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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3
votes
2answers
1k 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 ...
-1
votes
1answer
859 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.
85
votes
5answers
5k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
15
votes
2answers
439 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 ...
-1
votes
2answers
267 views

Tiling problem: 100 by 100 grid and 1 by 8 pieces

Why can't I tile a $100 \times 100$ table with $1$ by $8$ pieces? If we look at the number of tiles, $100^2$ is divisible by $8$. So this does not contradict existence of such tiling. The standard ...
13
votes
2answers
2k views

Tiling a 3 by 2n rectangle with dominoes

I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences ...
8
votes
5answers
1k views

Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?

I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that ...
12
votes
1answer
8k 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 ...
6
votes
3answers
999 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 ...
3
votes
1answer
2k views

Can a rectangle be cut into 5 equal non-rectangular pieces?

How to prove that the only figure of which 3 copies can be used to tile a rectangle is a rectangle? Is it possible to cut a rectangle into 5 equal (modulo rotations/reflections) non-rectangular ...
6
votes
2answers
249 views

Squaring the plane, with consecutive integer squares. And a related arrangement

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would ...
6
votes
3answers
322 views

uncountable number of Penrose tiling

I am interested in Penrose tiling, though I am not an expert, so I apologize in advance. It is known that that the number of different Penrose tilings is uncountable. This may be implied from de ...
16
votes
1answer
2k views

What is the mathematical significance of Penrose tiles?

I have a very limited understanding of groups and symmetry gained mostly from online videos (for eg. this one), so forgive me if this sounds ignorant. Particular parts of Penrose tilings exhibit ...
31
votes
1answer
2k 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 $C(p)...
13
votes
1answer
2k 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 ...
7
votes
3answers
542 views

Decidability of tiling of $\mathbb{R}^n$

Given a polytope of dimension $n$, is there some general way to determine if it can tile $\mathbb{R}^n$?
4
votes
1answer
394 views

Fill the board with dominoes

Imagine a 8x8 cell board that is missing two cells at the opposite corners, a domino takes up exactly two cells. How can you fill the board with dominoes so that none overlap or hang off the edge?
3
votes
1answer
466 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 ...
2
votes
1answer
378 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, ...
4
votes
2answers
345 views

Tiling Posters on a Wall

I'm a noob, and I'm not a mathematician (Although I will be a Math major next semester). My question is: I have 68 maps I would like to use as posters on my wall at home. They are all rectangles, ...
10
votes
3answers
687 views

a tiling puzzle/question

My teacher gave us a riddle that goes like this: You have a 7x7 square and 16 3x1 tiles. Of the 16 tiles, 15 are straight and 1 is crocked ("L" shaped). When you tile the square with these tiles you ...
5
votes
1answer
627 views

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
5
votes
2answers
463 views

An Olympiad Problem (tiling a rectangle with the L-tetromino)

An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks, ...
3
votes
2answers
77 views

non-trivial non-repetitive aperiodic tiling of the plane

Which is the less trivial example of non-repetitive aperiodic tiling of the plane you know? I cannot come up with a famous non-repetitive tiling. Are there any? A tiling is repetitive if for every ...
2
votes
1answer
67 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
3
votes
1answer
140 views

Not understanding this proof in Grünbaum-Shephard's Tilings and Patterns

I'm reading Grünbaum and Shephard's Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). ...
1
vote
1answer
36 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 $R_i,R_j$...
1
vote
0answers
161 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
2answers
54 views

How many face we could make regular convex polyhedron

I want to tile the sphere as many face as possible. And I want every face be the same size and shape. Is it possible to generate more than 100 or 1000 faces of regular convex polyhedron?