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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0
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0answers
123 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
326 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
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0answers
346 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
1answer
98 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 ...
12
votes
1answer
622 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
261 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 ...
1
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1answer
93 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
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2answers
97 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
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1answer
199 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.
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1answer
201 views

Tiling with three vetrices made from squares and equilateral triangles

Does anyone know how to tile the plane with squares and equilateral triangles and three vertices?
7
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3answers
345 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 ...
5
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3answers
534 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
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1answer
94 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
236 views

Another new periodic tiling of the plane from Pythagorian triplets

Lets have a square with sides equal to a=13. From this square we can construct a dodecagon and an octagon which cover the plane. Please see the diagram below. If you prefer all the polygons to be ...
2
votes
1answer
569 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
153 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
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0answers
80 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 ...
10
votes
1answer
367 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
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1answer
415 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 ...
6
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2answers
293 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 ...
12
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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
91 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
328 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
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1answer
212 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
3k 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, ...
1
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3answers
377 views

Tiling with polyominos

How to prove or disprove that if a polyomino tiles the plane, it must also be able to perfectly tile some larger polyomino, which also tiles the plane? A polyomino is finite set of unit squares ...
2
votes
1answer
142 views

Optimal polyomino induced coloring

Which polyominos (with orientation) of $n$ squares, requires the least number of different colors, $c(n)$, such that if this polyomino is placed anywhere on an optimally colored infinite square grid ...
5
votes
4answers
507 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 ...
3
votes
1answer
120 views

Planar kelvin problem

What is the minimal possible value of the maximal total sidelength shared by any two tiles in a tiling of the plane if all tiles have the same area A? total sidelength = Length-integral of the curve ...
6
votes
2answers
416 views

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 ...
4
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1answer
346 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?
4
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1answer
147 views

CH for tilings of the plane

Given any set of jordan curves that can tile the plane, how to prove that the number of possible tilings using tiles from this set is either in bijection with the real numbers or a (possibly infinite) ...
1
vote
1answer
125 views

Tile $\mathbb{R}^n$ with Primitive Cuboids

For every integer $n$ with $i$ prime factors associate a unique tile in $\mathbb{R}^m$ with $m \ge i$ as such, for every prime factor $p_j$ of $n$, the tile is a cuboid of dimension $m$ with a ...
2
votes
3answers
738 views

Summation in a recurrence relation

edited to reflect advice from the comments: While working on a generalization of a tiling problem, I generated a recurrence relation to describe the total number of possible tilings. The relation ...
2
votes
1answer
242 views

fill the board with dominos

Imagine a 8x8 cell board, but missing two cell at the opposite corners, a domino take up exactly two cell, how to fill the board with dominoes so that none overlap or hang off the edge?
3
votes
1answer
1k 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 ...
11
votes
2answers
469 views

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 ...
8
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1answer
193 views

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?
2
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2answers
193 views

Status of single aperiodic tile

What is known about the existence of a single tile, that tiles R^n only aperiodically? Has such a tile been found/proven to exist/not exist for any R^n?
6
votes
2answers
235 views

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 ...
75
votes
4answers
4k 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 ...
5
votes
1answer
852 views

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 ...
10
votes
3answers
411 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 ...
1
vote
1answer
149 views

Truncated octahedron is bipartite. Proof?

Any idea how to proof that when 3D space is tiled with truncated octahedra, all vertices can be colored black and white such that no 2 vertices sharing the same color are adjacent?
2
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1answer
245 views

Truncated octahedron tiles 3D space. Proof?

where can I find a proof that truncated octahedron tiles Euclidean 3D space?
7
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2answers
894 views

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 ...
12
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2answers
433 views

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 ...
6
votes
2answers
221 views

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 ...