# Questions tagged [tiling]

For questions about mathematical tiling.

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### On cutting tetrahedrons into mutually congruent pieces

Simple observations: A regular tetrahedron can be cut into 2 mutually congruent pieces (in 3 obvious ways which are all basically the same way, giving one and same pair of congruent pieces). The ...

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### Are there triangles that can be cut into 7 mutually congruent connected polygons?

First question below had appeared in a note at Triangles that can be cut into mutually congruent and non-convex polygons
Following the results of Beeson quoted in the answer at Subdivision of ...

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### Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"

I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...

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### Dividing a polyhedron into two similar copies

The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...

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### 'Self-similar and perfect' partitions of planar regions

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...

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### Perfect squaring of rectangles

A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...

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### Trying to extend a theorem on Tiling with mutually non-congruent triangles

In the light of Cubing the cube - as 'perfectly' as possible, We try to slightly 'relax' the main theorem proved by Kupaavski, Pach and Tardos here:
https://arxiv.org/pdf/1711.04504.pdf
...

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### Non-monotileable amenable groups

This is crossposted from MSE.
We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$.
In his article Monotileable Amenable Groups, B. Weiss ...

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### Cubing the cube - as 'perfectly' as possible

Ref: https://en.wikipedia.org/wiki/Squaring_the_square
A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...

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### A puzzle with magic Egyptian tilings

Background
I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...

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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 ...

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### 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 ...

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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 $n$ ...

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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 ...

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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 ...

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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 "...

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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 ...

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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 ...

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### 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 ...

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### 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 (...

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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 ...

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### 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 ...

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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 ...

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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?

Introduction
In a video released about a month ago, Pembesita describes1 a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling.
In the single-player game2, the player may employ ...

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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 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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### 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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### 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 ...