# Questions tagged [tiling]

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265
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### 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.

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### 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 ...

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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 ...

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### 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 ...

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### References and upper bounds for the SONNAT tiling game?

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 two rhombi. The ...

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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 ...

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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' ...

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### 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^{...

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### 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 ...

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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) ...

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### 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 ...

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### 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 $n$ ...

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### 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 "...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### On tiling the plane with non-congruent, equal area triangles with each edge having a unique length

Ref: Tiling with incommensurate triangles shows an approach for tiling with incommensurate triangles - all sides and all angles unique and also with different areas - with the perimeters of the tiles ...

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### 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), ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $x+2y\neq 0\neq y+2x$. ...

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### 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 ...

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### 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 ...

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### Tiling with non-congruent triangles all of which have an equal angle and equal area

Reference 1: an earlier question on tiling with pair-wise non-congruent tiles: Tiling with triangles of same circumradius and inradius
Reference 2: Triangulation of polygons with all triangles having ...

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### 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 ...

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### 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 ...

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### Tiling planar integer lattice by finite point sets

I am interested in the following question.
Are there nice characterizations of the finite sets $S\subseteq \mathbb{Z}\times\mathbb{Z}$ that tile $ \mathbb{Z}\times\mathbb{Z}$ by translations (i.e. $\...

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### Sufficient conditions for periodic tiling by Wang tiles

I'm recently interested in whether a sub-shift of finite type contains a doubly-periodic problem, when the set of configurations is of the sort $\mathcal{A}^{\mathbb{Z}^2}$. When $Q_2=\{0,1\}^2$, and ...

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### Possible weaker version of the Domino/Wang tiling problem

This may be a dumb question, but I was wondering whether the question of 'periodically tiling the plane from a finite set of tiles' is the same as the domino tiling problem or a weaker version. I ...

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### Aperiodic tilings of the plane by squares and rhombi

Consider tilings of the plane by unit squares and by rhombi of unit side length and angles $\pi/3$, $2\pi/3$. It is easy to come up with periodic tilings of the plane - consider the following:
(from ...

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### Square-and-equilateral-triangle aperiodic tiling with $\leq 4$ prototiles?

There exist aperiodic tilings composed of square and equilateral-triangle tiles of unit side length: see https://tilings.math.uni-bielefeld.de/substitution/square-triangle/ and https://hal.archives-...

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### An aperiodic hexagonal tile?

This hexagon-with-dents is a tile which, I think, tiles the plane in a necessarily aperiodic way:
...

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### Regarding fundamental domain of 2 genus surface

Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 ...

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### What are some other methods for partitioning an n-dimensional space based on a set of points in that space?

So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...

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### Polyomino that can tile itself

Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ ...

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### number of ways to cover an $m × n$ rectangle

Given a positive integer $k\ge2$, let be $f_k(m,n)$ the number of ways to cover an $m × n$ rectangle with $mn/k$ tiles ( $1×k$ or $k×1$)
$f_2(m,n)$ is kasteleyn formula
$f_k(m,n)$?

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### Tiling with triangles with same Steiner ellipses

We continue from Tiling with triangles of same circumradius and inradius .
Definitions: Given any triangle, its Steiner circumellipse is the unique circumellipse (ellipse that touches the triangle at ...