Questions tagged [tiling]
For questions about mathematical tiling.
295 questions
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Rigid monohedral tilers
Say that a tile $T$ that alone can tile the plane—a monohedral tile—is rigid
if it is not the case that $T$ can be slightly deformed to $T'$ so that:
$T'$ can also tile the plane
$T'$ is arbitrarily ...
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Which polygons tessellate the hyperbolic plane?
The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
It is well known that in Euclidean geometry, all triangles and all ...
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Are there any convex pentagonal rep-tiles?
A rep-tile is a shape that can tile larger copies of the same shape.
Question 1: Are there any convex pentagons that are also rep-tiles?
Remarks: 15 convex pentagonal tiles of the plane are known and ...
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Distribution over Penrose Tilings?
The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow ...
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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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Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?
A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...
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Objects in bijection with integer partitions (and lattices)
A partition of $n$ is a non-increasing sequence of positive integers of sum $n$. Several lattices are defined over integer partitions, in particular the dominance order and the Young lattice.
Several ...
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Cutting polygons into mutually similar and non-congruent pieces
It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) - for example, see https://en.wikipedia.org/wiki/...
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Fair cutting of the plane with lines
An infinite countable family $\cal{L}$ of straight lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:
$\bullet$ No circle intersects ...
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On Covering a Planar Region with Copies of a Tile of Different Shape
Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...
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Random domino tilings: Is this distribution well-defined, and how can it be sampled from?
I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means.
My first instinct was to do ...
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If two convex polygons tile the plane, how many sides can one of them have?
The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula ...
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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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Maximum number of colors for an optimal tiling which guarantees infinite paths
This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...
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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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How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?
When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...
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Possible cardinalities of spherical tiling
Suppose that we have a tiling of $n$-dimensional (I want to get answer for $n = 4$, but general result would be nice!) sphere by isometric tiles strictly contained inside the right-angled simplex. ...
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Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
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Absolute and relative tilings of the hyperbolic plane
In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it).
The ...
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Graph theory: Closed neighourhoods and generalized clustering coefficients
The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$.
The number of edges between neighbours divided by the number of pairs of neighbours is ...
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Vertex configuration to tile repeat unit
I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
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For what n and t can a square be partitioned into n similar rectangles in t congruence classes?
It is known that a square can be partitioned into three similar rectangles, all mutually non-congruent. I don't think it's possible with four. With what numbers of rectangles can this be achieved? And ...
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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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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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Distance spectra of uniform tilings
Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that ...
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Structures for random graphs with structure
Background[You may skip this and go immediately to the Definitions.]
Crucial features of a (random) graph or network are:
the degree distribution $p(d)$ (exponential, Poisson, or power law)
the mean ...
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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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Monotile that tiles when you apply a rubber band
My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...
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Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
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What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
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Domino tiling obtained from space-filling curves, is possible to predict basic properties?
Periodic and aperiodic domino tiling systems can be obtained by the following construction rules:
Draw a regular square grid n×n of n2 cells.
Select a space-filling curve that is consistent with ...
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Is there a rectangular tiling based on the Padovan sequence? [closed]
I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. ...
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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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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\...
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Can local flip moves connect dimer matchings on 'quadrangulated' planar bipartite graphs? (perfect matching reconfiguration problem)
I'm interested in the structure of dimer matchings on planar graphs with a bipartite structure. In particular, I'm interested in whether any two perfect matchings can be connected, i.e. transformed ...
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Generalizations of classical tiling problem
A classic problem using an inductive construction is to show that the $2^n \times 2^n$-square, with a missing corner, can be tiled with L-triominoes.
The proof goes like this:
It is true for $n=1$, ...
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Triangling the triangle
Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case ...
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Tradeoffs in translation-invariant tilings of $\mathbb{R}^3$
Suppose I tile $\mathbb{R}^3$ in a ($\mathbb{Z}^3$-)translation-invariant manner. If we insist on the tiling being regular, then we are left with only the cubic tiling. However, suppose that we ...
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Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?
This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
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Changing tiles by swapping squares
In an $n\times n$ table, initially there is a $1\times n$ tile in each row. A swap is an operation that involves choosing two tiles, move one square from the first to the second tile, and ...
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Aperiodic tile with rational area
Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
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Random Walk on Pentagonal Tiling
I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
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Graphs determined by monohedral, edge-to-edge tilings of the plane
Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
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Unbalanced colourings of Penrose tiles
It is known that both the rhombus and kite-and-dart Penrose tilings are three-colourable, from
Babilon, Robert. "3-colourability of Penrose kite-and-dart tilings." Discrete Mathematics 235, no. 1-3 (...
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Tiling rectangles using all squares of sides 1, 2, 3, ..., n
Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle?
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Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
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Tiling the surface of a hypersphere with regular simplices
Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...
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Hooks, monomers, dimers and Young diagrams: Part II
As promised, I've upgraded my last question.
Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...
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Hooks, monomers, dimers and Young diagrams: Part I
Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it.
Consider the one-line partition $\lambda_n=(n)$ and its ...
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monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
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