Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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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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Random walk on a Penrose tiling
Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the
origin, or, equivalently, returns to the origin infinitely often.
It was subsequently established that in $\mathbb{Z}^3$, ...
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The sparsest planar net that captures every unit segment
Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...
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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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tiling a rectangle with the smallest number of squares
This is based on another thread. For $m,n\in \mathbb N$, let $f(m,n)$ be the minimum number of squares with integer sides needed to tile a $m\times n$ rectangle. Recently, a table of values for $n\le ...
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Is this an instance of any existing convex pentagonal tilings?
Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
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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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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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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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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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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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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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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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"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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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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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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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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Can every tromino (including those with gaps) tile the plane?
I've generalized trominos to include "gaps", i.e. they are formed by removing all but $3$ squares from an $n$-omino where $n$ is finite.
The generalized trominos pictured above can tile the plane ...
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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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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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Name this periodic tiling
I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were previously ...
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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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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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Tiling a square with rectangles
Is it possible to completely tile a square with different rectangles of integer sides but all with the same area?
The original problem, not requiring integer sides for rectangles, was proposed by Joe ...
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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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quasicrystal and penrose tiling, mathematical introduction
Starting to research on quasicrystal from material science, I want to know more about how to understand quasicrystal from a purely mathematical (especially tiling) perspective (probably start from ...
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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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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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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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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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Penrose tiling substitution is bijective
Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles.
It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...
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Radial tilings with variable area ratios
I was looking at this neat page on logarithmic spiral tilings when a question popped up:
http://www.uwgb.edu/dutchs/symmetry/log-spir.htm
It seems that in all of the tilings shown, the area of each ...
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Tiling by regular simplices
The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional ...
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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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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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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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Inequality among domino tilings of large triomino shapes
Inspired by this question, which asks for what shape maximizes the number of domino tilings, I want to ask the following seemingly simpler question, which I have been thinking about for a while:
...
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Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$
When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...
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Bicoloring of $\mathbb{N}^2$, avoiding set of patterns, is the maximal limit density rational?
Consider a bi-coloring of $\mathbb{N}^2$, (black and white), where we wish to maximize the limit (limsup) of the density of black squares in $[n] \times [n]$ as $n \to \infty$. Here, we identify each ...
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What can we learn from the newly discovered monohedral convex pentagonal tiling?
Wikipedia: https://en.wikipedia.org/wiki/Pentagonal_tiling#Stein_.281985.29_and_Mann.2FMcLoud.2FVon_Derau_.282015.29
Media coverage: http://www.theguardian.com/science/alexs-adventures-in-numberland/...
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For which sidelengths are there polyominos composed of three squares that tile the plane?
Given three naturals $a<b<c$. We consider polyominos, connected or not, which are composed of three squares of sides $a,b,c$.
How can one characterize all triples $a,b,c$ for which such a ...
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Aperiodic graphs
The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph $G$...
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slick-proof-of-trick-for-counting-domino-tilings
The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
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Tiling with restricted overlap
Non-overlapping tilings of regions is a well-studied topic.
I wonder if the following variant has been considered:
A tile can be partitioned into several regions, where such regions from different ...
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Soft question: mathematics about truchet tiles
It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...
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Translative packing constant strictly larger than lattice packing constant
Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...
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Consecutive Integer Squared Square
Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.
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Optimal planar net for catching convex shapes
Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...
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