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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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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2answers
33 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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0answers
14 views

Numerically define tessellation / tiling?

Assuming that you know how to express and define a geometric shape, either with a Cartesian coordinate system or with trigonometric curves, in a discrete set or in a complex set, there is a way to ...
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18 views

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
15 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
27 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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1answer
35 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
52 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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0answers
66 views

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
32 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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1answer
46 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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23 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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0answers
23 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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1answer
31 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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0answers
29 views

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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0answers
79 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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1answer
151 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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0answers
73 views

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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0answers
41 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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2answers
244 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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1answer
52 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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1answer
55 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
66 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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6answers
274 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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59 views

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
43 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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0answers
133 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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60 views

Elementary proof that the complexity function of the Fibonacci word is $p(n)=n+1$

The Fibonacci word $f$ (or Fibonacci tiling) is a bi-infinite sequence (with prescribed origin) over the alphabet $\{0,1\}$ which is the unique sequence fixed by the substitution $1\mapsto 10, ...
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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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1answer
169 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
139 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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2answers
384 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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0answers
39 views

Layering two different sized hexagonal planes periodically

I have two planes stacked on top of each other. Each plane looks like this: Each plane, however, has a different hexagonal side length (the first plane has hexagons with side length $a$ and the ...
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1answer
110 views

“Simmetric” connected k-regular bipartite graph

Let $G$ be a k-regular bipartite graph with $k > 0$. Then it is known that the two sets which partition the vertex set of $G$ have the same cardinality. However I am interested in connected ...
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2answers
223 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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0answers
43 views

Given 4 tile types, what are the chances that there are no sets of 3 in a 6x6 array?

I know this question seems arbitrary, but it actually applies to a matching game that I'm writing. I randomly typed the following letters and created a 6x6 array using the letters A, S, D, and F. ...
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1answer
127 views

Tiling an $n\times n$ Grid

Given an $n\times n$ grid, and $2\times 2$ checkered tiles (white in the upper left and bottom right corners, and black in the upper right and bottom left corners), what is the smallest number of ...
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1answer
81 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
82 views

Giant Pufferfish skin pattern―how could that be generated

I just started my investigations about tesselations and tilings for some very special kind of design Project. I came over that image: It shows a part of the Giant Pufferfish's skin and I am very ...
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48 views

Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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1answer
43 views

Group partition with sets.

the following questions come from tiling problems. Maybe the answer is easy but so far I don't know how to start as I can't see whether I have to prove that it is true or to find a counter-example. ...
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0answers
87 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. How can I determine if a connected subset of these hexagons can tile the plane by translations that form a (Bravais) lattice? ...
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1answer
255 views

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the ...
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0answers
287 views

Generating evenly distributed points on a sphere

How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
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1answer
90 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
530 views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
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2answers
245 views

Prove or disprove a chessboard with diagonal corners removed, cannot be tiled with L shape pieces or size 2

I think this is impossible, but I don't know how to prove an integer solution doesn't exist for a given equation. Here's my approach: First, observations: The removed tile will be of the same color. ...
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3answers
1k views

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 ...
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1answer
87 views

Tiling a minimal perimeter region with $n$ unit squares

Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square. Is ...
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2answers
96 views

primality on tiles?

Call $S_n$ the square of area $n^2$. See it as a collection of $n^2$ unit squares. In the following, what I call tile is a collection of unit squares that are glued together. If $n$ is not prime, say ...