Questions tagged [tiling]

For questions about mathematical tiling.

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Absolute and relative tilings of the hyperbolic plane

In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it). The ...
Hans-Peter Stricker's user avatar
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Graph theory: Closed neighourhoods and generalized clustering coefficients

The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$. The number of edges between neighbours divided by the number of pairs of neighbours is ...
Hans-Peter Stricker's user avatar
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Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
Jim Z's user avatar
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2 answers
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For what n and t can a square be partitioned into n similar rectangles in t congruence classes?

It is known that a square can be partitioned into three similar rectangles, all mutually non-congruent. I don't think it's possible with four. With what numbers of rectangles can this be achieved? And ...
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8 votes
1 answer
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A claim on partitioning a convex planar region into congruent pieces

Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
Nandakumar R's user avatar
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6 votes
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How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?

My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
user2554's user avatar
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Distance spectra of uniform tilings

Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that ...
Hans-Peter Stricker's user avatar
4 votes
1 answer
271 views

Structures for random graphs with structure

Background[You may skip this and go immediately to the Definitions.] Crucial features of a (random) graph or network are: the degree distribution $p(d)$ (exponential, Poisson, or power law) the mean ...
Hans-Peter Stricker's user avatar
9 votes
6 answers
600 views

Tiling with similar tiles

Question 1: Is there a polygon $P$ that cannot tile the plane and tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used? ...
Nandakumar R's user avatar
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3 votes
1 answer
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Monotile that tiles when you apply a rubber band

My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied. Does there ...
Ville Salo's user avatar
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Tilings of lattice polytopes by transformations of lattice polytopes

A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
Display name's user avatar
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2 answers
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What does the extension theorem for tilings state?

I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space. E.g. in "The Local Theorem for Monotypic Tilings" one reads The Extension Theorem [......
M. Winter's user avatar
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Domino tiling obtained from space-filling curves, is possible to predict basic properties?

Periodic and aperiodic domino tiling systems can be obtained by the following construction rules: Draw a regular square grid n×n of n2 cells. Select a space-filling curve that is consistent with ...
Peter Krauss's user avatar
1 vote
1 answer
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Is there a rectangular tiling based on the Padovan sequence? [closed]

I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. ...
Cye Waldman's user avatar
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1 answer
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Tiling rectangle with trominoes — an invariant

There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes. EDIT: we do not admit ALL ...
Per Alexandersson's user avatar
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1 answer
404 views

Looking for an efficient way of maximising 'pair scores' for subset of 30 selected from 50 to 10 000 objects

Context: I have a tiling program that uses a directed breadth first search algorithm. It is 'directed' by what I call 'pair scoring'. There are $N$ polyforms (pieces) used in the tiling. I have an $N\...
theonetruepath's user avatar
5 votes
3 answers
415 views

Can local flip moves connect dimer matchings on 'quadrangulated' planar bipartite graphs? (perfect matching reconfiguration problem)

I'm interested in the structure of dimer matchings on planar graphs with a bipartite structure. In particular, I'm interested in whether any two perfect matchings can be connected, i.e. transformed ...
Joe's user avatar
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Generalizations of classical tiling problem

A classic problem using an inductive construction is to show that the $2^n \times 2^n$-square, with a missing corner, can be tiled with L-triominoes. The proof goes like this: It is true for $n=1$, ...
Per Alexandersson's user avatar
4 votes
1 answer
316 views

Triangling the triangle

Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case ...
theonetruepath's user avatar
2 votes
0 answers
140 views

Tradeoffs in translation-invariant tilings of $\mathbb{R}^3$

Suppose I tile $\mathbb{R}^3$ in a ($\mathbb{Z}^3$-)translation-invariant manner. If we insist on the tiling being regular, then we are left with only the cubic tiling. However, suppose that we ...
squiggles's user avatar
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Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
James Hanson's user avatar
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7 votes
1 answer
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Changing tiles by swapping squares

In an $n\times n$ table, initially there is a $1\times n$ tile in each row. A swap is an operation that involves choosing two tiles, move one square from the first to the second tile, and ...
pi66's user avatar
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Aperiodic tile with rational area

Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
Arun 's user avatar
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12 votes
2 answers
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Random Walk on Pentagonal Tiling

I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
Franklin Pezzuti Dyer's user avatar
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0 answers
176 views

Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
Joseph O'Rourke's user avatar
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Unbalanced colourings of Penrose tiles

It is known that both the rhombus and kite-and-dart Penrose tilings are three-colourable, from Babilon, Robert. "3-colourability of Penrose kite-and-dart tilings." Discrete Mathematics 235, no. 1-3 (...
Darren Ong's user avatar
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0 answers
155 views

Tiling rectangles using all squares of sides 1, 2, 3, ..., n

Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle?
Bernardo Recamán Santos's user avatar
11 votes
1 answer
389 views

Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.                     $50$ ...
Joseph O'Rourke's user avatar
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1 answer
204 views

Tiling the surface of a hypersphere with regular simplices

Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...
Chris Jones's user avatar
5 votes
0 answers
102 views

Hooks, monomers, dimers and Young diagrams: Part II

As promised, I've upgraded my last question. Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...
T. Amdeberhan's user avatar
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0 answers
154 views

Hooks, monomers, dimers and Young diagrams: Part I

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it. Consider the one-line partition $\lambda_n=(n)$ and its ...
T. Amdeberhan's user avatar
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0 answers
149 views

monomer-dimer tiling of a Young diagram

As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known. Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
T. Amdeberhan's user avatar
7 votes
3 answers
953 views

Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N

For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
Bernardo Recamán Santos's user avatar
4 votes
0 answers
145 views

Tiling squares with oblongs

An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so ...
Bernardo Recamán Santos's user avatar
4 votes
0 answers
188 views

Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...
Andrea Prunotto's user avatar
5 votes
0 answers
115 views

What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?

Here are two facts that are superficially similar. Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons. Feit-Higman Theorem: The only finite ...
GMB's user avatar
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11 votes
1 answer
481 views

Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not share an edge length or a vertex angle, and their areas differ. Suppose you'd like to tile the plane with pairwise incommensurate triangles. I ...
Joseph O'Rourke's user avatar
3 votes
0 answers
108 views

chromatic number of plane using Cairo pentagonal tiling

Scale the Cairo pentagonal tiling so the short side is of length 1. Then it is easy to colour the tiling with 8 colours, two parallel ribbons of four colours each, to establish that the chromatic ...
Michael Ruxton's user avatar
10 votes
1 answer
326 views

Is there a triangle which makes dense set of angles by drawing medians?

This problem is a restatement of this question, first announced in MathStackExchange. We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
Solar Galaxy's user avatar
2 votes
0 answers
107 views

dividing a square into unique rectangles with the same perimeter

There's a solution for dividing a square into unique rectangles with the same area which is the blanche dissection. There's also a solution for dividing a square into unique rectangles with the same ...
elbert k's user avatar
4 votes
0 answers
161 views

Tileability and computabilty

Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(...
Dominic van der Zypen's user avatar
1 vote
0 answers
195 views

Squares as sum of squares

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares? Of these n x n squares, which can be actually cut into n smaller squares?
Bernardo Recamán Santos's user avatar
1 vote
0 answers
121 views

Minimal period for a bounded Langton's ant moving on a tessellation

We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...
ahstat's user avatar
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1 vote
1 answer
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Order question about pentagonal tiling type 9 and type 10

People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see Pentagonal tiling. These 15 types of pentagonal was named by finding date except type 9 and type ...
John's user avatar
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5 votes
2 answers
397 views

Smallest tile to *isohedrally* tessellate the hyperbolic plane

Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane? In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
Christopher King's user avatar
20 votes
2 answers
2k views

Smallest tile to tessellate the hyperbolic plane

Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself. I think it will be a Triangle group, but I'...
Christopher King's user avatar
2 votes
1 answer
280 views

Aperiodic tiling of compact space by small number of basic tiles

Suppose we have compact space, like sphere or torus in particular dimension $d$. Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just ...
kakaz's user avatar
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3 votes
0 answers
106 views

How many positions of a tile can occur in a periodic tiling?

In my recent question about polygonal tilings where tiles can occur in infinitely many positions, both constructions given as solutions are of self-similar nature. This means in particular that there ...
Wolfgang's user avatar
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1 vote
1 answer
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Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?

My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...
Wolfgang's user avatar
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10 votes
1 answer
374 views

How many positions of a tiling polygon can occur simultaneousy?

Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$. My question: How many different positions can occur in ...
Wolfgang's user avatar
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