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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5
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2answers
66 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 ...
5
votes
0answers
96 views

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 ...
5
votes
0answers
37 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 $1\times1$ ...
1
vote
0answers
54 views

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 ...
1
vote
0answers
99 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 ...
4
votes
0answers
59 views

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 ...
3
votes
1answer
29 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 ...
3
votes
2answers
83 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 ...
1
vote
1answer
30 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 ...
0
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0answers
17 views

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 ...
2
votes
0answers
46 views

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 ...
2
votes
2answers
45 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 ...
0
votes
0answers
19 views

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 ...
0
votes
0answers
27 views

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?
10
votes
2answers
213 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 ...
2
votes
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 ...
13
votes
4answers
251 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 ...
3
votes
1answer
46 views

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 ...
1
vote
0answers
46 views

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 ...
1
vote
2answers
193 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 $1 \times 3$. $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ ...
2
votes
1answer
31 views

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 ...
3
votes
1answer
25 views

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 ...
5
votes
1answer
40 views

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 ...
5
votes
2answers
694 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.
0
votes
0answers
39 views

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. ...
8
votes
1answer
158 views

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 $a$, $b$ ...
4
votes
0answers
47 views

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 ...
0
votes
1answer
123 views

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 ...
3
votes
2answers
72 views

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 ...
-1
votes
2answers
261 views

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 ...
0
votes
1answer
76 views

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)?
5
votes
1answer
85 views

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 $\pi$).
2
votes
2answers
174 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 ...
0
votes
2answers
51 views

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?
3
votes
3answers
154 views

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 ...
3
votes
2answers
401 views

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 ...
16
votes
1answer
170 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 ...
0
votes
1answer
28 views

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 ...
2
votes
1answer
85 views

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 ...
5
votes
0answers
109 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 ...
0
votes
0answers
22 views

Tessellations of Flower like Functions of the form k + Sin[mx]/n

When, if ever, does the shape of the polar plot of k + Sin[mx]/n from $0\leq x\leq 2 \pi$, where k,m,n $\in$ N form a monotiling in the Euclidean Plane? If some values of k,m, and n allow for this ...
2
votes
1answer
69 views

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 ...
3
votes
3answers
298 views

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 $n$? ...
0
votes
1answer
46 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

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 ...
0
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0answers
27 views

Is there any two edge colour tiling of the plane with regular polygons?

Is it possible to tile the plane with regular polygons such that every edge is one of two colours, and no two adjacent edges are the same colour?
0
votes
0answers
35 views

Determine if a rectagle is fully “compatible” with a given Polyomino

Recently, I came across a unique problem for which I couldn't find a complete solution. I want to determine if a given rectangle is fully "compatible" (for the lack of a better word, please suggest ...
2
votes
1answer
93 views

Number of algebraic solutions to a formula related to a square tiling problem

How can many different sets of prime-factors fit together so well in this formula? I am curious about the number of solutions to the following equation: $$ r_3 = \sqrt{2}\; \frac{ 1 + r_1 (r_2 ...
1
vote
1answer
37 views

How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$?

According to what I found on Wikipedia[1,2], you can represent any quasi-crystal structure in $\mathbb{R}^n$ by cutting a space $\mathbb{R}^N, N>n$ at an angle with the $\mathbb{R}^n$ space and ...
0
votes
1answer
40 views

Covering the plane with convex polygons?

I have got the following task here: Prove, that you can't cover the "Plane" with convex polygons, which have more than $\,6\,$ vertices! The answer is pretty obvious for $\,n=3\,$ vertices, because ...
0
votes
1answer
50 views

How to prove if this is false or not?

Can you cover the planar with regular pentagons and decagons(all of their sides are 1 unit long), without any holes or overlaps? I think that the answer is no, but can't really prove it. Any ideas? ...