Questions tagged [tiling]

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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 ...
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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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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 ...
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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 ...
28 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 ...
163 views

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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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 ...
202 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 ...
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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? ...
155 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 ...
111 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 ...
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Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I originally asked this on math.stackexchange where it was well received and we made some good progress. Here's ...
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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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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 ...
378 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 ...