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1
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0answers
48 views

Maximal product of rectangular tiles [closed]

Description: Given a rectangle that is $R_w$ by $R_h$, tile it with $n$ rectangles such that the product of the areas of the tiles is maximal. Let $R_w$, $R_h$, and $n$ be natural numbers. Let ...
14
votes
2answers
1k views

“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 ...
4
votes
0answers
108 views

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 ...
6
votes
1answer
219 views

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 ...
4
votes
2answers
109 views

(non-)existence of the aperiodic monotile

The aperiodic monotile problem asks whether there exists a single tile that every tiling made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? ...
6
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1answer
111 views

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 ...
-2
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1answer
88 views

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 ...
15
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1answer
222 views

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., ...
5
votes
1answer
122 views

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 ...
10
votes
3answers
746 views

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, ...
5
votes
2answers
181 views

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 ...
9
votes
1answer
327 views

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 $n \ge 2$ is ...
2
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0answers
70 views

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 ...
8
votes
1answer
119 views

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 ...
8
votes
1answer
173 views

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 $\mathbb{Z}^3$, ...
1
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0answers
102 views

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 ...
9
votes
3answers
356 views

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 ...
3
votes
1answer
215 views

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 ...
1
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0answers
76 views

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 ...
9
votes
1answer
284 views

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 ...
3
votes
1answer
122 views

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 ...
22
votes
3answers
2k views

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 ...
3
votes
1answer
116 views

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 ...
0
votes
1answer
143 views

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 ...
2
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0answers
47 views

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 ...
4
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0answers
294 views

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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1answer
557 views

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 ...
1
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1answer
78 views

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 ...
2
votes
1answer
125 views

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 ...
3
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0answers
228 views

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 $J$ ...
6
votes
1answer
136 views

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 ...
5
votes
1answer
310 views

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 ...
2
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0answers
54 views

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 ...
1
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0answers
116 views

Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size

Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares of length ...
4
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3answers
551 views

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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0answers
265 views

Relation Between Truncated Braid Groups and Regular Tilings of the Complex and Hyperbolic Plane

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question on math.SE. There exists a rather remarkable ...
5
votes
1answer
244 views

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 ...
2
votes
1answer
368 views

Minimizing the Perimeter of a polyomino

Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...
1
vote
1answer
270 views

Frustrating the number of possible common edges between two connected components composed of square Penrose tiles

Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...
3
votes
1answer
373 views

tiling a rectangle with squares: how unique are the minimal solutions?

This is a follow-up of my recent thread about tiling a $m\times n$ rectangle with squares: I'm wondering to what extent a minimal tiling is essentially unique, that is, up to reflections of the whole ...
19
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0answers
574 views

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 ...
2
votes
1answer
280 views

What are some properties of Delone sets that come from Barlow packings of spheres?

Given a Barlow packing of $\mathbb{R}^n$ by balls with at most a finite number of different radii, the centers of the balls will form a Delone set in $\mathbb{R}^n.$ For a highest density sphere ...
1
vote
1answer
147 views

Cut and Project Sets Using Hyperbolic Space

One strategy for creating aperiodic sets in $\mathbf{R}$ is to take a line $L$ of irrational slope in $\mathbf{R}^2$ along with a compact window $W \subset \mathbf{R}$ which is thought of as a subset ...
13
votes
1answer
318 views

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 ...
8
votes
3answers
796 views

What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little. Background Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...
19
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1answer
639 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
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1answer
215 views

one-dimensional (sort of) tilings

Consider the following one-dimensional tiling problem. Each "tile" is a sequence of nonnegative integers. A "region" is also such a sequence. I can shift the "tiles", or reverse them. A tiling is ...
6
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0answers
173 views

Possible structures for minimal tiling sets

Inspired by Col. Sicherman's results here, my speculations have so far outrun my expertise that I thought I might pass my question along to others who might find it equally intriguing, but perhaps ...
12
votes
7answers
1k views

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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2answers
646 views

Fractal Tiling of Rhombic Dodecahedra

Hello, this is my first question on Math Overflow... Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres. Consider a "nucleus" ...