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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Coverings of a rectangle

How many coverings of the rectangle with height $1$ and length $n$ exist, if we use only tiles with height $1$ of the following 6 types: The solution should be in a closed form (formula).
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Wallpaper groups

I am trying to understand the wallpaper groups and their symmetries. For example consider the following tiling : I believe that the symmetries are two translations, a $120^\circ$-rotation and a ...
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5answers
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Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?

I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that ...
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Tiling pentominoes into a 5x5x5 cube

I have this wooden puzzle composed of 25 y-shaped pentominoes, where the objective is to assemble them into a 5x5x5 cube. After spending quite a few hours unsuccessfully trying to solve the puzzle, I ...
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1answer
50 views

Plane tesselation, using stairs $n\times n$, is it possible?

The other day I was constructing new mathematical problems for my pupils and thought of something like this: Given the infinite sequence of "stairs" $n\times n$, constructed from $1\times1$ ...
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2answers
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Aperiodic tessellations of the plane

Here are some examples of non-periodic tessellations of the plane. Sir Roger Penrose is the expert in that field. How could someone go about proving that a certain tiling of the infinite plane with ...
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3answers
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Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.

Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that $T(kx,ky)...
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Edge-matching icosahedron puzzle

Color the edges of an icosahedron with 4 colors so that all 20 triangles have a distinct set of colors. Color the edges of an icosahedron with 6 colors so that all 20 triangles have a distinct set ...
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1answer
146 views

Prime number proof for tiling a rectangle

The following theorem has many proofs, several of which are highlighted in this document. Whenever a large rectangle is tiled by rectangles, each of which has at least one integer side - the ...
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A tesselation where four parallel rectangles meet at each vertex

If we tile the plane with parallel rectangles that are translated copy of the same rectangle, then each point is either inside a tile, or on a segment common to two adjacent tiles, or a corner common ...
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What type of aperiodic tiling is used by Turkish Airlines on their bathroom walls?

The walls and bulkheads of Turkish Airline flights are decorated with a pattern that appears to be some sort of aperiodic tiling. They are most prominent on the bulkheads of flights, and are also used ...
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1answer
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Making bigger penny triangles from smaller

I thought up a problem a long time ago, and googling doesn't even close to turn up the answer. It is this: Given unlimited 3-penny triangles (e.g. triangles with 3 pennies touching each other,) for ...
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3answers
161 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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1answer
51 views

Pentagonal tiling

I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I ...
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2answers
1k views

How to tile a sphere with points at an even density?

I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon ...
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Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
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2answers
67 views

Can we cover the entire plane with the square with area 1/n for each positive integer n?

We have one square with area 1/n for each positive integer n. Is it possible to place these squares in the xy-plane in such a way that they completely cover the entire plane. If Yes, can you describe ...
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1answer
466 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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Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
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1answer
253 views

Determining if a set of hexagons on a grid can tile the plane

Suppose I have a regular grid of identical hexagons that tile the plane, that is a hexagonal lattice. How can I determine if a connected subset of these hexagons (i.e. a poly-hex) can tile the plane ...
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0answers
102 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
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1answer
171 views

Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...
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1answer
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penrose tilings and symmetry

After going through the following question on Penrose Tiling and reading de Bruijn's papers on the subject, I came accross Grünbaum and Shephardbook "Tilings and Patterns", p. 543, where they say that ...
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3answers
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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 ...
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1answer
78 views

Program for playing around with aperiodic tilings

I know this is technically a software question, but I figure the mathematicians here would know more about it: I'm looking for software to play around with different "paintings" (different patterns ...
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Is this a 16th type of a convex pentagon that can tile a plane?

As you can read here: http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem there are now 15 known convex pentagons, or ...
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1answer
34 views

3d equivalent geometric shape of a 2d tiled space

In case anyone remembers the old game Comets, it was about this: You had a spaceship which you could move around the screen and various meteors appeared and you had to shoot them up. When you moved ...
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98 views

how to divide a hexagon into regular polygons

I want to cut a hexagon paper into regions of equal areas (more precisely either into squares of side c or into regular hexagons of side c). In both cases some of the papers will be wasted. Is it ...
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2answers
181 views

Prove that it is not possible to completely cover a 6 × 6 chessboard by tiles which have dimensions 1 × 4.

I think I have some sort of understanding of how to solve this but I'm not sure. I would colour the board with 4 colours such that every 1x4 rectangle would cover one of each colour. Then cover the ...
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1answer
36 views

Scalene rectangulation of a square: let me count the ways

A rectangulation of a square is a dissection of the square $S$ into smaller rectangles $R_i$, $i=1,\ldots,n$ with the usual caveats: $S = \cup_i R_i$ and the interiors of distinct rectangles $R_i,R_j$...
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An infinite tiling problem

Assume we have an infinite area of hexagonal close tiling with hexagon side $s$ metres with corners of the hexagons marked by straight narrow trees of $d$ metres diameter and height $h$. Assume the ...
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How far can the plane be tiled by congruent regular pentagons?

What is the limit, as the radius of the disk increases, of the greatest area, in proportion to the area of the disk, of the region covered by regular pentagons of the same fixed size, all lying within ...
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2answers
47 views

Placing tetrominos in square, maximum size

I am currently coding an algorithm which places a list of Tetrominos (tetris pieces) in the smallest square possible. My question is : is there a mathematical way to know the maximum size (upper ...
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2answers
232 views

Tiling of a $9\times 7$ rectangle

Can a rectangle $9\times 7$ be tiled by "L-blocks" (an L-block consists of $3$ unit squares)? Although the problem seems to be easy, coloring didn't help me. The general theory is interesting, but I'...
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2answers
468 views

An Olympiad Problem (tiling a rectangle with the L-tetromino)

An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks, ...
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5answers
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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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4answers
253 views

Maximal tiling without any 3-in-a-rows

You are given an arbitrarily large grid, where each square can either be off or on (think Game-of-life type board). You need to tile such a grid to maximize the number of "on" squares without there ...
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112 views

Graph Relatives for Tessellation of the Hyperbolic Plane

I'm trying to get into the theory about the Modular group. Among the "Paracompact hyperbolic uniform tilings in [∞,3] family" in the section "Tessellation of the hyperbolic plane" I found the Order-3 ...
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Isohedral polygons that tile space in Voronoi tessellations

Besides the three uniform tilings of space in $\mathbb{R}^2$, are there other regular or semiregular tilings, by one or more types of isohedral polygon, that are also Voronoi tessellations of some ...
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Is the “Fibonacci square tiling” of Fibonacci-sided rectangles always optimal?

Is an optimal square tiling of a rectangle with side lengths of successive Fibonacci numbers always the sequence of Fibonacci numbers, as in the picture below?
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1answer
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Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly $C(p)...
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249 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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2answers
194 views

Generating function for number of different tessellation checkered rectangle

Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$. Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$. $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $...
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2answers
111 views

Tiling $\mathsf{L}$-shaped spaces using the well-ordering principle

Prove, using the well-ordering principle, that, for all $n\geq 1$, an $\mathsf{L}$-shaped space with two sides of length $2n$ and four sides of length $n$ can be tiled using some number of 3 square $\...
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Filling a rectangle with congruent squares in two columns

I have a rectangle. This rectangle is divided into two columns; the widths of these columns are not necessarily equal, and are not known. I want to fill the rectangle with squares. The number of ...
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1answer
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Is there a convex polygon such that it cannot be tiled with some number of congruent connected pieces?

So the title says it all. I assume that polygons have straight line segments as their edges and that they have finite number of edges. The number $n$ of pieces is, of course, $n>1$, to avoid ...
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1answer
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Tiling the concave polygons with non-polygons

Suppose that we reside in the set of all concave polygons (that is, polygons which are non-convex and simple, simple means that the boundary of the polygon does not cross itself). Let us denote that ...
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1answer
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Equilateral polygon plane tiling

From playing around with some toothpicks and peas, I think that it should be possible to prove that the plane cannot be tiled by a possibly infinite set of equilateral polygons with the same number of ...
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2answers
698 views

How do you build a square from this figure?

If you can use only this figure, what is the LEAST number of such figures that you can use to build a square ? You can use any isometry to build a square and it must be full from the inside.
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1answer
394 views

Fill the board with dominoes

Imagine a 8x8 cell board that is missing two cells at the opposite corners, a domino takes up exactly two cells. How can you fill the board with dominoes so that none overlap or hang off the edge?