Questions tagged [tiling]

For questions about mathematical tiling.

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Hyponontiling Wang tiles

Call a finite collection of tiles that can tile the plane if we have to use each tile at least once tiling. Is there a collection of at least 3 tiles that is not tiling, but such that after removing ...
domotorp's user avatar
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10 votes
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330 views

Examples of games developed purposely to analyze players' strategies for mathematics research

Background This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
Max Muller's user avatar
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1 answer
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To place copies of a planar convex region such that number of 'contacts' among them is maximized

A contact between two planar convex regions obviously happens either along a line segment or at a single point. Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
Nandakumar R's user avatar
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1 vote
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Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?

We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles and Cutting polygons into mutually similar and non-congruent pieces A (non square) rectangle can obviously ...
Nandakumar R's user avatar
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4 votes
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Can a square be partitioned into mutually non-congruent triangles all of same area and perimeter?

It is known that the plane cannot be tiled by pair-wise non-congruent triangles all having same area and same perimeter (https://arxiv.org/abs/1711.04504). Question: Can a square be partitioned into ...
Nandakumar R's user avatar
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To tile the plane with mutually non-congruent rational triangles of equal area

We add a little to Tiling the plane with pairwise non-congruent rational triangles Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "...
Nandakumar R's user avatar
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2 votes
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Is this an actual solution for centroidal Voronoi tiling, or just a visual approximation? [closed]

For the capstone project at the end of my graduate Data Science studies in late 2022, I needed an algorithm that would converge to something close to centroidal Voronoi tiling, while tiling contiguous ...
Florin Andrei's user avatar
1 vote
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Does a substitution tiling being FLC depend on starting seed?

I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
Keen-ameteur's user avatar
1 vote
1 answer
106 views

Recognizability/unique composition property for substitution tiling

This may be a very basic question, but I have not found an answer to it so far in my search. The question is whether there is an "algorithmic" way to check unique-composition/recognizability ...
Keen-ameteur's user avatar
12 votes
0 answers
160 views

Can the optimal packing density in $\mathbb{Z}^d$ be irrational?

For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
Arsenii Sagdeev's user avatar
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Aperiodic SFT equal to a substitution subshift

I was wondering whether there are primitive symbolic substitutions over $\mathbb{Z}^d$ and alphabet $\mathcal{A}$ whose associated subshift is equal to an aperiodic SFT. By SFT here I mean a subshift ...
Keen-ameteur's user avatar
4 votes
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131 views

What is the tiling semigroup for an Einstein "hat" tiling?

My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
Shaun's user avatar
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Chains in tilings with the aperiodic monotile

This is starting with any given infinite tiling of the plane by the aperiodic "hat" monotile, where chains of similarly oriented tiles are colored as in Figure 2.2 on p. 10 in the original ...
Wolfgang's user avatar
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7 votes
1 answer
239 views

Decidability of completing Penrose tilings

Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
interstice's user avatar
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1 answer
207 views

Does $\mathbb{Z}\times\mathbb{Z}$ have an aperiodic monotile?

For any set $S\subseteq \mathbb{Z}\times\mathbb{Z}= \mathbb{Z}^2$ and $a\in \mathbb{Z}^2$, we set $a+S = \{a+s: s\in S\}$, where $+$ is the componentwise addition in $\mathbb{Z}^2$. Moreover, for any ...
Dominic van der Zypen's user avatar
4 votes
0 answers
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Draw an arbitrary line on a Penrose tiling. Determine a sequence of tiles can it intersect

Let us consider a Penrose tiling of $\mathbb R^2$. Starting with an arbitrary point on the tiling, draw an arbitrary straight line. Assume that this straight line never overlaps perfectly with a ...
Darren Ong's user avatar
0 votes
1 answer
91 views

Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent? Remark: Every trapezoid is affine equivalent to ...
Nandakumar R's user avatar
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4 votes
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267 views

References and upper bounds for the SONNAT tiling game?

Introduction In a video released about a month ago, Pembesita describes a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling. In the single-player game, the player may employ ...
Max Muller's user avatar
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Periodic tilings in finite type tiling spaces and substitution tiling spaces

I was reviewing the following statement from a survey by E. Arthur Robinson about tilings in $\mathbb{R}^d$ to better understand geometric tiling rather than tilings over symbols. I consider the ...
Keen-ameteur's user avatar
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Local complexity of tilings under substitutions

I am trying to read this survey of E. Arthur Robinson, about tilings of $\mathbb{R}^d$. I have some familiarity with 'symbolic' tilings, but I don't think I have a good intuition on 'geometric' ...
Keen-ameteur's user avatar
1 vote
0 answers
57 views

Counting problem, tiling rectangle with two types right isosceles triangle

How many ways are there to tile a rectangle of size $m\times n$ with two types of isosceles triangle, type 1 having area $\frac{1}{2}$ and type 2 having area 1? I know with only type 1 there are $2^{...
mendalan lor's user avatar
1 vote
0 answers
95 views

Tiling the plane with pair-wise non-congruent and mutually similar triangles

Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints. Note 1: Reg requirement 3 above: since any ...
Nandakumar R's user avatar
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Tiling the plane with pair-wise non-congruent rational triangles of bounded size and unique sides

We add a bit to Tiling the plane with pairwise non-congruent rational triangles. The solutions given there show tilings of the plane with pairwise non-congruent rational triangles that are either (1) ...
Nandakumar R's user avatar
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9 votes
4 answers
1k views

Tiling the plane with pairwise non-congruent rational triangles

A rational triangle is one in which all side lengths are rational numbers. Question: Can we tile the Euclidean plane with rational triangles that are pairwise non-congruent? No further requirements on ...
Nandakumar R's user avatar
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0 votes
0 answers
116 views

On partitioning convex planar regions into congruent pieces - 2

We add a bit to A claim on partitioning a convex planar region into congruent pieces . Definition: A perfect congruent partition of a planar region $C$ is a partition of it into some finite number $n$ ...
Nandakumar R's user avatar
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6 votes
2 answers
612 views

Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?

Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...
Wolfgang's user avatar
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1 vote
1 answer
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Tiling the hyperbolic plane by non-regular quadrilaterals

We add a bit to Which polygons tessellate the hyperbolic plane?. Question: Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the ...
Nandakumar R's user avatar
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5 votes
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401 views

On the aperiodic monotile

One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
Jim Conant's user avatar
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22 votes
1 answer
1k views

Aperiodic monotile without reflections?

The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
Timothy Chow's user avatar
17 votes
1 answer
567 views

Aperiodic monotile in $\mathbb{R}$

Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
Dominic van der Zypen's user avatar
14 votes
5 answers
1k views

How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?

This is motivated by the new paper of Smith, Myers, Kaplan, and Goodman-Strauss, wherein they define the existence of an aperiodic monotile. Clearly their tiling is not three-colorable, so we have ...
Lucas Blakeslee's user avatar
9 votes
0 answers
846 views

Are aperiodic monotiles generalizable to higher dimensions?

This question is motivated by a recently released paper written by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. It constructs the first topological disk that tiles the ...
Nicholas James's user avatar
34 votes
1 answer
3k views

Can we three-color a tiling of the plane with Smith, Meyers, Kaplan, and Goodman-Strauss's einstein?

The tiling world is a bit aflutter recently with the drop of Smith, Meyers, Kaplan, and Goodman-Strauss's paper showing an einstein - a simply-connected polygon - that must aperiodically tile the ...
Mark S's user avatar
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2 votes
2 answers
101 views

Reference request: Cut-and-project method gives rise to a fiber bundle over the torus

I apologize in advance for how vague this request is. A few weeks ago, I came upon a paper that (if I recall correctly) proves that the hull of a cut-and-project tiling is a fiber bundle over a torus. ...
Kyle's user avatar
  • 243
3 votes
2 answers
388 views

An "incomplete" tiling?

Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them? When each square of the board is covered by a domino this ...
Jiyuan Zhang's user avatar
1 vote
1 answer
65 views

Computing admissible patches of a substitution

I have been recently trying to look at substitution tilings with finite local complexity by examining their admissible patch\pattern atlas, which is sometimes called their language. I have also seen ...
Keen-ameteur's user avatar
1 vote
0 answers
95 views

What is the example of a circle being filled with congruent tiles (not pie slices), with no overlap of the tiles and and no space left?

I think I read somewhere that at one time it was thought the only way to lay tiles that would fill a circle with no overlap of the tiles and no exposed space in the cirlce, was to lay pieces that ...
Peter's user avatar
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1 vote
0 answers
37 views

Tiling the hyperbolic plane with mutually-non congruent equal area triangles

This post continues On tiling the plane with non-congruent, equal area triangles with each edge having a unique length Can the hyperbolic plane be tiled by pair-wise non-congruent equal area ...
Nandakumar R's user avatar
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1 vote
0 answers
46 views

Kissing behavior of planar regions

This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$. Background: Given a 2D region $C$ (not necessarily convex), ...
Nandakumar R's user avatar
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8 votes
1 answer
234 views

For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?

I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture). I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...
Andreas Rüdinger's user avatar
1 vote
2 answers
90 views

A variation of domino tiling problem with fusions

I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to ...
Keen-ameteur's user avatar
2 votes
1 answer
117 views

Reference on relation between SFTs and Wang-tiles

I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a ...
Keen-ameteur's user avatar
4 votes
3 answers
358 views

Tiling with ten-fold symmetry and (unoriented) Penrose tiles?

Consider tilings of the plane made out of rhombi of side 1 and either angles $\pi/10$ and $2\pi/5$ or angles $\pi/5$ and $3\pi/10$. If we give a certain orientation to the edges and respect that ...
H A Helfgott's user avatar
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5 votes
1 answer
332 views

How much of an aperiodic tiling is needed to force aperiodicity?

Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
H A Helfgott's user avatar
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14 votes
2 answers
654 views

How to characterize the regularity of a polygon?

In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
Caio Tomás's user avatar
4 votes
1 answer
141 views

Squarefree parts of integers of the form $xy(x+2y)(y+2x)$

The motivation for this question comes from Theorem 3.3 of the 1995 paper Tilings of Triangles by M. Laczkovich, which states: Let $x$ and $y$ be non-zero integers such that $x+2y\neq 0\neq y+2x$. ...
RavenclawPrefect's user avatar
7 votes
0 answers
87 views

Tiling the plane with mutually non-congruent equal area rectangles

Question: Is it possible to tile the plane with mutually non-congruent rectangles all of equal area? Note 1: If the answer is "yes" then, there could be constrained versions of the question ...
Nandakumar R's user avatar
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1 vote
1 answer
155 views

Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts

For a multidimensional subshift $X$ over $\mathbb Z^d$, the topological full group $[X]$ is the set of homeomorphisms $f$ of $X$ that can be written as $f : x \mapsto \sigma_{c(x)}(x)$ with $c : X \to ...
Numbra's user avatar
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16 votes
0 answers
387 views

Is "Escherian metamorphosis" always possible?

$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
Noah Schweber's user avatar
6 votes
2 answers
208 views

Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
Nandakumar R's user avatar
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