Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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Are there aperiodic tilesets with 7-fold rotational symmetry?
In my explorations, I have seen non-periodic tilings with 7-fold rotational symmetry, and I've seen substitutions for such tilings, however I haven't seen anywhere a tileset which enforces such a ...
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For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
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Can you see through a cannonball packing?
More precisely, in a regular sphere packing, either the
HCP or FCC lattice packing,
does there exist a line $L$ disjoint from every sphere,
i.e., not touching any sphere?
If so, one could "look ...
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Tiling with one of each 3D shape
Encouraged by the positive solutions to my question,
Tiling with one of each shape,
I'd like to pose the $\mathbb{R}^3$ equivalent:
Q. Is there a tiling of $\mathbb{R}^3$ by (bounded) polyhedra, one ...
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Tiling with one of each shape
Q. Is there a tiling of the plane by one each of simple polygons of $n$ vertices:
one triangle, one quadrilateral, one pentagon, $\ldots$ ,
one simple polygon of $n$ vertices, $\ldots$ ?
Here a ...
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Tilings of $\mathbb{R}^n$ and Riemannian manifold that is uniformly locally isometric to a ball in $\mathbb{R}^n$
Suppose that we have a Riemannian manifold $(M, g)$ that is uniformly locally isometric to a ball in $\mathbb{R}^n$, that is, there exists $r > 0$ such that for every $x \in M$ ball $B(x,r)$ in $M$ ...
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Tilings in finite (not necessarily Abelian) groups
Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that
$$ G = \bigsqcup_{b\in B} bA.$$
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Computing the language of an $S$-adic shift
I have been looking online for how or if one can compute the language of an $S$-adic subshift generated by finitely many substitutions. I know that one can compute the language of a substitution ...
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Truchet tiles with non-periodic tiling from finite group multiplication tables (Thue-Morse plane)?
Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups:
$$
\pi : G \rightarrow S_n, \quad g \mapsto \pi(g)
$$
where ...
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Is there mathematical significance to the LaGuardia floor tiles?
I noticed that the new terminal at LaGuardia Airport (in New York) has an intriguing design for the tiles on at least one of their floor areas. It bears a superficial resemblance to aperiodic tilings ...
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Chromatic tiling complexity and the chromatic number conjecture
Let $T$ be a finite set of tiles in $\mathbb{R}^d$. A tiling of $\mathbb{R}^d$ by $T$ is a collection of disjoint translates of tiles in $T$ whose union is $\mathbb{R}^d$. A tiling is $k$-chromatic if ...
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Hexagon tiling and affine Weyl group $\widetilde{A}_2$
Let $H$ be a regular hexagonal room centered at the origin. Let $W$ be the group generated by reflections about the six sides of $H$. It's well known that $W$ is the affine Weyl group of type $\...
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On convex polygons that can be cut into convex and mutually congruent pieces in exactly one way
Observations: any thin isosceles triangle has exactly 1 partition into 2 congruent pieces - only 1 line, bisector of its apex, does it.
By attaching a right triangle with base 1 and altitude 2 to an ...
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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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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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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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