Questions tagged [tessellations]

For question on Tessellations, the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps.

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consistent vertex configurations for Archimedean tiling

I am trying to find if there is a simple rule for vertex configurations $n_1.n_2\ldots n_k$ that define Archimedean tilings (equivalently called uniform/semi-regular/vertex-transitive tilings). In ...
Tomáš Bzdušek's user avatar
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Non-monotileable amenable groups

We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss gives lots of examples of amenable ...
Saúl RM's user avatar
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How to build a $6\times 6\times 6$ cube using 4-unit T-shaped pieces?

I am trying to solve a 3D block puzzle. It consists of 54 T-shaped pieces, each made up of 4 units, shaped as follows: $$ \square \\ \square\square\square $$ The goal of the puzzle is to build a $6\...
Evyenia Coufos's user avatar
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Question on counting the number of triangles formed by 1999 points in a square

I was reading an explanation for a solution to an Olympiad problem as follows: Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another ...
deadskull16's user avatar
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Maximum and minimum distance distributions in Poisson Voronoi Tessellations

Consider an homogeneous PPP in $\mathbb{R}^2$ with intensity parameter $\lambda > 0$, and denote the set of generator seeds in the spatial domain as $P = \{p_1, p_2, \dots, p_N\}$. At the same ...
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Create a 2D plane tessellation starting from given centroids

I have a problem: I want to find a method which, given a set of centroids as x,y coordinates (the number of the centroids does not matter, more than one) it returns ...
rafiki's user avatar
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Rigorous version of the Symmetry of Things

I’m reading through The Symmetry of Things and I love it, but some of the arguments are handwavey and it’s hard for me to know if I fully understand. I’m curious if there is a rigorous reference that ...
jrudd's user avatar
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Is it really impossible to use hexagons for mixed-resolution cover?

"To cover" and, more strictly, subpaving is a set of nonoverlapping "boxes" of R⁺. A subset X of R² can be approximated by two subpavings X⁻ and X⁺ such that  X⁻ ⊂ X ⊂ X⁺. ...
Peter Krauss's user avatar
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Can "the hat" tessellation be recreated using hexagons marked with "hat" borders and some simple rules?

An aperiodic tiling shape, "the hat" was discovered recently. If you split a tiled planed into hexagons, containing the borders of the hat, would there be a finite number of unique hexagons. ...
Megasaur's user avatar
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Find vertices of hyperbolic tilings in Poincare disk. [closed]

In the hyperbolic plane, there can be tilings of regular $p$-gons with $q$ of them around a vertex. In the Poincare disk model, given that the origin is a vertex, what is the euclidean distance from ...
nonhuman's user avatar
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what is the exact relation between Dedekind tessellation and modular group

In the wiki for the Modular Group the group ($\Gamma$) is described as a $(2,3,\infty)$. The page has a diagram for "a typical fundamental domain for the action of $\Gamma$ on the upper half-...
unknown's user avatar
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Square to octagon dissection - how to cut the square?

How to cut the square which tessellates to octagon using straightedge and compass? What are the exact measures of colored sides? What is the angle marked with red color? Edit (I added vertices): Edit....
Przemyslaw Remin's user avatar
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The Schläfli Symbol for tilings of regular polygons with irregular tiles

A type of notation that is useful for describing regular tilings is the Schläfli symbol. It is stated as {# edges per polygon, # polygons meeting at a vertex}. In this notation, the three regular ...
Astrid's user avatar
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Diversity of edge numbers of space filling polyhedra

I am trying to find out if there is at least one polyhedron that tessellates for each valid edge number. I have found one for all edge numbers except 10 and 13. Here is my thought process so far. ...
Michael Toth's user avatar
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Proving that no tile can fill both squares and equilateral triangles

Cut up a square into a finite number of identical tiles. Here is one possibility: How do I prove that the tiles could never be rearranged to form an equilateral triangle (with filled interior and no ...
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Good books on geometric recursion

I was looking at my bathroom tiles, which were an interesting sort of repeating pattern of different squares and rectangles, and wondering how to model them as a recursion formula. Does anyone know ...
Flying Spaghetti's user avatar
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Questions about aperiodic tilings with the hat tile

As a curious bystander fascinated by aperiodic tilings, I skimmed the more informal parts of the (very nice!) paper about the hat monotile and have a few questions about the tilings it produces, ...
apirogov's user avatar
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Aperiodic cube tilings in all dimensions $n\geq3$

In a paper the following statement is made (link): "We note that in high dimensions there are many "exotic" cube tilings. There are aperiodic cube tilings in all dimensions $n\geq3$, ...
DaveGoneRogue's user avatar
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Can a patch of an aperiodic tiling of the plane be mapped onto / glued into a closed surface such as a torus?

Basically, as the title says. Maybe this is trivially true or false, but I have not enough intuitions about topological surfaces or aperiodic tilings. To make it a bit more precise - I mean the kind ...
apirogov's user avatar
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Reflection lines in cube tilings

When considering tilings of the plane by the unit cube. That is, a tiling by the unit square $I^2=[0,1]^2$ which covers $\mathbb{R}^2$. Usually one says that the unit cube tiles the plane using ...
DaveGoneRogue's user avatar
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Proof only three types of hexagonal, convex monohedral tessellations

Where can I get a proof that there exist exactly three types of convex, hexagonal, monohedral tessellations of the plane? I'm already aware of Reinhardt thesis and Bollobás paper (1963). I was ...
mps94's user avatar
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Why don't these hexagons tile seamlessly-- without space in between? Where in my math have I gone silly?

Intro to the Problem Hi there! I've been trying to seamlessly tile hexagons on an XY plane-- without spaces in between. I've currently accomplished this, however: which as you can tell, isn't ...
Alex Moulton's user avatar
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What does it mean for a tiling (in particular, one involving the recently discovered "Hat" monotile) to be "aperiodic"?

In these articles "Mathematicians Excited About New 13-Sided Shape Called 'the Hat'" (Gizmodo), "An 'einstein' tile? Mathematicians discover pattern that never repeats" (...
Moo's user avatar
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Tiling the hyperbolic plane.

I found this theorem online and I have no idea how to prove it. I can't find a proof anywhere; if anyone knows one that would be great. Theorem 6.10 Let $D$ be a hyperbolic triangle with angles $\frac{...
simon1976's user avatar
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Do Schläfli symbols unambiguously represent gemetric shapes?

According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization: However, when looking at it,...
HelloGoodbye's user avatar
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Explanation of using wallpaper groups with vertex figure for k-uniform tilings

In my project I need to implement various uniform tiling of a 2D-plane, so some time ago I started to dig a little bit into sources related to subject. From what I understand, any k-uniform ...
Carpetman's user avatar
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1 answer
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Tessellation of n-gons into parallelograms

Which regular n-gons can be tiled by (or tessellated into) a finite number of non-overlapping parallelograms? I have enumerated a few cases below where I know the answer: Case $n = 3$, there is ...
Thomas's user avatar
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How many tiling?

Recently, I study tiling of the plane with regular polygon which is edge-to edge. If we restrict to the two triangles and two hexagons, then we can slide rows of tiles so that in adjacent rows, we ...
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How to obtain translations vector in periodic tiling

A tiling of the plane, $\mathcal{T}$, is a family of sets- called tiles- that cover the plane without gaps or overlaps. Assume that tiles are regular polygons and tiling is edge-to-edge. A tiling is ...
user479859's user avatar
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Filling space with polycube snakes

One of Martin Gardner's "Mathematical Games" columns (Scientific American June 1981, pp24–29; reprinted in The Last Recreations (1997), pp274–283, and The Colossal Book of Mathematics (2001) ...
Quuxplusone's user avatar
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Is "Escherian metamorphosis" always possible?

This question is motivated by Escher's series of Metamorphosis woodcuts (see e.g. here), where one tesselating tile is gradually transformed into another. Basically, this is a precise way of asking ...
Noah Schweber's user avatar
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Tiling of a $2n \times 2n$ board

Is there a way to tile a $2n \times 2n$ square with dominoes such that two rectangles cannot be partitioned and slide along each other (interlocking)? I was able to show that for $2 \times 2$ and $4 \...
user1106787's user avatar
2 votes
1 answer
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Can we algorithmically map a Delaunay tesselation in $\mathbb{R}^n$ to a planar embedding in $\mathbb{R}^2$?

Preamble Suppose I have a finite collection of points in $\mathbb{R}^n$, and I compute a Delaunay tesselation on them. In this $n$-dimensional space the straight-line edges are non-intersecting line ...
Galen's user avatar
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12 votes
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Finding all 15-ominoes that tile the plane and have distinct internal adjacencies

Problem Description: This problem oddly came up in Minecraft with some friends. Not sure what the best terms are; but that's partly why I'm here. So a polyomino is built up from squares. This problem ...
WhiteStoneJazz's user avatar
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Equally spaced rays - Tessellating a sphere?

I was trying to develop a basic path-tracing algorithm for one of my projects when I encountered the following problem. Suppose I have a spherical light source at the origin. A uniform arrangement of ...
kichapps's user avatar
4 votes
2 answers
187 views

Does any edge-to-edge tiling of the Euclidean plane by convex regular polygons have only demiregular vertex configurations?

In the Euclidean plane, a vertex figure of an edge-to-edge tiling by convex regular polygons is called demiregular if and only if its vertex configuration is $3.3.4.12, 3.3.6.6, 3.4.3.12,$ or $3.4.4.6$...
mathlander's user avatar
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Can anyone prove why this hyperbolic tiling works?

Let's start with the following hyperbolic tiling. It has four equilateral triangles and two squares meeting at every vertex. Its edge is about 1.06 absolute units (it's actually identical to the edge ...
Marek14's user avatar
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Isolated edges in hyperbolic Delaunay triangulation

I have played with the C++ library CGAL to do some hyperbolic Delaunay triangulations. Sometimes (often) the triangulation has some isolated edges, as in this example: Is it theoretically normal to ...
Stéphane Laurent's user avatar
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1 answer
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Tesselation of hyperbolic plane are hyperbolic

Let’s take a tesselation $T$ of the hyperbolic plane (not necessarily regular), my intuition tells me that clearly $T$ should be hyperbolic itself (in the sense of Gromov or using $\delta$-slim ...
TeemoJg's user avatar
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How do I define the correct arc of a circle?

A start and end point as well as another point on the arc are given, which together define an arc of a circle. The points are given in polar coordinates. When defining such an arc, how do I ensure ...
Dawid's user avatar
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1 answer
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How to divide a sphere into many equally sized triangular tiles?

I am inspired by spherical tilings: How do you take a sphere, such as one roughly the size of earth, which has a radius (assuming perfect sphere) of 6,356,000 meters, and divide it into triangular ...
Lance's user avatar
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2 votes
1 answer
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Transform point in the Poincaré disc to point in tile

I have a set of image files that I want to use to use to texture a set of tiles with (one image per tile), in order to render a textured version of the Poincaré disc with a specific tiling by using ...
HelloGoodbye's user avatar
2 votes
1 answer
370 views

11 sided irregular shape that tessellates

My friend was fiddling around on the triangle when he created an irregular heptagon with it not able to tessellate. He then asked me if I could create an 11 sided irregular polygon that is able to ...
Stevo's user avatar
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2 votes
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Expected semi-perimeter in Mondrian tessellation process

I'm working on Mondrian Process [paper], which in few words splits a boxed region in $R^d$ by axis-aligned hyperplanes, uniformly located on a random axis, chosen proportionally to the lenght of the ...
fdallac's user avatar
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3 votes
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How to find a minimal '3D polyhedron' which is similar to A Minimal Circle in the planer graph?

Greetings all and thank you. I'm a Ph.D. candidate working on a 3D tessellation project and get stuck. I've simplified the system into a set of lines linked together which form a Line-Framework (...
Simon Shi's user avatar
7 votes
1 answer
178 views

What mapping does Escher use?

In Escher's hyperbolic tesselations, he takes (effectively) a tesselation of the plane and maps it to a tesselation of the unit disk, by a mapping that takes straight lines to circles meeting the disk ...
rihartley's user avatar
1 vote
1 answer
96 views

Distance labels in regular hyperbolic tilings

Consider the order-4 pentagonal tiling of the hyperbolic plane (shown in the figure Hyperbolic plane tiling with pentagons). Pick a vertex $s$ (in white), label it with $0$ and then label all the ...
Matteo Pariset's user avatar
7 votes
1 answer
175 views

How many fixed polyominoes does it take to force an aperiodic tiling of the plane?

A longstanding open problem asks whether there is a single connected tile that only tiles the plane aperiodically. As far as I know, this is still open even in the case of polyominoes: we do not know ...
RavenclawPrefect's user avatar
6 votes
1 answer
170 views

If a (possibly nonconvex) pentagon tiles the plane, can it do so periodically?

From the classification of monohedral tilings with convex pentagons, we know that all convex pentagons which tile the plane can do so periodically; I'd like to know whether the same result is known to ...
RavenclawPrefect's user avatar
7 votes
2 answers
324 views

Smallest non-space-filling polycube?

The title nearly says it all: what is the fewest number of cubes that can be fused face-to-face into a polyhedron that does not fill space? The smallest that seemed like a sure non-tiler to me was 9: ...
Glen Whitney's user avatar
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