Questions tagged [tiling]

For questions about mathematical tiling.

Filter by
Sorted by
Tagged with
15 votes
2 answers
718 views

Tiling survey that updates "Tilings and patterns"?

Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one. I am ...
3 votes
1 answer
222 views

Is there a formula for a number of one-sided N-ominoes?

As we all know, Polyominoes are shapes which consist of certain number of squares connected together. A famous videogame - Tetris - has a gameplay based around tetraminoes - polyominoes with 4 squares ...
4 votes
0 answers
131 views

Chromatic number of rectangle tilings

Suppose we have a region of the plane tiled by finitely many rectangles. We want to color the rectangles so that two rectangles have different colors if they share a part of an edge or if they share ...
2 votes
1 answer
78 views

What is the average component size of a coloring?

Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
1 vote
1 answer
196 views

Tiling a square with similar non-congruent rectangles. What is the aspect ratio of the rectangles as n grows large?

I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a ...
3 votes
2 answers
300 views

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 ...
1 vote
0 answers
2k views

How to fill a rectangle with smaller rectangles of given sizes?

I have a problem. I try to find an algorithm to fill up a given rectangle with smaller ones. Something like in this picture: I know the size of the big rectangle, the size of all the little ...
4 votes
0 answers
156 views

Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!

Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...
2 votes
1 answer
146 views

Packing densities of non-centrally symmetric planar convex regions

Reference: https://en.wikipedia.org/wiki/Smoothed_octagon Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...
20 votes
4 answers
2k views

Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?

Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical? It is a nice exercise with rational generating functions (or equivalently, ...
2 votes
0 answers
62 views

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 ...
1 vote
0 answers
362 views

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 ...
2 votes
0 answers
127 views

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/...
3 votes
1 answer
171 views

Construct by compactness (Pentagonal tiling – Rao paper)

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that: "... We keep a connected component $H_d'$ of $H_{d}$ such ...
5 votes
2 answers
297 views

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 ...
24 votes
1 answer
1k views

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 ...
2 votes
1 answer
274 views

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 ...
27 votes
3 answers
12k views

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 ...
0 votes
0 answers
87 views

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 ...
5 votes
1 answer
261 views

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 ...
5 votes
0 answers
137 views

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 ...
1 vote
0 answers
49 views

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 ...
79 votes
6 answers
4k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
14 votes
0 answers
379 views

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 ...
4 votes
0 answers
91 views

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. ...
4 votes
1 answer
162 views

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 ...
15 votes
1 answer
927 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 \...
0 votes
0 answers
137 views

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 ...
0 votes
0 answers
36 views

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 ...
3 votes
0 answers
81 views

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 ...
4 votes
1 answer
271 views

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 ...
9 votes
6 answers
600 views

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? ...
3 votes
1 answer
202 views

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 ...
4 votes
1 answer
130 views

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 ...
3 votes
1 answer
77 views

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 ...
1 vote
1 answer
202 views

Overlaying two domino-like constructions such that all individual pairs of domino-like cells in the overlay have matching symbols

Imagine I have two $n$ x $m$ assemblies of $P = (n*m)$ unit square cells on the plane, $(c_{(a,1)}, ..., c_{(a,P)}) \in A$ and $(c_{(b,1)}, ..., c_{(b,P)}) \in B$, where every cell, $c_k$, in a ...
4 votes
2 answers
556 views

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 ...
9 votes
2 answers
896 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 (...
1 vote
1 answer
113 views

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. ...
35 votes
5 answers
3k views

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 ...
4 votes
1 answer
1k views

What Islamic tiling patterns are constructible?

Eric Broug in his book Islamic Geometric Patterns gives straightedge and compass construction of some simpler patterns. It is clear his techniques will provide constructions for many Islamic patterns. ...
5 votes
3 answers
416 views

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 ...
4 votes
1 answer
316 views

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 ...
1 vote
0 answers
358 views

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 $n=1$, ...
2 votes
0 answers
140 views

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 ...
7 votes
1 answer
302 views

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 ...
3 votes
0 answers
137 views

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 ...
12 votes
2 answers
438 views

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 ...
11 votes
1 answer
481 views

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 ...
7 votes
3 answers
954 views

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?