Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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Tiling a square with rectangles whose areas or perimeters are 1, 2, 3, ..., N
For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?
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Tiling squares with oblongs
An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so ...
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Generating a Penrose tessellation around a given tile
Given a starting Penrose tile, I need to build a "spiraling" tessellation around it.
The following picture illustrates the request:
In this example, the starting tile is a "thin rhombus" (the pink ...
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What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
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Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
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chromatic number of plane using Cairo pentagonal tiling
Scale the Cairo pentagonal tiling so the short side is of length 1. Then it is easy to colour the tiling with 8 colours, two parallel ribbons of four colours each, to establish that the chromatic ...
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Is there a triangle which makes dense set of angles by drawing medians?
This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
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dividing a square into unique rectangles with the same perimeter
There's a solution for dividing a square into unique rectangles with the same area which is the blanche dissection.
There's also a solution for dividing a square into unique rectangles with the same ...
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Tileability and computabilty
Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(...
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Squares as sum of squares
For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?
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Minimal period for a bounded Langton's ant moving on a tessellation
We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...
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Order question about pentagonal tiling type 9 and type 10
People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see Pentagonal tiling.
These 15 types of pentagonal was named by finding date except type 9 and type ...
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Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
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Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
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Aperiodic tiling of compact space by small number of basic tiles
Suppose we have compact space, like sphere or torus in particular dimension $d$.
Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just ...
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How many positions of a tile can occur in a periodic tiling?
In my recent question about polygonal tilings where tiles can occur in infinitely many positions, both constructions given as solutions are of self-similar nature. This means in particular that there ...
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Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?
My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...
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How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
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Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
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What rectangles can a set of rectangles tile?
(I asked this question first on math.stackexchange, but did not get any responses so I thought I would try here.)
If we have a set of $p_i \times q_i$ rectangles ($p_i, q_i \in \mathbf{N}$), which $m \...
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Are there unique additive decompositions of the reals?
Given $b\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+bU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?
Here is ...
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Tiling of polygons in $\mathbb{R}^2$ by squares
Let $X\subset \mathbb{R}^2$ be a polygon (possibly nonconvex, but not intersecting itself) with all the sides parallel to one of the axes.
I am interested on whether $X$ can be tiled by (finitely ...
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♢ ⧫ ⬠: the fourth kind of Penrose tiling?
It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There ...
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Partitioning a rectangle into different isosceles triangles
After all the discussion raised by this old question, I am wondering about a somewhat complementary one:
For any given rectangle, does there exist a finite set of pairwise different isosceles ...
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Covering the plane with line segments with local hexagonal constraints
Can we characterize the following kinds of plane coverings? (Open-ended, but provide some description more "useful" than the constraints given.) For a more answerable question, is there an effective ...
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Game theoretic aspects of Wang tiles?
Wang tiles are interesting in that they can simulate Turing machines. My question is whether anyone has studied their game theoretic properties?
In particular, we could imagine a game in which you ...
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Does one pieces of every kind of connected polyominoes P in $\mathbb{R}^2$ which has no hole cover a plane?
Or polyominoes with no hollow in $\mathbb{R}^3$?
I created this conjecture and tried to make counterexample, but it doesn't work well. Thank you for any answer or correcting question.
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How hard is it to tell when a finite set tiles the integers?
Given a nonempty set $B$ of integers between 1 and $n$, we wish to determine whether or not $\mathbb{Z}$ can be tiled with translates of $B$ (that is, covered by disjoint translates of $B$). I know an ...
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"Transcendental tilings": Do they exist?
Let $T$ be a tiling of the plane.
Fix an origin and shoot a ray $r$ from the origin.
Mark off points $p_i$ along $r$ separated by unit distance.
Compute from $r$ a binary number $0 < b(r) < 1$ ...
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Can a row of five equilateral triangles tile a big equilateral triangle?
Can rotations and translations of this shape
perfectly tile some equilateral triangle?
I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
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Number of labelings of symmetric hexagonal tilings P(a,b,c) with j descents
I am searching for the Number of labelings of symmetric hexagonal tilings
If the hexagon is of the form P(n,n,n) then the coefficients can be found here
A217311
I am looking for the coefficients of ...
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Periodic tilings of the plane by regular polygons
Let $A$ be a tiling of $\mathbb{R}^{2}$ using regular polygons. Assume that the tiling is edge-to-edge. Assume also that there are two directions of periodicity, so that $\mathbf{u},\mathbf{v}\in \...
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Space-tiling convex prisms
A convex prism is a subset of $\mathbb{R}^3$ congruent to the Cartesian product of a convex polygon (the prism's base) with the interval $[0,1]$.
Question. If a family of congruent convex prisms ...
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A class of tilings with amazing visual qualities
For more examples please see my related question on MSE:
Interesting tiling with a lot of symmetrical shapes
This is achieved by rotation of square grid over itself by atan(3/4).
Resulting ...
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Terrible tilers for covering the plane
Let $C$ be a convex shape in the plane.
Your task is to cover the plane with copies of $C$, each under any rigid motion.
My question is essentially: What is the worst $C$, the shape that forces the ...
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16-cell honeycomb (4D tiling by cross-polytopes)
A 4-dimensional cross-polytope (also called 16-cell) is a regular polytope whose vertices are all permutations of $(\pm1,0,0,0)$. It is known that this body tiles the space $\mathbb{R}^4$ by ...
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Tilings of the plane and meromorphic functions on the plane
This question has three up-votes on m.s.e. but isn't getting any answers.
Every textbook says every doubly-periodic meromorphic function on $\mathbb C$ has a fundamental domain that is a ...
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Decidability of (restricted) periodicity of Wang tilings
Consider a Wang tiling (given a subset of $C^4$ for a finite set $C$ of colours, e.g.). It's well-known to be undecidable whether there exists a tiling, and also whether there exists a periodic tiling....
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Minimal covers instead of tilings in Maxwell Allman's problem
The question I'm going to ask is inspired by this thread. I wonder what happens if instead of tilings we consider minimal covers, i.e., families of convex closed polygons that cover the square and ...
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Convex polygonal tiling of the square such that every line intersects at most k polygons
Consider a tiling of a square by convex polygons, such that every line through the square intersects at most $k$ polygons. Let $n$ be maximum number of polygons such a tiling can have. What is the ...
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Is this a new type of convex pentagonal tiling? [duplicate]
The following pentagon produces a tiling that does not appear to belong to any of the existing 15 categories:
Here's the tiling:
Specifically, it is not Type4 or Type6 because those are edge-to-edge ...
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How is the Penrose tiling decapod count of 62 calculated?
From Martin Gardner's 'From Penrose Tiles to Trapdoor Ciphers'
From page 14, Chapter 1;
https://www.maa.org/sites/default/files/pdf/pubs/focus/Gardner_PenroseTilings1-1977.pdf
"Any spoke of the ...
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Decidability of convex rearrangements of polygons
Triggered by the MO question,
"How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question:
Q. Given $n$ polygons in a set $S$, say each with integer ...
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Reference requests for tiling easiness [closed]
For Wang tile problem, is there some general statements in a paper stating that the more tiles (supposed provided by random) available, the easier it is for these tiles to tile the plane? Thank you.
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Torsion-free, normal subgroups of certain Coxeter groups
Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...
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Classification of symmetries of tilings in surfaces?
Is there a general study of the symmetries of tilings on surfaces?
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
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Tileable subsets of $\mathbb{Z}\times\mathbb{Z}$
For $t\in \mathbb{Z}\times\mathbb{Z}$ and $A\subseteq\mathbb{Z}\times\mathbb{Z}$ we set $t+A :=\{t+a: a\in A\}$.
Call $A\subseteq\mathbb{Z}\times\mathbb{Z}$ tileable if there is $T\subseteq\mathbb{Z}\...
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Complexity of $\mathbb{Z}^n$ tilings
Let $\mathcal{T} \subset \mathbb{Z}^n$ be a finite set. Let $\Lambda \subset \mathbb{Z}^n$ be a full rank lattice.
We say that $\mathcal{T}$ is a $\Lambda$-tile for $\mathbb{Z}^n$ if the following ...
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Integer Triples and Reflection tiling $1,2,\ldots,n$
$\forall a,b\in\mathbb Z,\ \exists n\in \mathbb N$ such that the numbers $1,2,\ldots,n$ can be tiled using translates of $\{0,\ a,\ a+b\}$ and $\{0,\ -a,\ -(a+b)\}$ ?
In other words for every integer ...
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Generating function for number of different tessellation checkered rectangle
Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$.
Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$.
$\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $...
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