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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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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3answers
95 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 ...
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2answers
185 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 ...
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1answer
21 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 ...
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1answer
56 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 ...
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0answers
25 views

Graph Relatives for Tessellation of the Hyperbolic Plane

I'm trying to get into the theory about the Moduar 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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0answers
16 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 ...
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1answer
55 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 ...
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3answers
282 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 $n$? ...
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0answers
32 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 ...
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0answers
20 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?
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0answers
27 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 ...
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1answer
79 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 ...
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1answer
31 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 ...
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1answer
28 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 ...
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1answer
47 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? ...
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0answers
35 views

enumerating paths in a hex tiling

assume a hexagonal tiling in the shape of a circle with radius $r$ nodes. given two nodes on the perimeter, $a$ and $b$, such that $a \ne b$, a set of paths $P_{ab}$ connect the two. what is the total ...
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1answer
21 views

Tiling of $R^3$ with fixed number of neighbors $n$, limit on $n$?

$R^3$ can be tiled regularly with cubes, where each tile has six neighbors. If I do not care for the form of the tiles, but only for the number of neighbors, is there a limit on the number of ...
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0answers
67 views

Integer hexagonal grid variations for Harborth

Harborth's conjecture states that every planar graph has a planar drawing in which all edge lengths are integers. I was looking at that, and I wondered what was known about hexagonal grid graphs. For ...
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0answers
43 views

Tiling with exactly 2013 different ways

A rectangle with side lengths integers $a$ and $b$ will be covered with tiles, rectangular with a length of one side so that a portion of the rectangular area will be covered with black tiles and the ...
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0answers
38 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 ...
3
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1answer
93 views

Prime number proof for tiling a rectangle

The following theorem has many proofs, several of which are highlighted in this document. Whenever a rectangle is tiled by rectangles which has at least one integer side, then the rectangle has ...
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0answers
56 views

Tile a Box With Translates of Two Given Rectangular Bricks

What is the layman explanation for the theorem explained in this paper? Lets say I have a rectangle of $25 \times 25$. What bricks $B_1(1 \times a)$, $B_2 (1 \times b)$ will be able to completely tile ...
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2answers
323 views

If you know that a shape tiles the plane, does it also tile other surfaces?

For instance there is a hexagonal tiling of the plane. There is also one using quadrilaterals. It seems intuitive that both of these tilings also apply on a torus. Is it the case that anything that ...
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1answer
97 views

Number of ways to fill a $2\times n$ grid with $1\times 2$ and $2\times 2$ tiles

How many ways are there to fill a $2\times n$ grid with $1\times 2$ and $2\times 2$ tiles? Rotating is allowed. Progress Let $T_n$ be the number of ways; then $T_n = T_{ n-1} + T_{ n-2} + 1 $ ...
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3answers
102 views

What's the best polygon for tiling the plane?

We want to cover the whole plane by tiles, shaped as a polygon with equal-length sides, such that there is not overlapping and any gap (Note that all the tiles are similar to each other). which ...
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1answer
61 views

Semiregular tilings of the hyperbolic plane

Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid: I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where ...
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1answer
66 views

Showing that the number of ways to cut a 200 x 3 board into 1 x 2 dominoes is divisible by 3.

Showing that the number of ways to cut a 200 x 3 board into 1 x 2 dominoes is divisible by 3. My only idea is to assume the opposite, make some needed arrangement, and to show that changing the ...
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1answer
59 views

How can I compute angles & lengths of the following tiling shape

a while back I created a tile made of arrows: I did it using a vector graphics software, without really understanding the properties of this shape. Now, let's say I want to write a program to ...
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1answer
48 views

Calculating invariant for T shape tetrominos on rectangular board

The question is from Roland Backhouse Algorithmic Problem Solving. Suppose a rectangular board can be covered with T-tetrominoes. Show that the number of squares is a multiple of 8. The ...
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0answers
59 views

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
228 views

uncountable number of Penrose tiling

I am interested in Penrose tiling, though I am not an expert, so I apologize in advance. It is known that that the number of different Penrose tilings is uncountable. This may be implied from de ...
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1answer
98 views

Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?
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1answer
45 views

What does “uniqueness of composition” mean here? (Grünbaum-Shephard's Tilings and Patterns)

This is a follow-on from a previous question, in which I paraphrased Statement 10.1.1 of Grünbaum and Shephard's Tilings and Patterns. The original statement is shown below: where Figures 10.1.3 to ...
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2answers
154 views

MNTILE SPOJ Tiling patterns

We have tiles of size 2 * 1. We need to arrange the tiles to get the floor size of m * n. For ...
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1answer
89 views

Not understanding this proof in Grünbaum-Shephard's Tilings and Patterns

I'm reading Grünbaum and Shephard's Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). ...
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1answer
70 views

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
191 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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0answers
87 views

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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2answers
88 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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1answer
100 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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0answers
36 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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2answers
196 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
95 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
121 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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1answer
1k 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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1answer
54 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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5answers
293 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
36 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
117 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?