# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Tetromino coloring problem on a grid

Problem: All the tiles on an $n\times n$ grid are black or white. Every possible T-tetromino of tiles on the grid is examined. A T-tetromino is 'happy' if has 2 white tiles and 2 black tiles. ...
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### Tiling of squares in instances of Pythagoras Theorem

The Pythagorean Theorem (PT) states that in a right triangle, the square on the hypotenuse equals the sum of the squares on the legs or $a^2 + b^2 = c^2$. Is there an instance of PT (a given $a$, $b$ ...
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### When are convex polygon tilings Voronoi?

A square is divided into convex polygons. Is this always a Voronoi diagram? If not, what are some simple examples of non-Voronoi tilings? Which of the pentagon tilings are Voronoi? I took a look ...
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### Tiling a rectangle with L-tromino [duplicate]

Consider a $2^{1999} \times 2^{1999}$ square, with a single $1 \times 1$ square removed. Show that no matter where the small square is removed it is possible to tile this "giant square minus tiny ...
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### non-trivial non-repetitive aperiodic tiling of the plane

Which is the less trivial example of non-repetitive aperiodic tiling of the plane you know? I cannot come up with a famous non-repetitive tiling. Are there any? A tiling is repetitive if for every ...
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### Tiling problem: 100 by 100 grid and 1 by 8 pieces

Why can't I tile a $100 \times 100$ table with $1$ by $8$ pieces? If we look at the number of tiles, $100^2$ is divisible by $8$. So this does not contradict existence of such tiling. The standard ...
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### covering chess board with 3*1 dominoes [duplicate]

Why is it not possible to cover the chess board with 3X1 dominoes if one of the corner squares is missing (e.g. the top right square)?
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### Covering any rectangle with this shape is not possible

Why can I not tile any rectangle without gaps with the given shape? http://i.stack.imgur.com/9oxO4.png You can mirror the shape (i.e. turn it around an axis in its own plane by $\pi$).
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### 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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### How many face we could make regular convex polyhedron

I want to tile the sphere as many face as possible. And I want every face be the same size and shape. Is it possible to generate more than 100 or 1000 faces of regular convex polyhedron?
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### Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt. I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
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### On pentagonal tilings

The following image has been in the news recently: My understanding is that these are all the known (to-date) tilings of the plane using convex pentagons. Can someone explain to me why the ...
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### 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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### Wrong number of $4\times 4$ domino tilings, but why?

From the internet, I know that a domino tiling of a $4\times4$ checker board can be arranged in $36$ different ways. With the following reasoning, I conclude that it must be $37$, which is one more ...
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### Shapes for tiling a circular disk?

Does anyone know of a catalog of sorts for what shapes are allowed for tiling a circular disk? For example, if you are allowed one piece to tile the disk, are all the possibilities essentially "pie"-...
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### 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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### Dominoes on chessboard

A $2016\times 2016$ chessboard is tiled with $2 \times 1$ dominoes. I can prove that there is a grid line that pass through at least $505$ dominoes. But how to prove or disprove that there is a ...
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### Cover a cicular hole with planks

A friend of mine asked me the following question. Whats the minimum number of rectangular planks of unit width (and infinite length) needed to cover a circular hole with diameter $n$? ...
Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...