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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Is there an aperiodic tiling consisting of deformed hexagons?

The typical Penrose tiling consists of two deformed quadrangles. But it's there any aperiodic tiling consisting entirely of two or more deformed hexagons? Maybe even one that shares some properties of ...
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1answer
121 views

Tetromino Proof

Prove that an 8 x 8 board cannot be covered by 15 L-tetrominos and one square tetromino (an L-tetromino is a plane figure shown below, constructed from four unit squares arranged in the form of L; a ...
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258 views

Pictorial puzzle

Can someone suggest a pictorial representation of a mathematical puzzle or problem that could be represented in coloured tiles on an orangery floor $4489\times7525$mm with an insert of $907 ...
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343 views

What is the mathematical significance of Penrose tiles?

I have a very limited understanding of groups and symmetry gained mostly from online videos (for eg. this one), so forgive me if this sounds ignorant. Particular parts of Penrose tilings exhibit ...
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Tiling a $23 \times 23$ square by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles

A $23 \times 23$ square is tiled by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles. Prove that at least one 1 x 1 tile must be used. Find such a tiling with exactly one $1 \times 1$ tile. Hint: ...
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Tiling an L-shape with “almost square”s

ABSTRACT: Define an "almost square" as a rectangles with aspect ratio in $[{1 \over 2},2]$. What is the minimal number of interior-disjoint almost-squares required to tile the following L-shape (where ...
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1answer
37 views

5d Basis Vectors of Penrose's Tilings

I have been writing some software to display/render Penrose tilings. I was hoping to use the approach of projecting a 5-dimensional lattice into 2d and apply some coloring based on regions etc. I ...
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What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
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Tiling a rectangle and tensor products

Consider the following theorem: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. There is a paper ...
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1answer
66 views

Is a “nice” plane tiling possible where each tile has 7 (8, 9, …) neighbors?

Is a "nice" plane tiling possible where each tile has 7 (8, 9, ...) neighbors? With "nice" I mean: The tiling is (preferably) periodic. The tiles are from a finite set The tiles themselves are ...
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68 views

Tiling arbitrarily large portions of the plane implies tiling the plane?

Suppose that one has a finite collection of types of (polygonal) tiles. We say that we can cover a region of the plane if we can place tiles in a non-overlapping fashion (except for edges of tiles ...
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135 views

Squaring the plane, with consecutive integer squares. And a related arrangement

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would ...
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1answer
30 views

Conventional unit mesh

I'm trying to find and outline a non-primitive conventional unit mesh, I'm not sure how to go about it. I'd also like to find any mirrors of planes and rotional symmetry axes. Would this look ...
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0answers
34 views

Squaring the plane with consecutive integer squares. And a related arrangement. [duplicate]

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2 $squares, with sides $1,2\ldots n^2$ (n odd). Which seems like it would work ...
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1answer
34 views

Penrose tilings as a cross section of a $5$-dimensional regular tiling

Could somebody explain to me how a penrose tiling , which is not periodic, can be a cross section of a regular tiling in $5$ dimensions, which is periodic? It does not make sense to me how a periodic ...
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1answer
26 views

Tiling an $m\times n$ grid.

For natural numbers $m$ and $n$, an $m\times n$ grid of squares can be tiled with tiles of the form completely filling the grid, without overlapping, if and only if $m,n\geq2$ and $6\mid mn$. It ...
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3answers
738 views

Summation in a recurrence relation

edited to reflect advice from the comments: While working on a generalization of a tiling problem, I generated a recurrence relation to describe the total number of possible tilings. The relation ...
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1answer
49 views

How to convert a Turing Machine program to a tiling using Wang Tiles?

To illustrate my question I provide the following example. The website Online Turing Machine provides a Turing Machine simulator. The following program adds 1 to any binary number. q0,1 q0,1,> ...
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1answer
63 views

Polygonal tilings: differentiating between tiles and tilings, and their edges and vertices?

I'm just starting to study tilings in a groups and geometry module, and I'd like some confirmation of my understanding of precisely what it is which differentiates a single tile, from a tiling- when ...
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2answers
89 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
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1answer
84 views

What are the simple Heesch-2 polyforms?

At the Tiling Database: There are 3, 20, 198, 1390 non-tiling polyominoes of order 7 to 10. There are 4, 37, 381, 2717 non-tiling polyhexes of order 6 to 9. There are 1, 0, 20, 103, 594, 1192, 6290 ...
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Name of Generalized/Generic Polyominoes

I was just wondering whether there were any names for all "polyomino-like" objects. Preferably, objects in this set would satisfy some basic condition (such as being composed of 'tiles' and the ...
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1answer
22 views

What are the parameters for the edge-match-restricting arcs drawn on Penrose tilings?

I would like to try generating some computer art involving Penrose tilings. I'm looking into the layouts algorithms separately, this question concerns the decoration. Here is a schematic of P2: No ...
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1answer
49 views

What is the dual lattice of Kagome lattice?

We know that the dual lattice of a triangular lattice is the honeycomb lattice. What is the dual lattice of Kagome lattice?
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Tiling a 3 by 2n rectangle with dominoes

I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences ...
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How many subsets of NxM rectangle are tileable by dominos?

There are a lot of articles, formulas and algorithms for the number of domino tilings for some region, but I couldn't find anything about number of tileable regions. Is there any exact formula or at ...
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1answer
36 views

How to prove that a particular polyiamond tiles the Euclidean plane?

I read that among the 24 heptiamonds there is one piece that does not tile the Euclidean plane. My question is the following, given a particular polyiamond how do you prove that the piece does tile ...
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438 views

Dissection puzzle for area 49 to area 50

49 and 50 are close, as are 288 and 289. That allows a grid illusion. If cut out of wood, perhaps with coloring on the border as an "assistance", the pieces could be dumped out of the tray, flipping ...
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1answer
52 views

Tiling m by n rectangle game

Consider an $m$ by $n$ rectangle. On this rectangle, two players take turns placing either $1$ by $2$ tiles or $3$ by $4$ tiles. The player who is able to place the last tile wins. Which player has a ...
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27 views

Wang's Tile Reference

I am working on projects in solving ground state of generalized ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results: ...
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32 views

What are interesting ways to tile a square image? Or a transformation that make an image tilling-able?

I want a method to tile arbitrary square image. For most cases the boundaries do not agree. So I are looking for a transformation from a square image to a square image whose boundaries agrees. One ...
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300 views

Covering a chess board with $2$ missing places with $31$ dominoes

I am reading a book that is intended to a wide audience (and not just mathematicians etc'), the book is, of course, about mathematics (Its still not clear about what exactly, I am only in page $2$). ...
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1answer
96 views

Repetitive tiling implies finite local complexity

My question probably needs to include the definitions of the terms in the title so I will first ask the question and then introduce the necessary definitions. The following Theorem is stated without ...
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1answer
36 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
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1answer
166 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
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How get closest vertex in triangular tiling from coordinates on plane?

Currently I have a plane with square tiling. It pretty trivial to get point on plane mapped to vertex of square tiling: plane point (x; y) -> vertex of square tiling (x div A; y div A). How to get ...
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82 views

What is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree 2, 3,..., n (i.e. a tile of an arbitrary degree) would be? Asked another way- what is it that ...
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2answers
276 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
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42 views

Tile a spline with bricks

I have a series of identically sized virtual Lego bricks. I need to "tile" these bricks along the length of a Hermite spline to create a curved road. The spline is two-dimensional. E.g. the curve only ...
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1answer
56 views

Does the fact that a tiling is tile-uniform always guarantee that it is also vertex-uniform?

It seems to me that if a tiling is tile-uniform, then it must be vertex-uniform as well. But is this the case? How would one go about devising a proof? By 'tile-uniform', I mean a tiling whose ...
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2answers
74 views

Dicrete Math Interesting question about Tromino

Prove that for a m$\times$n rectangle, if this rectangle can be covered completely by trominoes of the shape indicated in the picture, then mn is divisible by 3. I came up with a tentative way to ...
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Tilings of the Hyperbolic plane

Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions: $f(r) =$ number of graph nodes contained within the a ...
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1answer
44 views

In search of a symmetric homogeneous graph with a pivotal origin

I'm trying to design a computer game and I need a symmetric homogeneous graph with a pivotal origin which will act as the map of the game (players will walk according to it). Here's an example of ...
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1answer
98 views

Substitution tilings with parallelograms

I'm looking for a substitution tiling made with parallelograms, that is, a tiling of the plane with parallelograms (which do not have to be of the same shape) such that we can take one parallelogram ...
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163 views

Squaring rectangles

it is a nice high-school exercise to prove that a square can be tiled with n squares if and only if n=1, 4 or is any integer greater or equal to 6. A direct consequence is that any rectangle that can ...
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182 views

$2013\times2013$ Board with no trominoes.

Let A be a $2013\times2013$ board with $k$ black squares and containing no $L$ shaped black trominoes(in any rotation) and such that if any white square is dyed black then $A$ contains a black $L$ ...
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1answer
1k views

Algorithm to get the maximum size of n squares that fit into a rectangle with a given width and height

I am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of ...
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3answers
2k views

how to generate tesselation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's ...
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1answer
175 views

Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino

Suppose we have an $2^n$x$2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely. You can mix different rotations in the same tililng.
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221 views

Getting started with aperiodic tiling

I spent a little time looking around the Wikipedia and Wolfram articles on Penrose Tiling, the Domino Problem, Wang Tiles, etc., but I'm having a little trouble getting into them. A lot of these ...