Use this tag for questions about (not necessarily periodic) tilings of metric spaces, their combinatorial, topological and dynamical properties, as well as basic definitions and concepts.

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What is the dual lattice of Kagome lattice?

We know that the dual lattice of a triangular lattice is the honeycomb lattice. What is the dual lattice of Kagome lattice?
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Covering the plane with convex polygons?

I have got the following task here: Prove, that you can't cover the "Plane" with convex polygons, which have more than $\,6\,$ vertices! The answer is pretty obvious for $\,n=3\,$ vertices, because ...
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47 views

Semiregular tilings of the hyperbolic plane

Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid: I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where ...
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56 views

How can I compute angles & lengths of the following tiling shape

a while back I created a tile made of arrows: I did it using a vector graphics software, without really understanding the properties of this shape. Now, let's say I want to write a program to ...
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33 views

What are the parameters for the edge-match-restricting arcs drawn on Penrose tilings?

I would like to try generating some computer art involving Penrose tilings. I'm looking into the layouts algorithms separately, this question concerns the decoration. Here is a schematic of P2: No ...
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1answer
120 views

What are the simple Heesch-2 polyforms?

At the Tiling Database: There are 3, 20, 198, 1390 non-tiling polyominoes of order 7 to 10. There are 4, 37, 381, 2717 non-tiling polyhexes of order 6 to 9. There are 1, 0, 20, 103, 594, 1192, 6290 ...
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98 views

Tilings of the Hyperbolic plane

Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions: $f(r) =$ number of graph nodes contained within the a ...
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53 views

Integer hexagonal grid variations for Harborth

Harborth's conjecture states that every planar graph has a planar drawing in which all edge lengths are integers. I was looking at that, and I wondered what was known about hexagonal grid graphs. For ...
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312 views

Squaring rectangles

it is a nice high-school exercise to prove that a square can be tiled with n squares if and only if n=1, 4 or is any integer greater or equal to 6. A direct consequence is that any rectangle that can ...
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Tiling a rectangle and tensor products

Consider the following theorem: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. There is a paper ...
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54 views

How get closest vertex in triangular tiling from coordinates on plane?

Currently I have a plane with square tiling. It pretty trivial to get point on plane mapped to vertex of square tiling: plane point (x; y) -> vertex of square tiling (x div A; y div A). How to get ...
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49 views

Given 4 tile types, what are the chances that there are no sets of 3 in a 6x6 array?

I know this question seems arbitrary, but it actually applies to a matching game that I'm writing. I randomly typed the following letters and created a 6x6 array using the letters A, S, D, and F. ...
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71 views

Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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Tiling with exactly 2013 different ways

A rectangle with side lengths integers $a$ and $b$ will be covered with tiles, rectangular with a length of one side so that a portion of the rectangular area will be covered with black tiles and the ...
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26 views

Program for playing around with aperiodic tilings

I know this is technically a software question, but I figure the mathematicians here would know more about it: I'm looking for software to play around with different "paintings" (different patterns ...
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51 views

penrose tilings and symmetry

After going through the following question on Penrose Tiling and reading de Bruijn's papers on the subject, I came accross Grünbaum and Shephardbook "Tilings and Patterns", p. 543, where they say that ...
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111 views

How many subsets of NxM rectangle are tileable by dominos?

There are a lot of articles, formulas and algorithms for the number of domino tilings for some region, but I couldn't find anything about number of tileable regions. Is there any exact formula or at ...
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40 views

What are interesting ways to tile a square image? Or a transformation that make an image tilling-able?

I want a method to tile arbitrary square image. For most cases the boundaries do not agree. So I are looking for a transformation from a square image to a square image whose boundaries agrees. One ...
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83 views

What is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree 2, 3,..., n (i.e. a tile of an arbitrary degree) would be? Asked another way- what is it that ...
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130 views

Tiling an L-shape with “almost square”s

ABSTRACT: Define an "almost square" as a rectangles with aspect ratio in $[{1 \over 2},2]$. What is the minimal number of interior-disjoint almost-squares required to tile the following L-shape (where ...
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488 views

Generating evenly distributed points on a sphere

How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
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85 views

Minimial Tiling Problem on a sphere

This question is a revision of the math exchange post found here. Consider the following: A sphere, $S$, with radius $r_1$. N regions projected onto $S$, whose projections, $\left\lbrace E_i ...
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22 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
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Is there any two edge colour tiling of the plane with regular polygons?

Is it possible to tile the plane with regular polygons such that every edge is one of two colours, and no two adjacent edges are the same colour?
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21 views

Determine if a rectagle is fully “compatible” with a given Polyomino

Recently, I came across a unique problem for which I couldn't find a complete solution. I want to determine if a given rectangle is fully "compatible" (for the lack of a better word, please suggest ...
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enumerating paths in a hex tiling

assume a hexagonal tiling in the shape of a circle with radius $r$ nodes. given two nodes on the perimeter, $a$ and $b$, such that $a \ne b$, a set of paths $P_{ab}$ connect the two. what is the total ...
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49 views

Tile a Box With Translates of Two Given Rectangular Bricks

What is the layman explanation for the theorem explained in this paper? Lets say I have a rectangle of $25 \times 25$. What bricks $B_1(1 \times a)$, $B_2 (1 \times b)$ will be able to completely tile ...
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46 views

Tile a spline with bricks

I have a series of identically sized virtual Lego bricks. I need to "tile" these bricks along the length of a Hermite spline to create a curved road. The spline is two-dimensional. E.g. the curve only ...
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184 views

Determining if a set of hexagons on a grid can tile the plane

Suppose I have a regular grid of identical hexagons that tile the plane. How can I determine if a connected subset of these hexagons can tile the plane by translations that form a (Bravais) lattice? ...