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.
Questions tagged [tiling]
745 questions
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Question about the proof of the Ornstein-Weiss tiling lemma
In the original paper of Ornstein and Weiss "entropy and isomorphism theorems for actions of amenable groups" I have trouble understanding the proof of the quasi-tiling lemma for countable groups. In step (2) one wants to cover a set $D$ that is…
mathemagician99
- 904
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mapping an aperiodic tiling into a periodic tiling
I came across an aperiodic tiling yesterday, and I was wondering, being just a stupid engineer and not a mathematician: Is it possible to map this into a periodic tiling using a continuous function? My guess is "No", but I thought I'd still ask,…
NNN
- 228
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Tiling a rectangle with the maximum number of squares of side length greater than $s$
I am 3D modelling a small parts storage organizer when I came across the idea of having variously sized cubic compartments tiling a rectangular drawer. A thought this would make for an interesting design, and would allow me to store a variety of…
Ryan Rudes
- 345
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Mapping "valid" tile sequences to number series
I'm interested in understanding the maths behind a satisfying time-waster game called Noodles!. In one of the game varieties there's a grid of tessellating hexagons with pipes drawn on them, and you must make all the pipes line up to win:
Each of…
JP.
- 123
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How many isohedral ways are there to stack $1\times 1\times 2$ blocks?
If I have blocks of size $1\times 1\times 2$ cubes, how many ways there are to stack them isohedrally in 3D space? I now have pretty robust system for solving this kind of problem in 2D, but 3D escapes me.
I presume that it could be solved by simply…
Marek14
- 883
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Color a 1000 by 1000 grid with 0s and 1s
Show that one could remove 990 rows with a 1 remaining in every column, OR delete 990 columns with 0 in remaining in every row.
My approach: let $r_i$ be the number of 1s in row i and $c_i$ be the number of 0s in column i but then I am stuck. Any…
Kai
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Is there a word for a polyomino with $n$ connected cells in each row?
Here are some examples for $n = 4$.
Herman Tulleken
- 3,608
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Tiling with different rectangle
If a rectangle R can be covered by non-overlapping rectangular tiles, each of which has at least one side with integer length, is it true that R must also have a side with integer length?
horatio
- 11
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Wrong number of $4\times 4$ domino tilings, but why?
From the internet, I know that a domino tiling of a $4\times4 $ checker board can be arranged in $36$ different ways.
With the following reasoning, I conclude that it must be $37$, which is one more than the correct result. Where is my…
Daniel S.
- 530
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Determine if a rectagle is fully "compatible" with a given Polyomino
Recently, I came across a unique problem for which I couldn't find a complete solution.
I want to determine if a given rectangle is fully "compatible" (for the lack of a better word, please suggest if there is any) with all members of a given…
Ravi M Patel
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Navigating in Z-Order curve with different ordering
I want to navigate in a Z-order curve by moving up, down, left or right.
The Wikipedia page has this section :
This property can be used to offset a Z-value, for example in two dimensions the coordinates to the top (decreasing y), bottom…
Charles
- 258
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3D Ising model and hexagonal domino tillings
I'm preparing an exam and in a preparation sheet there is an exercise that I just don't know how to deal with. Could someone please explain it to me?
a) Explain why the 3D Ising model on the cubic box $ \{0,\ldots,n\}^3 $ with $ \beta \to +\infty $…
3m0o
- 783
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What is an elegant algorithm for tiling a rectangle with identical squares based on 2 constraints (ratio of rectangle and minimizing uncovered area)?
Squares should all have the same dimension. The ratio constraint is that the length of the rectangle should be equal to its width (a square) or no more than two times its width (2:1 ratio). The algorithm should minimize the uncovered area. For the…
jeromecc
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Clarifying the meaning of the tiling/tessellation notations?
I would like to come up with a final list of "tilings", but am having hard determining what the name or even a standard representation of the tiling is. Sidenote, it appears that the terms "tiling" and "tessellation" can be interchanged, where some…
Lance
- 3,698
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Tiling a 4xN board with L shaped 3x2 tiles
I need to find the number of the possible combinations of tilings of a 4xN board, with L shaped tiles that are 3x2. The tiles can be rotated freely. Any solution? Thanks.
