The tiling tag has no wiki summary.
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1answer
44 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 ...
0
votes
2answers
80 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
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0answers
19 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.
...
5
votes
1answer
91 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 ...
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0answers
33 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
54 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 ...
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0answers
27 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 ...
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0answers
28 views
The number of vertices, edges and faces(as in a tiling) inside a circular region is proportional to the area.
I'm trying to prove that the Euler characteristic of a surface generated by modding out R^2 by a wallpaper group has characteristic zero.
Conway in his book gives a proof of this, which uses the fact ...
2
votes
1answer
34 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.
...
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0answers
43 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? ...
5
votes
1answer
149 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 ...
0
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0answers
127 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' ...
2
votes
1answer
53 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
296 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
139 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. ...
12
votes
3answers
947 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 ...
0
votes
1answer
66 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
83 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
124 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
159 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
258 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
votes
4answers
325 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
65 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 ?
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1answer
204 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 ...
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0answers
48 views
Minimal Tiling versus Sphere Packing
Are there any significant differences with sphere packing and minimal tiling?
I have been researching the problem for a good while and I have yet to come up with a good answer as to any significant ...
1
vote
1answer
303 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
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1answer
121 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 ...
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0answers
51 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 ...
9
votes
1answer
226 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
189 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
votes
2answers
264 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 ...
25
votes
1answer
849 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 ...
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votes
3answers
595 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
82 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
224 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
153 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 ...
4
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1answer
1k 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, ...
0
votes
3answers
307 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
125 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 ...
4
votes
4answers
295 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
113 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
392 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 ...
3
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1answer
169 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
138 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
117 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
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2answers
267 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
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1answer
177 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
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1answer
758 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
364 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
180 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?
