All Questions
Tagged with tiling mg.metric-geometry
84 questions
10
votes
1
answer
151
views
For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
- 263
9
votes
1
answer
542
views
Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"
I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...
15
votes
1
answer
528
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
1
vote
0
answers
52
views
'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
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4
votes
1
answer
438
views
Perfect squaring of rectangles
A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...
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0
votes
0
answers
41
views
Trying to extend a theorem on Tiling with mutually non-congruent triangles
In the light of Cubing the cube - as 'perfectly' as possible, We try to slightly 'relax' the main theorem proved by Kupaavski, Pach and Tardos here:
https://arxiv.org/pdf/1711.04504.pdf
...
- 5,979
9
votes
0
answers
186
views
Cubing the cube - as 'perfectly' as possible
Ref: https://en.wikipedia.org/wiki/Squaring_the_square
A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
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1
vote
1
answer
98
views
To place copies of a planar convex region such that number of 'contacts' among them is maximized
A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
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7
votes
1
answer
248
views
Decidability of completing Penrose tilings
Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
- 509
6
votes
2
answers
822
views
Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?
Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...
- 13.4k
6
votes
1
answer
435
views
On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
- 4,898
10
votes
0
answers
924
views
Are aperiodic monotiles generalizable to higher dimensions?
This question is motivated by a recently released paper written by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. It constructs the first topological disk that tiles the ...
- 377
1
vote
0
answers
100
views
What is the example of a circle being filled with congruent tiles (not pie slices), with no overlap of the tiles and and no space left?
I think I read somewhere that at one time it was thought the only way to lay tiles that would fill a circle with no overlap of the tiles and no exposed space in the cirlce, was to lay pieces that ...
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1
vote
0
answers
46
views
Kissing behavior of planar regions
This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...
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5
votes
1
answer
397
views
How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 20.1k
15
votes
2
answers
778
views
How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
- 161
16
votes
0
answers
391
views
Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
- 21.1k
6
votes
2
answers
215
views
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
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3
votes
1
answer
152
views
Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
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1
vote
0
answers
40
views
Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius.
The following are known about infinite sets of triangles that can be parametrized with one variable:
from an infinite set ...
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5
votes
0
answers
177
views
Tiling with triangles of same circumradius and inradius
Consider a pair of positive real numbers $r$ and $R$ with $r<R/2$. Then we can form infinitely many triangles all with circumradius $R$ and inradius $r$.
For any such pair, the resulting triangles ...
- 5,979
24
votes
3
answers
3k
views
Polyomino that can cover an arbitrarily large square but not the entire plane
https://userpages.monmouth.com/~colonel/nrectcover/index.html
For a polyomino with no holes that cannot tile the plane, we may ask what are the maximal rectangles and infinite strips that it can ...
- 359
7
votes
0
answers
227
views
Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
- 151k
2
votes
1
answer
158
views
On sets of rectangles that can all together form at least one big rectangle
Let us say a set of $n$ rectangles is rectifiable if all $n$ rectangles together form a big rectangle without gaps or overlaps.
Question: How hard computationally is the question of deciding whether a ...
- 5,979
2
votes
1
answer
84
views
What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 13.4k
4
votes
0
answers
171
views
Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!
Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...
- 5,349
2
votes
1
answer
164
views
Packing densities of non-centrally symmetric planar convex regions
Reference: https://en.wikipedia.org/wiki/Smoothed_octagon
Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...
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8
votes
1
answer
353
views
Are there any convex pentagonal rep-tiles?
A rep-tile is a shape that can tile larger copies of the same shape.
Question 1: Are there any convex pentagons that are also rep-tiles?
Remarks: 15 convex pentagonal tiles of the plane are known and ...
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3
votes
2
answers
279
views
Construct by compactness (Pentagonal tiling – Rao paper)
In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:
"… We keep a connected component $H_d'$ of $...
- 133
24
votes
1
answer
1k
views
Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?
A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...
- 151k
2
votes
0
answers
131
views
Cutting polygons into mutually similar and non-congruent pieces
It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) - for example, see https://en.wikipedia.org/wiki/...
- 5,979
31
votes
5
answers
1k
views
Fair cutting of the plane with lines
An infinite countable family $\cal{L}$ of straight lines in the plane $\mathbb{R}^2$ forms a fair cutting of the plane if the following conditions are satisfied:
$\bullet$ No circle intersects ...
- 7,276
0
votes
0
answers
90
views
On Covering a Planar Region with Copies of a Tile of Different Shape
Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...
- 5,979
4
votes
0
answers
92
views
Possible cardinalities of spherical tiling
Suppose that we have a tiling of $n$-dimensional (I want to get answer for $n = 4$, but general result would be nice!) sphere by isometric tiles strictly contained inside the right-angled simplex. ...
- 4,600
12
votes
1
answer
373
views
A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
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10
votes
6
answers
700
views
Tiling with similar tiles
Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...
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3
votes
1
answer
212
views
Monotile that tiles when you apply a rubber band
My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...
- 6,652
3
votes
1
answer
84
views
Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
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1
vote
2
answers
232
views
What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 13.6k
4
votes
1
answer
139
views
Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?
This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
- 12.4k
3
votes
0
answers
137
views
Aperiodic tile with rational area
Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
- 745
2
votes
0
answers
182
views
Graphs determined by monohedral, edge-to-edge tilings of the plane
Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
- 151k
11
votes
1
answer
392
views
Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
- 151k
11
votes
1
answer
499
views
Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
- 151k
1
vote
0
answers
196
views
Squares as sum of squares
For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?
- 3,707
3
votes
0
answers
106
views
How many positions of a tile can occur in a periodic tiling?
In my recent question about polygonal tilings where tiles can occur in infinitely many positions, both constructions given as solutions are of self-similar nature. This means in particular that there ...
- 13.4k
1
vote
1
answer
163
views
Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?
My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...
- 13.4k
10
votes
1
answer
401
views
How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
- 13.4k
9
votes
1
answer
282
views
Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
- 151k
9
votes
3
answers
1k
views
What rectangles can a set of rectangles tile?
(I asked this question first on math.stackexchange, but did not get any responses so I thought I would try here.)
If we have a set of $p_i \times q_i$ rectangles ($p_i, q_i \in \mathbf{N}$), which $m \...
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