# 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 ...
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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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### 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? ...
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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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### 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 ...
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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 ...