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
10
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Is this a new pentagonal tiling?
I discovered this while thinking about the pentagonal tiling of type 15. Is this a new type of tiling? If it is, then I think I have found several other new pentagonal tilings like this one and the pentagonal tiling of type 15. They all have…
six
- 317
10
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Covering any rectangle with this shape is not possible
Why can I not tile any rectangle without gaps with the given shape?
https://i.stack.imgur.com/9oxO4.png
You can mirror the shape (i.e. turn it around an axis in its own plane by $\pi$).
Tamiel Farken
- 111
9
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3 answers
Properly drawing a Penrose tiling using the pentagrid method
As part of my work, I create tools for artists to make various types of patterns for artistic purposes. I am trying to make a tool to create a Penrose tiling and I would like to use the pentagrid method of generating it, as it seems like the easiest…
user1118321
- 295
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Penrose tilings with physical tiles
I have a question:
If one has a sufficient supply of Penrose tile and starts from a nice five-symmetric start and then continues putting down tiles outwards in a spiral fashion, is it relatively easy/likely to continue without problems or would it…
Phira
- 20,860
8
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1 answer
Is it known whether strips cannot be tiled aperiodically with a single tile?
As far as I know, it is an open problem whether there exists a single connected tile that tiles the plane only non-periodically.
Is the situation different for a strip? (Or, for that matter, a half strip, or quadrant, or bent strip.) I mean, is it…
Herman Tulleken
- 3,608
7
votes
1 answer
Penrose tiling "nesting number"
I discovered a simple way to generate assign a "nesting number" for each tile in a Penrose "kites and darts" tiling, which results in a really nice way to visualize the tiling (see image above).
Is there a way to generate this "nesting" number from…
Tom Sirgedas
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6
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Can we enforce a P3 Penrose tiling using unmarked rhombi with a small set of local matching rules?
Suppose I've got some rhombi that I want my friend to construct a P3 Penrose tiling out of:
However, the edge markings on my tiles have worn off, so I need to give my friend instructions about how to place unmarked tiles edge-to-edge such that they…
RavenclawPrefect
- 17,299
6
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1 answer
How much of a Penrose tiling must be specified to uniquely determine the tiling?
Every Penrose tiling contains every valid finite patch of tiles, as shown e.g. in Theorem 8 here. So in order to figure out exactly which of the uncountably many Penrose tilings one is looking at, we must examine some infinite set of tiles - I'm…
RavenclawPrefect
- 17,299
5
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2 answers
$2^n \times 2^n$ chessboard with one square removed - Is the tiling unique?
It is well-known that a $2^n \times 2^n$ chessboard with an arbitrary square removed is tilable by an L-shaped tromino (piece composed of three squares). The standard proof is by induction, and is constructive (gives an algorithm for producing such…
Samuel Li
- 785
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Do we need 17 monominoes to tile a $9\times 9$ with square tetrominoes?
Or more generally, is it true that to tile a $(2k+1) \times (2k + 1)$ square with square tetrominoes and monominoes, we need at least $4k + 1$ monominoes?
Each row (having an odd number of cells) need at least one monomino in each row, and similarly…
Herman Tulleken
- 3,608
5
votes
1 answer
Does every domino tiling have at least two "exposed" dominoes?
In a domino tiling, a domino is exposed if it has a long edge that does not neighbor another domino. A domino that shares an edge with a long edge of another domino covers the edge (or the domino, if we understand which edge we mean) and prevents it…
Herman Tulleken
- 3,608
5
votes
0 answers
minimal oblique rectangle to sample wallpaper
From a rectangular wallpaper pattern with periods $(w,0)$ and $(0,h)$, you must cut a rectangle with aspect ratio $m$, to include every part of the pattern-unit; in other words, you must cover the (flat) torus. You may rotate this sample-piece at…
Anton Sherwood
- 628
4
votes
1 answer
Aperiodic Monotile with hole in it.
Here is A partial tiling of the plane in the style of Heesch tiling, but with the new Monotile discovered and encircling a hexagonal hole.
Call such a combination a quasi-Heesch tiling, where you use a single tile to build a tiling in encircling…
4
votes
1 answer
Periodic Wang tiling implies existence of doubly periodic tiling.
I'm struggling to understand how to prove a theorem stated by Karel Culik in his paper on 13 aperiodic Wang tiles. He defines a periodic tiling over a finite tile set T as being a function $f:\mathbb{Z}^2 \rightarrow T$ such that $f(x,y)=f(x+h,y+v)$…
R Suth
- 41
4
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3 answers
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 and replicate the tiling inside it in a smaller…
lhf
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