Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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How to divide a square into three similar rectangles
Preparing some exercises for my High School pupils I came across this question: How can you tile a square into three similar (ie., same shape, different size) rectangles?
With a bit of algebra it can ...
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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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Tiling the plane with incongruent isosceles triangles
It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
...
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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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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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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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Lights Out game over GF(p)
On Jaap's Puzzle Page
http:// www.jaapsch.net/puzzles/lomath.htm#domtilings
Theorem 7 says:
If standard Lights Out is played on a m x n grid-like board, ...
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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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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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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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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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Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...
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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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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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"a shape that ... lies halfway between a square and a circle"
An article in the
Notices of the AMS, Volume 61, Issue 10, 2014
(PDF download link),
on Khot's Unique Games Conjecture, says this:
Another group ... found a
shape that in a certain sense lies ...
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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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Thinnest 2-fold coverings of the plane by congruent convex shapes
It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...
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(non-)existence of the aperiodic monotile
The aperiodic monotile problem asks whether there exists a single tile that every tiling of the plane made with it results non-periodic. What is known about this problem? If this tile exists, how can ...
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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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What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]
I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...
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Higher dimensional generalization of: Any quadrilateral tiles the plane?
Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., $\mathbb{R}^d$ ...
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Aperiodic set of corner Wang Tile [closed]
There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...
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What exact number of domino tilings cannot be realizable?
Inspired by some other questions, (this and this),
I wonder what numbers $n$ there are that satisfy
$$p(n)=\text{there is no region that admits exactly } n \text{ domino tilings}.$$
If this is true, $...
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What is known about tiling a rectangle in an irreducible way by smaller rectangles?
Given two naturals $s<t$. Is there always a square (or at least a bigger rectangle) that can be tiled with $s\times t$ rectangles in an irreducible way (i.e. any grid line splitting it cuts at ...
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Are there irregular tilings by L-polyominoes?
I wonder if one can tile the plane with an order-$n$ L-polyomino
in a fundamentally irregular manner.
I seek help in defining what should constitute "irregular."
An L-polyomino of order $n \...
- 151k
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Mixing time for dimers on the square-octagon graph
Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). It's been known empirically for twenty years that if one turns the set of ...
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Exact enumerations from two-dimensional stat mech models
Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge ...
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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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Arctic Circle Theorems and the Wave Equation
I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function $\mathcal{H}...
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Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...
user46976
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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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Periodic Tiling of Wang tile [duplicate]
Without considering any aperiodic tiling, is there any established (or better efficient) algorithms that tries to determine whether a set of tile can tile Wang tile periodically (or better determine ...
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A question about tiling Hilbert Space
Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
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Domino Shuffling and Warren's process
In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...
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Can a unit square be cut into rectangles that tile a rectangle with irrational sides?
For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...
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Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space?
In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose ...
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Can aperiodic tilings be non-hierarchical? and confusion over domino problem
Anyone experienced with the undecidability of aperiodic tiling?
It's related to the halting problem which Turing proved was undecidable in the 30's and basically superimposes tiles onto other tiles ...
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What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?
I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of Weaire–...
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Reference for Wang tile
I am working on projects in solving ground state of generalized Ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results:
...
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Unique domino tiling
Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset $S$ of the $xy$-plane is star-convex if there ...
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Can the sphere be partitioned into small congruent cells?
On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
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Question on Conway tilings
Conway in http://olympiads.mccme.ru/lktg/2009/4/articles/conway.pdf provided some elegant techniques for identifying tiling of simply connected regions. Are there similar techniques for regions that ...
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Recognizing parallelogram tilings from their vertex set
Suppose I have a tiling of the plane with parallelograms where the sides of the parallelograms come from a specified finite set of vectors. If I only have access to the vertices of this tiling I may ...
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Tiling a rectangle with weighted cells (min-max problem)
I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...
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2-layer tilings with a center-of-gravity constraint
I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...
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Poisson Summation Formulas for Cut and Project Quasicrystals
In Lagarias' paper "Mathematical Quasicrystals and the Problem of Diffraction" http://www.math.lsa.umich.edu/~lagarias/doc/diffraction.pdf he discusses various ways one might get Poisson summation ...
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Is there a Fourier transform for principal r-discrete groupoids?
I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...
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