\circ$ triangles in non-trivial substitution tiling?

Let base = $45^\circ$–$60^\circ$–$75^\circ$ triangle. Over at What is the smallest number of bases that a square can be divided into? it was determined that 23 base were needed to make a $45^\circ$–$...
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2answers
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Does there exist a partition of an L to create a square?

Background: Consider the following collection of tiles. These can be arranged to form a "difference of two squares" which I call an "L" (shown above), or a "square" (shown below). In this particular ...
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2answers
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Regular tilings of n-simplex

Consider a regular n-simplex (the n-dimensional generalisation of a triangle/tetrahedron). A triangle will tile the plane in a triangular pattern. In 4, 8 and 24 dimensions. Can we tile the volume ...
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1answer
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Permutations for tilings / tessellations

I'm an artist, many years past my maths a-level - so apologies for any idiocy up front. At the moment, I'm working with tilings/tessellations of right-angled isosceles triangles. I have two designs, ...
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1answer
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How Wang's conjecture implies decidability of The Domino Problem?

Wang stated following conjecture about Wang tiles (which was proven false by R. Berger): A finite set of plates [Wang's tiles] is solvable if and only if it has at least one periodic solution. ...
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1answer
31 views

How many colors are necessary for a W-polyomino to never cover a color more than once?

A W-polyomino is a polyomino with 2 cells in each row (except possibly the last, which may have one cell), and each row offset once cell to the right. Below are the first few W polyominoes. How many ...
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1answer
91 views

Four Dragon Curves are Edge-covering/Plane-tiling

Four Dragon curves generating outwards from the same vertex will traverse every edge of a grid exactly once (and as a consequence will be plane-tiling as well). I am captivated by this fact, and ...
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1answer
96 views

Tiling a 7x9 rectangle with 2x2 squares and L-shaped trominos

It's possible to cover a 7x9 rectangle using 0 2x2 squares and 21 L-shaped trominos, for example: ...
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1answer
47 views

Tiling Problem with Patterns and Colors

My wife proposed an interesting tiling problem to me. The specific problem she proposed is: I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size ...
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3answers
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Can there be more than four types of polygons meeting at a vertex?

Can there be more than four different types of polygons meeting at a vertex? How? (The polygons must be convex, regular and different) There are two ways to fit 5 regular polygons around a vertex, ...
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1answer
67 views

Build a wall with three types of bricks. What is the max length of wall less than 1000 cm you won't be able to lay?

You need to build a wall of length no longer than $1000$ cm. You can use bricks of three sizes: $23$ cm, $27$ cm or $36$ cm, and you are not allowed to cut bricks. What is the maximum length ...
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1answer
101 views

Partially tiling a square with parallelograms

I found the following puzzle on reddit, and am struggling to find the solution: You have an $n\times n$ square, and a supply of parallelogram tiles with side lengths $1$ and $\sqrt{2}$ and angle $...
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3answers
225 views

Maximum Area Covered by an S-Shaped Tiling

Define an s-tile as a path of squares that makes two turns in opposite directions. For instance, if one chooses the lower left corner, the middle square of the bottom side, the center, and the ...
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1answer
65 views

In how many ways can a $ 2 \times n $ rectangle be tiled by $ 2 \times 1 $ or $ 1 \times 1 $ tiles?

This problem is from the book "Problem Solving Strategies" by Arthur Engel (Chapter 9, problem 64) and the solution given there is $\ a_0 = 1, \ a_1 = 2,\ a_2 = 7$ and the recurrence relation being $\ ...
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2answers
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$2^n \times 2^n$ chessboard with one square removed - Is the tiling unique?

It is well-known that a $2^n \times 2^n$ chessboard with an arbitrary square removed is tilable by an L-shaped tromino (piece composed of three squares). The standard proof is by induction, and is ...
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4answers
313 views

How many colors are necessary for a rectangle to never cover a color more than once?

If we have an infinite grid, and we color each cell, how many colors do we need so that a $m \times n$ rectangle always covers at most 1 of each color no matter how it is placed? (Rotation of the ...
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0answers
29 views

“Extra” step in counting the number of ways to have a domino tiling of a 4 by n rectangle?

The picture above explains a method for doing this via generating functions & finite state machines, but what I do not get is why we must record the number of dominoes used to make a state ...
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0answers
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Finding all rectangles with fault-free tilings of the P-pentomino

I am trying to find all rectangles with fault-free tilings of the P pentomino. (A fault is a vertical or horizontal line inside the rectangle that is not crossed by any tiles; a fault-free tiling is ...
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2answers
171 views

What's wrong with this Penrose pattern?

I programmed the Penrose tiling by projecting a portion of 5D lattice to 2D space, by the "cut and project" method described in Quasicrystals: projections of 5-D lattice into 2 and 3 dimensions, H. ...
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How does one generalize the dual of a tiling by regular polygons to other tilings?

This question is motivated by some work in curriculum design for outreach math. Normal tilings of the Euclidean plane are a popular topic since people of all ages can play with it, but on the other ...
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Is there an aperiodic tiling such that approximates space?

Many aperiodic tilings still have some preferred directions. Think of penrose tilings. Imagine turning a tiling into a network and finding geodesic paths through the network. Are any tilings in 2 3 or ...
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2answers
295 views

Rosenfeld's $7 \times 7$ square puzzle

A $7 \times 7$ square puzzle may be described as following. Start with a $7 \times 7$ square divided into $7 \cdot 7$ unit squares. First select a unit square and mark it. And then, in each ...
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A conjecture about the connection between a Penrose tiling and the Fibonacci word fractal

Consider the Penrose tiling $P3$, inflated up to $6$ generations: We draw a line passing through the center of the tiling (red dot) and the outer vertex of the rightmost starting tile (black dot). ...
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1answer
42 views

Bounds on $d$ for tiling $\mathbb{Z}^d$ with subset of $\mathbb{Z}^n$?

According to this remarkable paper by Gruslys, Leader and Tan, given any subset $T$ of $\mathbb{Z}^n$, $\exists d$ s.t. $T$ tiles $\mathbb{Z}^d$. This immediately became one of my favourite ...
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Quasi-crystal/aperiodic pattern?

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. A ...
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1answer
98 views

Tiling a board with domino pieces

Let $m$ and $n$ be natural numbers such that $m\geqslant n>1$ and that the numbers $m$ and $n$ aren't both odd. Consider a board with $m$ columns and $n$ rows. Obviously, the board can be tiled ...
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0answers
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Determine the tiles intercepted by a given function in an irregular tessellation

Consider an irregular tessellation of the plane composed of convex tiles, such as the following one. For each tile, the coordinates of each vertex (black dots) are known up to a finite precision. ...
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1answer
1k views

Rolling icosahedron Hamiltonian path

A cube has 24 orientations. By rolling the cube on its edge within the perimeter of a $2\times4$ rectangle 3 times, all 24 orientations are reached and the next roll returns the cube to both the ...