Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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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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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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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^{...
CommunityBot
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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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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 ...
- 3,068
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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 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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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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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 ...
CommunityBot
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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\...
CommunityBot
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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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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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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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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 ...
- 3,032
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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?
...
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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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Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
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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 ...
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Decidability of periodic tilings of the plane
I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (...
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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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Construct by compactness (Pentagonal tiling – Rao paper)
In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:
"… We keep a connected component $H_d'$ of $...
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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 ...
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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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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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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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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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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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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 ...
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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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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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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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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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Computational approach deciding whether a set of Wang Tile could tile the space up to some size
As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
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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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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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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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Tiling a rectangle with all simply connected polyominoes of fixed size
For which values of $n$ can we tile some rectangle with one copy of each free simply-connected $n$-omino (that is, each polyomino with $n$ squares that has no holes)?
It appears that it is possible ...
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Does every 5-celled animal tile the plane?
An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) One may ...
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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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Mathematics of quasicrystals
I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.
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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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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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