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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Placing tetrominos in square, maximum size

I am currently coding an algorithm which places a list of Tetrominos (tetris pieces) in the smallest square possible. My question is : is there a mathematical way to know the maximum size (upper ...
2
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1answer
46 views

Pentagonal tiling

I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I ...
2
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1answer
76 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 ...
2
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1answer
84 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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1answer
29 views

Scalene rectangulation of a square: let me count the ways

A rectangulation of a square is a dissection of the square $S$ into smaller rectangles $R_i$, $i=1,\ldots,n$ with the usual caveats: $S = \cup_i R_i$ and the interiors of distinct rectangles ...
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1answer
242 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, that is a hexagonal lattice. How can I determine if a connected subset of these hexagons (i.e. a poly-hex) can tile the plane ...
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1answer
45 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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1answer
39 views

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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1answer
77 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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1answer
66 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 ...
0
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1answer
44 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
144 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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0answers
32 views

Plane tesselation, using stairs $n\times n$, is it possible?

The other day I was constructing new mathematical problems for my pupils and thought of something like this: Given the infinite sequence of "stairs" $n\times n$, constructed from $1\times1$ ...
5
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0answers
108 views

Graph Relatives for Tessellation of the Hyperbolic Plane

I'm trying to get into the theory about the Modular group. Among the "Paracompact hyperbolic uniform tilings in [∞,3] family" in the section "Tessellation of the hyperbolic plane" I found the Order-3 ...
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0answers
117 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 ...
4
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0answers
55 views

Is this a 16th type of a convex pentagon that can tile a plane?

As you can read here: http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem there are now 15 known convex pentagons, or ...
4
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0answers
44 views

When are convex polygon tilings Voronoi?

A square is divided into convex polygons. Is this always a Voronoi diagram? If not, what are some simple examples of non-Voronoi tilings? Which of the pentagon tilings are Voronoi? I took a look ...
3
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0answers
93 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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0answers
28 views

Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$ ($d=1.15096...$) that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
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0answers
44 views

How far can the plane be tiled by congruent regular pentagons?

What is the limit, as the radius of the disk increases, of the greatest area, in proportion to the area of the disk, of the region covered by regular pentagons of the same fixed size, all lying within ...
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0answers
71 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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0answers
60 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. ...
2
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0answers
86 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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0answers
51 views

Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
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0answers
91 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
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0answers
43 views

Filling a rectangle with congruent squares in two columns

I have a rectangle. This rectangle is divided into two columns; the widths of these columns are not necessarily equal, and are not known. I want to fill the rectangle with squares. The number of ...
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0answers
47 views

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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0answers
138 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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0answers
53 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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0answers
88 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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0answers
158 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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0answers
677 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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0answers
86 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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0answers
16 views

An infinite tiling problem

Assume we have an infinite area of hexagonal close tiling with hexagon side $s$ metres with corners of the hexagons marked by straight narrow trees of $d$ metres diameter and height $h$. Assume the ...
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0answers
19 views

Isohedral polygons that tile space in Voronoi tessellations

Besides the three uniform tilings of space in $\mathbb{R}^2$, are there other regular or semiregular tilings, by one or more types of isohedral polygon, that are also Voronoi tessellations of some ...
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0answers
27 views

Is the “Fibonacci square tiling” of Fibonacci-sided rectangles always optimal?

Is an optimal square tiling of a rectangle with side lengths of successive Fibonacci numbers always the sequence of Fibonacci numbers, as in the picture below?
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37 views

Tetromino coloring problem on a grid

Problem: All the tiles on an $n\times n$ grid are black or white. Every possible T-tetromino of tiles on the grid is examined. A T-tetromino is 'happy' if has 2 white tiles and 2 black tiles. ...
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0answers
22 views

Tessellations of Flower like Functions of the form k + Sin[mx]/n

When, if ever, does the shape of the polar plot of k + Sin[mx]/n from $0\leq x\leq 2 \pi$, where k,m,n $\in$ N form a monotiling in the Euclidean Plane? If some values of k,m, and n allow for this ...
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0answers
27 views

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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0answers
34 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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0answers
65 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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0answers
52 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 ...