# Tagged Questions

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 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 ...
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### 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 "...
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### Maximal tiling without any 3-in-a-rows

You are given an arbitrarily large grid, where each square can either be off or on (think Game-of-life type board). You need to tile such a grid to maximize the number of "on" squares without there ...
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### 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 ...
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### Decomposing the plane into intervals

A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to ...
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### 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 ...
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### 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 ...
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### Penrose Tile generator

Does anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site? I found this http://stephencollins....
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### 3D picture of the 38-sided Engel space-filling polyhedron

On page 220 of Peter Engel's Geometric Crystallography, he describes a 38-sided convex polyhedron that can fill space. I've seen this this accepted as the record in various places, but I've never ...
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### 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 ...
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### 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 ...
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### Tiling of a $9\times 7$ rectangle

Can a rectangle $9\times 7$ be tiled by "L-blocks" (an L-block consists of $3$ unit squares)? Although the problem seems to be easy, coloring didn't help me. The general theory is interesting, but I'...
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### $2013\times2013$ 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 $L$ ...
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### If you know that a shape tiles the plane, does it also tile other surfaces?

For instance there is a hexagonal tiling of the plane. There is also one using quadrilaterals. It seems intuitive that both of these tilings also apply on a torus. Is it the case that anything that ...
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### 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 ...
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### The “XX XX” polyomino

The polyomino XX XX (two blocks of two squares with a gap) does not tile any rectangle, how to prove/disprove that it tiles the plane?
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### Tiling of squares in instances of Pythagoras Theorem

The Pythagorean Theorem (PT) states that in a right triangle, the square on the hypotenuse equals the sum of the squares on the legs or $a^2 + b^2 = c^2$. Is there an instance of PT (a given $a$, $b$ ...
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### Getting started with aperiodic tiling

I spent a little time looking around the Wikipedia and Wolfram articles on Penrose Tiling, the Domino Problem, Wang Tiles, etc., but I'm having a little trouble getting into them. A lot of these ...
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### Decidability of tiling of $\mathbb{R}^n$

Given a polytope of dimension $n$, is there some general way to determine if it can tile $\mathbb{R}^n$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Checkerboard-Coloring $\mathbb{Z}^2$

If every square of the unit square lattice in the plane is colored black or white according to a set of rules, is there a way to find the maximum asymptotic ratio $r_n$ of the number of black squares ...
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### 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 ...
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### Is there such a thing as the “edge-face dual” of a polyhedron, and is the “edge-face dual” of a cube a rhombic dodecahedron?

The dual of a polyhedron is a polyhedron where the vertices of one correspond to the faces of the other, and vice versa. Is there always a similar correspondence between a pair of polyhedra where the ...
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### Prime number proof for tiling a rectangle

The following theorem has many proofs, several of which are highlighted in this document. Whenever a large rectangle is tiled by rectangles, each of which has at least one integer side - the ...
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### 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 ...
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### Easy proofs of the undecidability of Wang's tiling problem?

Wang tiles are (by Wikipedia): "equal-sized squares with a color on each edge which can be arranged side by side (on a regular square grid) so that abutting edges of adjacent tiles have the same color;...
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### 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 ...
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### What type of aperiodic tiling is used by Turkish Airlines on their bathroom walls?

The walls and bulkheads of Turkish Airline flights are decorated with a pattern that appears to be some sort of aperiodic tiling. They are most prominent on the bulkheads of flights, and are also used ...
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### 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 $2$). ...
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### How do you build a square from this figure?

If you can use only this figure, what is the LEAST number of such figures that you can use to build a square ? You can use any isometry to build a square and it must be full from the inside.
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### 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?
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### 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, ...
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### Equilateral polygon plane tiling

From playing around with some toothpicks and peas, I think that it should be possible to prove that the plane cannot be tiled by a possibly infinite set of equilateral polygons with the same number of ...
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### Tiling arbitrarily large portions of the plane implies tiling the plane?

Suppose that one has a finite collection of types of (polygonal) tiles. We say that we can cover a region of the plane if we can place tiles in a non-overlapping fashion (except for edges of tiles ...
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### Can we cover the entire plane with the square with area 1/n for each positive integer n?

We have one square with area 1/n for each positive integer n. Is it possible to place these squares in the xy-plane in such a way that they completely cover the entire plane. If Yes, can you describe ...
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### 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 ...
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### 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, ...
### 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 $1\times1$ ...
Why can I not tile any rectangle without gaps with the given shape? http://i.stack.imgur.com/9oxO4.png You can mirror the shape (i.e. turn it around an axis in its own plane by $\pi$).