# 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### Surfaces from Self-Intersecting Regular Tilings

It is well-known that there are precisely three ways to tile the plane with regular polygons joined edge to edge: with triangles, squares and hexagons. But why stop there? If we ignore the issue of ...
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### 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 ...
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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 ...
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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 (...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### Are there tessellations of the plane with curved $C^1$ shapes?

Does there exist a $C^1$ curve $\alpha:\mathbb{S}^1 \to\mathbb{R}^2$ such that the plane can be tessellated by a union of congruent shapes, each has boundary identical to $\text{Image}(\alpha)$ up to ...
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### Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb

Drake Thomas and I have proposed a sequence A343909 to the On-Line Encyclopedia of Integer Sequences (OEIS), which counts "generalized polyforms": generalizations of free polyominoes (Tetris ...
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### What is a minimum number of $1\times 3$ tiles that can be put on a table $5\times 5$ so that no more tiles $1\times 3$ can be put on it?

What is a minimum number of $1\times 3$ tiles that can be put on a table $5\times 5$ so that no more tiles $1\times 3$ can be put on it? It is 5 but I can not prove that if we put 4 tiles there is ...
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### Proof that higher-order Voronoi diagrams subdivide space

The Wikipedia article on Voronoi diagrams claims that higher-order Voronoi diagrams subdivide space. This is far from obvious. Can anyone point me to a proof?
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### Haberdasher problem of Henry Dudeney - is 4-piece hinged dissection of equilateral triangle into square possible?

Is it possible to divide the equilateral triangle into 4 pieces to build a square with those four pieces, provided that one or two pieces are flipped over to the other side? If possible, I wish to ...
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### Can any 2D tessellation be uniquely determined by for each n the number of cycles of length n containing any given vertex?

Basically what it says in the title. The title was actually intended to be: "Can any 2D tessellation be uniquely determined by a partition of its vertices into types and for each type a sequence ...
1 vote
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### Edge relation between Archimedean tessellations

I have run a simple script that numerically computes edge lengths of various hyperbolic tessellations and compares them. It seems that if you fit two squares and two 2n-gons to each vertex (...
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### Can a monohedral tiling have more than $21$ corner-adjacencies at every tile?

Suppose we tile the plane with a simply-connected tile $T$, and we wish to ensure that every tile touches at least $k$ other tiles. If we restrict "touching" to require sharing a positive-...
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1 vote
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### How to find a tessellation from a basis of linear transformations?

Sorry if this is a repeat. It seems like something that should be well known, but I have never found a really good way to write computer code to solve this problem, despite having ad hoc solutions for ...
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### If an animal tiles the plane via translation, can it do so in a lattice configuration?

It is known that if a polyomino tiles the plane using only translated copies, then it has at least one such tiling where the centroids of each tile form a lattice; see for instance the paper Arbitrary ...
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1 vote
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### Which tessellating 4-polytope has the largest symmetry group?

If we consider the set of all 4-polytopes which can tessellate four-dimensional space, which has the largest symmetry group? The tesseract clearly tessellates four-dimensional space and its symmetry ...
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### Tesselation beeing used in a game board

I like to know what is the math behind the outlines of these boards https://www.boardgamegeek.com/image/4383021/spirit-island https://www.boardgamegeek.com/image/4880004/spirit-island https://www....
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### Does a tiling being monohedral and monogonal imply that it is isohedral or isogonal?

I’m working through Grünbaum and Shepard’s Tilings & Patterns and I have been unable to make much progress on this problem. Does there exist a tiling that is monohedral and monogonal, but is ...
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### Does every 5-celled animal tile the plane?

An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) In this ...
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### Tessellation "open" problems for middle/high-schooler

I am a teacher and am currently tutoring middle/high-school students for research initiation, I gave small groups of them topics and let them advance by trials and error with essentially no guidance (...
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### Radius of hyperbolic circles in a tessellation

Killing my boredom, I played a bit on Geogebra and started this uniform hyperbolic pattern consisting of 8 circles per vertex. Considering my Poincaré disk model is of Gaussian curvature -1, and using ...
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### Tilings of the plane and quotient spaces

A square lattice has two discrete translation symmetries, isormorphic to $\mathbb{Z}^2$. If we take the quotient of the plane by this group, $\mathbb{R}^2/\mathbb{Z}^2$, we get the square with ...
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### "tessellation" of set of points in 2D with convex polygons

I have a set of points in 2D, that I want to 'triangulate' with the lowest number of convex polygons. Is there an algorithm to do this? (like Delaunay triangulation, but with polygons) Remarks: I ...
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### Which pairs of hyperbolic tessellations share edge length?

For any set of regular polygons, it's possible to find edge length that allows to fit these polygons around a vertex in either spherical, Euclidean (for edge $0$), or hyperbolic plane. If we disregard ...
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### Is there a tessellation with no corners?

Is it possible to have a tessellation of the plane with no corners? Here's what I mean precisely: Is it possible to tile the plane with tiles, each of which is the region enclosed by a smooth simple ...
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### Seeking intuition for why tesselations of space by hypercubes in dimensions 8+ need not have a face-to-face pair (Keller's conjecture counterexample)

According to Keller's conjecture: In any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. Perhaps surprisingly, this is false for every dimension greater ...
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### Why do circles (of the same sized circumference) not tesselate?

If our test for whether a regular polygon can tesselate with itself is whether the degrees of an individual interior angle can divide 360 to yield an integer (some examples of these integers are 6 [...
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1 vote
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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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### 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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1 vote
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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?
1 vote
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### 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 ...