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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What is the representation of the triangle group for the uniform (4 4 4) tiling of hyperbolic disk in terms of Mobius transformations?

I wonder how can one descibe the generators of the triangle group for the tesselation of Poincare unit disk by triangles with angles $\pi/4, \pi/4 , \pi/4 $ in terms of the action of the modular ...
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Is there any way to produce a random Voronoi diagram with a specific length and width? [closed]

A Voronoi diagram is an approach to the tessellation of medium. In this diagram, there are many points in a plane that divide the medium to many specific regions by their bisector. Any region is ...
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Tessellation of a parallelogram

I have a given parallelogram and a sequence of parallelograms in the plane $\mathbf{Z}^2$. Their vertices have integers coordinates. The parallelograms of the sequence get bigger and bigger in this ...
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Distribution of areas of Voronoi cells on a sphere

I have two sets of, respectively, N and M points, which are independently, randomly allocated on a sphere. I consider the Voronoi tessellation of the sphere by the N points, and I want to find how ...
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What regular or semiregular hyperbolic tiling has the smallest average tile area?

I have noticed that hyperbolic tilings tend to be rather "sparse" in that each tile takes up a lot of space. If I remember correctly, for a given curvature the area of any tile in a given hyperbolic ...
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Can the 2D plane be tessellated with a finite set of closed curves that have Geometric continuity > 1?

When I think of tessellating the 2D plane, all examples I can think of (squares, hexagons, rhombuses, polygonal stars...) have at least one point that is G1 continuous, even these fish: Is it ...
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Packing/tessellating 4 dimensional space fully by polytopes? Give examples.

What is a shortlist of first few simplest (say 5~10 simplest) possible shapes of polyhedra/polytopes (with a minimum number of edges shared) to pack the 4-dimensional flat space (say $\mathbb{R}^4$) ...
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Packing/tessellating 3 dimensional space fully by polytopes? Give examples.

What is a shortlist of first few simplest (say 5~10 simplest) possible shapes of polyhedra/polytopes (with a minimum number of edges shared) to pack the 3-dimensional flat space (say $\mathbb{R}^3$) ...
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Properties of Thiessen Polygons

Given a convex polygon, how does one test whether it's part of a Voronoi tesselation? In other words, what is a quick way to test if a polygon is a Thiessen polygon? I suspect one test could involve ...
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Triangle interpolation with 6 control points?

Through a costly simulation, I am able to calculate the value of a function at several discrete points on a plane. My task now is to interpolate, to find the values at all points of the grid. (It is a ...
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How heptagon tesselation layout is computed in hyperboloid model?

I want to try using heptagon-hexagon tesselation grid in hyperbolic plane like the one used in HyperRogue. This is how I understand it: First, graph representing tesselation is generated. (done) ...
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Is it possible to tile two planes, which have some line segments identified, in ways which can't be replicated on just one?

Suppose you have two Euclidean planes, and there is some set of pairs of points (themselves pairs of coordinates) such that for each such pair ((ax,ay),(bx,by)), the line segment connecting the point (...
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Formula for number of squares fitting into a right angle triangle

Is there a known formula for the number of squares of a certain size that would fit into a right angled triangle?
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31 views

Finding points for Conway Tessellation

First time on Math SE, so any feedback appreciated I'm working on some generative art that will draw Pinwheel tiles with svgs. To that end I'm building a function that given a set of three points, ...
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Seeking name of $90-135-90-45$ deg angled kite which forms a non-periodic Fractal Tessellation

In Barnsley and Vince's paper Self-Similar Polygonal Tiling https://people.clas.ufl.edu/avince/files/SSPfinal.pdf fig $6$ shows a non-periodic tiling made from a kite in 6 sizes. The angles of the ...
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Substitution scheme in Ammann–Beenker tiling

On wiki page about Ammann–Beenker tiling is described the substitution scheme $R → R r R ; r → R$ that introduces the ratio as a scaling factor: its matrix is the Pell substitution matrix, and the ...
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Higher dimensional plane tessellation

Let $n$ be some finite positive integer value. Now consider the plane $z=n-x-y$, so the points $(n, 0, 0), (0, n, 0), (0, 0, n)$ bound the region of the plane with strictly non-negative coordinates. ...
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Is there an algorithm to design 3 and n dimensional honeycombs or tessellations?

I have looked into creating multiple dimensional honeycombs, and I can't find an algorithm or methodology for 3 or more dimensions. Is there a general algorithm or heuristic that can be used?
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How to call complementary sides in tiling shapes

How mathematically can we describe the relation between two shapes which fit to each other? Is there a word in geometry for expressing that two sides of a tiling are complementary? How to describe two ...
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“Gaps in the Mats” problem

Problem Background* The mat at your karate dojo composed of 160 square interlocking foam tiles. Along each edge of each tile, there are has five "teeth" (10cm long) and five spaces-for-teeth (again ...
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Tessellations of 3-sphere

How many are there regular geodesic tessellations of the 3-sphere? What kind polyhedrons are used in those?
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Neighboring solids in tetrahedral-octahedral honeycomb

In the tetrahedral-octahedral honeycomb, each vertex seems to be incident to 6 octahedra and 8 tetrahedra: Such simple combinatorial fact is probably well-known, or perhaps even obvious. However, ...
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Substitution matrix for the Ammann-Beenker tiling

On the wikipedia page for the Ammann-Beenker tiling, it says the following: I am trying to derive the substitution matrix for and show that its eigenvalues are $(1\pm\sqrt{2})^1$... so far the things ...
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Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)

I had a professor who once introduced us to Wallpaper Groups. There are many references that exist to understand what they are (example Wiki, Wallpaper group). The punchline is $$There \,\, are \,...
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Proof for replicating tiles

A rep-tile is a shape which can be divided into n smaller shapes that are similar E.g. a square can be reproduced by 4 smaller squares rep-n means that a shape can be created from n smaller copies. ...
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Permutations for tilings / tessellations

I'm an artist, many years past my maths a-level - so apologies for any idiocy up front. At the moment, I'm working with tilings/tessellations of right-angled isosceles triangles. I have two designs, ...
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How does one generalize the dual of a tiling by regular polygons to other tilings?

This question is motivated by some work in curriculum design for outreach math. Normal tilings of the Euclidean plane are popular since people of all ages can play with them, but on the other hand the ...
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Prove that the shortest path between two points in a Delaunay triangulation minimizes angle at each step.

Say we have a set of points $p_{k (k \in K)}\in P$ Poisson distributed in a real coordinate plane $X$ residing in ${\rm I\!R}^{2}$ and with Euclidean distance function $d$ and $K$ is a set of indices. ...
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Self-Similar Polygon Tessellations

It is well-known that the only regular polygons which tessellate the plane (using only one shape) are the triangle, square, and hexagon. However, there are many more tessellations of the plane by ...
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Determine the tiles intercepted by a given function in an irregular tessellation

Consider an irregular tessellation of the plane composed of convex tiles, such as the following one. For each tile, the coordinates of each vertex (black dots) are known up to a finite precision. Now,...
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What is the best way to tessellate sphere into equal area in any level of detail? HEALPix or Geodesic Grid or another method?

I want to tessellate sphere into a grid in my 3D world map. There was 2 ways that I was consider right now, HEALPix and Geodesic If I use it specifically for world map that could be zoom into any ...
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Moscow Seven Sisters

Fix $n$ points in the plane in generic position, i.e. no three of them on the same line, etc. The number of lines joining two of them is ${n \choose 2}$. The number of regions in which $\ell$ lines ...
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Finding the right grid tessellation, for a chess board with elliptical geometry

I came up with the idea of non-euclidean chess(the chess board I'm working on will have 2D elliptical geometry) but I came across a problem. The question: What kind of grid do I put on it(what shapes ...
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Spiral path on a Penrose tiling

I would like to color a Penrose tiling by following a "spiral path", painting each tile according to a given color sequence. In this picture, I illustrate what I am looking for: The dashed line ...
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One element space tessellations

Disclaimer: i am bioinformatician and programmer, please excuse if my wording and definitions are far from elegant and occasionally imprecise. Intro: I am interested in space tessellations of n ...
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Are there any known plastic constant triangles?

I am trying to determine if there are any known plastic constant triangles. By this I mean specifically triangles for which all sides are powers of the plastic constant, $p\approx1.324717957244746$. ...
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Multiple use of Fermat's Principle , or Simple, Finite Tessellation (Unfolding)

First consider a specific question. Given a triangle and two interior points, a person at Point-1 is required to 'touch' all the 3 sides of the triangle exactly once and return to Point-2. Find the ...
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What determines if a hyperbolic tiling with ideal vertices is regular?

I'd like to say I need this community's help in clearing my mind of the clutter that leads me to this contradiction: As an example, the Wikipedia article for the Infinite-order triangular tiling ...
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Name and number of “equilateral tessellations with same angles on all vertexes”

Longer background, shorter questions below: Tessellations of 2D plane consisting of regular polygons are usually described with vertex configurations such as "3.4.6.4" meaning that there are a ...
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How to discretize a sphere?

I would like to discretize a sphere into icosahedra whose vertices are equidistant, i.e., I want to plot $n$ equidistant points on the surface of a sphere. I am familiar with R, Python, and Matlab. ...
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Is there a name or a reference for these aperiodic rhomboidal tilings?

Fill space with unit cubes and then remove all cubes that are not completely within a given half space. An isometric view of the remaining cubes will look like the following image. This is in ...
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Uniform distribution on infinite binary grid

Is there an infinite binary array, so that the distribution of all possible binary $N\times N$ arrays on it will be uniform? In other words, randomly picked $N\times N$ square will be $\sim\mathrm{...
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Is it possible to tile the pane with a semi-regular tesselation given only the vertex type?

I want to write a computer program that tessellates the plane with semi regular tiling, i.e these tilings: https://en.wikipedia.org/wiki/List_of_convex_uniform_tilings If I start with the vertex ...
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A series that converges to π/3 [duplicate]

While surfing on YouTube, I stumbled into this video which gave me a new insight about the well-known series $$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}+ \ldots $$ The idea shown ...
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How to construct an algorithm to move along a discrete grid at constant steps?

Assume we have a tessellation of cubes of length one of a 3D space. We have a starting point $P$ and a direction $\vec V$ of length one. Our goal is to move the point $P$ by direction $\vec V$ such ...
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Do the loops “Snakes” by M.C. Escher correspond to a regular tilling of the hyperbolic plane?

In M.C. Escher's Snakes, you have three snakes going through some loops. I'm more interested in the loops though. In this image, a ring model of the hyperbolic plane is given. It is given by $w=e^{za}...
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Prove that only triangles, quadrilaterals, and hexagons will Tesselate the plane

Prove that only triangles, quadrilaterals, and hexagons will Tesselate the plane So, I have almost completed the proof, I will write it down all the way to the part I'm stranded in. Theorem: only ...
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How many shapes with Euler characteristic 2 of which all faces border the same amount of faces (through edges and vertexes combined)?

I'm trying to create a shape that I can map to both a plane and a sphere. I don't care about distortions at all, so, for example, platonic solids work. However, I need a solution that works for a ...
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Is this Escher artwork a tessellation of the half-plane model of hyperbolic space?

One Escher's prints look like this. A similar one is this. These look suspiciously like Poincaré half-plane models of the hyperbolic plane (there are pieces of artwork by Escher specifically based on ...
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Why do matching rules make a substitution tiling aperiodic?

Wikipedia hath written: Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution ...