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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7
votes
2answers
465 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
121 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 ...
1
vote
2answers
274 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
45 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
138 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
98 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
94 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
61 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
44 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
137 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
370 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
394 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
109 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
903 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
277 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
118 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
98 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
214 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.
-4
votes
1answer
204 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
votes
3answers
363 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
607 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
97 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
241 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
643 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
157 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
81 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
383 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
446 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
297 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
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
92 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
339 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
217 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
vote
3answers
389 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
143 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
529 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
418 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
votes
1answer
367 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
votes
1answer
148 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 ...
3
votes
3answers
869 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
257 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
482 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
votes
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
votes
2answers
198 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?