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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2
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0answers
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 ...
1
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0answers
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 ...
2
votes
1answer
228 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
1
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0answers
126 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 ...
0
votes
0answers
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 ...
13
votes
2answers
320 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
2
votes
1answer
65 views

Does the fact that a tiling is tile-uniform always guarantee that it is also vertex-uniform?

It seems to me that if a tiling is tile-uniform, then it must be vertex-uniform as well. But is this the case? How would one go about devising a proof? By 'tile-uniform', I mean a tiling whose ...
1
vote
1answer
80 views

Polygonal tilings: differentiating between tiles and tilings, and their edges and vertices?

I'm just starting to study tilings in a groups and geometry module, and I'd like some confirmation of my understanding of precisely what it is which differentiates a single tile, from a tiling- when ...
1
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2answers
109 views

Dicrete Math Interesting question about Tromino

Prove that for a m$\times$n rectangle, if this rectangle can be covered completely by trominoes of the shape indicated in the picture, then mn is divisible by 3. I came up with a tentative way to ...
5
votes
6answers
441 views

Covering a chess board with $2$ missing places with $31$ dominoes

I am reading a book that is intended to a wide audience (and not just mathematicians etc'), the book is, of course, about mathematics (Its still not clear about what exactly, I am only in page $2$). ...
5
votes
0answers
95 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 ...
2
votes
1answer
45 views

In search of a symmetric homogeneous graph with a pivotal origin

I'm trying to design a computer game and I need a symmetric homogeneous graph with a pivotal origin which will act as the map of the game (players will walk according to it). Here's an example of ...
3
votes
0answers
296 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 ...
6
votes
1answer
3k views

Algorithm to get the maximum size of n squares that fit into a rectangle with a given width and height

I am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of ...
10
votes
1answer
192 views

$2013\times2013$ Board with no trominoes.

Let A be a $2013\times2013$ board with $k$ black squares and containing no $L$ shaped black trominoes(in any rotation) and such that if any white square is dyed black then $A$ contains a black $L$ ...
-1
votes
1answer
431 views

Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino

Suppose we have an $2^n$x$2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely. You can mix different rotations in the same tililng.
7
votes
2answers
568 views

Dissection puzzle for area 49 to area 50

49 and 50 are close, as are 288 and 289. That allows a grid illusion. If cut out of wood, perhaps with coloring on the border as an "assistance", the pieces could be dumped out of the tray, flipping ...
0
votes
1answer
126 views

“Simmetric” connected k-regular bipartite graph

Let $G$ be a k-regular bipartite graph with $k > 0$. Then it is known that the two sets which partition the vertex set of $G$ have the same cardinality. However I am interested in connected ...
2
votes
2answers
353 views

Pictorial puzzle

Can someone suggest a pictorial representation of a mathematical puzzle or problem that could be represented in coloured tiles on an orangery floor $4489\times7525$mm with an insert of $907 ...
2
votes
0answers
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. ...
7
votes
1answer
150 views

Tiling an $n\times n$ Grid

Given an $n\times n$ grid, and $2\times 2$ checkered tiles (white in the upper left and bottom right corners, and black in the upper right and bottom left corners), what is the smallest number of ...
1
vote
1answer
112 views

Repetitive tiling implies finite local complexity

My question probably needs to include the definitions of the terms in the title so I will first ask the question and then introduce the necessary definitions. The following Theorem is stated without ...
5
votes
1answer
114 views

Giant Pufferfish skin pattern―how could that be generated

I just started my investigations about tesselations and tilings for some very special kind of design Project. I came over that image: It shows a part of the Giant Pufferfish's skin and I am very ...
1
vote
0answers
67 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 ...
2
votes
1answer
45 views

Group partition with sets.

the following questions come from tiling problems. Maybe the answer is easy but so far I don't know how to start as I can't see whether I have to prove that it is true or to find a counter-example. ...
0
votes
0answers
180 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? ...
7
votes
1answer
470 views

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the ...
1
vote
0answers
466 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' ...
3
votes
2answers
122 views

Substitution tilings with parallelograms

I'm looking for a substitution tiling made with parallelograms, that is, a tiling of the plane with parallelograms (which do not have to be of the same shape) such that we can take one parallelogram ...
13
votes
1answer
1k views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
2
votes
2answers
305 views

Prove or disprove a chessboard with diagonal corners removed, cannot be tiled with L shape pieces or size 2

I think this is impossible, but I don't know how to prove an integer solution doesn't exist for a given equation. Here's my approach: First, observations: The removed tile will be of the same color. ...
14
votes
3answers
1k views

Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.

Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that ...
2
votes
1answer
130 views

Tiling a minimal perimeter region with $n$ unit squares

Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square. Is ...
5
votes
2answers
102 views

primality on tiles?

Call $S_n$ the square of area $n^2$. See it as a collection of $n^2$ unit squares. In the following, what I call tile is a collection of unit squares that are glued together. If $n$ is not prime, say ...
2
votes
1answer
239 views

Aperiodic hexagonal tiling?

Is there any known aperiodic tiling of the plane using hexagons? Wang tiles are a known aperiodic tiling using squares. I'm looking for something similar using hexagons.
7
votes
3answers
389 views

How to put eggs in baskets

A farmer has c chickens who have each laid e eggs, which she will put into b baskets. Each basket has a probability p(d) of being dropped, which breaks all the eggs in the basket. How should the ...
6
votes
3answers
740 views

When chessboards meet dominoes

You probably have heard about the following brainteaser : Consider a 8×8 chessboard. Remove two extreme squares (top-left and bottom-right e.g.). Can you fill the remaining chessboard with 1×2 ...
1
vote
1answer
100 views

Explicit formula for the position of “bent wedge” tiles

Is there an explicit formula to compute the position and the angle of the $n$th tile of a bent wedge tiling ?
2
votes
1answer
768 views

How to tile a sphere with points at an even density?

I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon ...
0
votes
1answer
165 views

A new periodic tiling of the plane

It is known if we use two convex polygons with equal sides we can cover the plane periodically in few ways. One new way to cover the plane periodically is if we use rhombuses and octagons of equal ...
1
vote
0answers
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 ...
11
votes
1answer
431 views

Covering points on a sphere with a disk

Suppose $m$ points ("sites") are selected on the unit sphere $S^2$. For a given radius $r < \pi$, we can define a disk around any point on the sphere as the set of points at geodesic distance at ...
1
vote
1answer
544 views

Hexagon into 12 identical hexagons

Puzzle: Divide a regular hexagon into 12 identical non-convex hexagons. I found this at Jaap Scherphuis' Tiling Applet, and it looks new to me. Are there any solutions other than the one answer ...
7
votes
2answers
305 views

Tiling pythagorean triples with minimal polyominoes

Given a Pythagorean triple $(a,b,c)$ satisfying $a^2+b^2=c^2$, how to calculate the least number of polyominoes of total squares $c^2$, needed, such that both the square $c^2$ can be build by piecing ...
28
votes
1answer
1k views

Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
13
votes
3answers
2k views

how to generate tesselation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's ...
0
votes
1answer
95 views

Tiling the Cantor space

Given the Cantor space $2^\mathbb{N}$, and a set of disjoint open sets $D$, are there any non-trivial upper bounds on the number of further open sets needed to complete the tiling of the space?
3
votes
1answer
359 views

Is there a way to tessellate an area using triangles and minimize/specify the number of unique triangles?

Is it possible to tessellate a planar surface from triangles but with the following constraints: density (average number of triangles) can be varied. a finite set of unique triangles are used for ...
1
vote
1answer
245 views

Cube nets hexomino tilings.

I am looking for an ~12x12 rectangle (small holes and small obtrusions are okay) made entirely of cube net hexominos. It is my understanding that perfect rectangles, in general, are not possible ...
5
votes
1answer
4k views

How to Divide a Square or Rectangle into squares of different sizes?

For an art project I want to compute a division of space into random squares. I have a number of applications where this would be a pleasant visual layout, I'm trying to figure this out for myself, ...