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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Tesselation of the upper half plane via Ford Circles

I have a question about the tesselation of the upper half plane via Ford Circles. Wikipedia says By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the ...
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Covering an area equally with layers of non-tesselating polygons

A series of hexagons on an hexagonal lattice means that the every point in the entire area is covered by one polygon only. A grid of octagons will not tesselate, leaving square holes such that 4/18 ...
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Simulation of typical cell in Poisson Voronoi tessellation

I would like to simulate a typical cell in Poisson-Voronoi tessellation model. I want to save the Cartesian coordinates of all vertices of the typical cell for each realization. How to do it? Thank ...
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How big do squares need to be to fit a box, tesselating, with minimal remainder?

A geometry question that I feel utterly defeated by. I'm trying to design a responsive user interface that efficiently fits a variable number of square elements on a screen, by adjusting the size of ...
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A combinatorial proof by tesselation of the plane.

Some days ago the following problem was posed in the site: given a set of $N$ points in the plane such that for each pair of points $p,q$ we have $\lVert p-q\rVert >1$, prove there is a subset of ...
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From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
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Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
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Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like the square and the ...
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The maths behind 'Sky and Water 1' by Escher

I've been inspired by Eschers 'Sky and Water1' woodcut, his work is so mindblowing, it got me thinking whats the maths behind it. How did he do it?
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Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set ...
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What is the optimal tiling of a regular n-gon in the plane?

I want to tile the plane with equal-sized regular polygons of $n$ sides. Obviously for some $n$, the tiles will be able to tessellate and cover the whole plane (e.g triangles, squares, hexagons) I ...
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What honeycomb has the highest volume to edge length ratio?

This question is analagous to the Kelvin Problem where the solution, the Weaire-Phelan Structure, has the highest volume to surface area ratio; however, the cell volume is compared to edge length ...
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Equiareal Voronoi tessellation

I'm interested in even (or "proportional") disrtributions of points on 2D areas. Here is the initial question, but many others ideas appeared later. Centroidal Voronoi tessellation is well known ...
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Can the lack of obstruction to deforming a checkerboard tessellation be seen as part of a larger picture?

All quadrilaterals, even the nonconvex ones, tessellate the plane. One way to see this is to start from a checkerboard tessellation made by repeatedly flipping squares across their edges, and then ...
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Plane tessellation $6^2*3^2$

An article I am reading mentioned "the plane tessellation $6^2*3^2$", I tried looking it up and I found all sort of plane tessellations - but not $6^2*3^2$. However, I did find information about ...
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Extension of Planar Algorithms to Higher-Dimensional Voronoi Diagrams

Voronoi diagrams are not new, and there are many established algorithms (Fortune's, Lloyd's) for generating them (or their duals, the Delaunay triangulation). There are many recent-ish papers too, ...
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Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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How to find the center of of rotational symmetry in a Penrose tiling?

The center of Helsinki sports a gorgeous Penrose tiled pavement: Could anybody give me an algorithm that will permit me to find its 'center' - that is, the point around which the five-fold ...
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Proof that $12$ in a row tic-tac-toe is a tie game?

How can be it proved that tic-tac-toe on an infinite grid (winning with $12$ in a row, a column or a diagonal) can always end in a tie (with optimal strategies of both players)? There is a hint: to ...
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Infinite sums and squaring the plane, or sort of

This question is regarding two algorithms for squaring/almost squaring the plane. the Henles' method of squaring the plane. pdf here my method of tiling $n^2$ squares. I worked out* a simple gap-...
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A tesselation where four parallel rectangles meet at each vertex

If we tile the plane with parallel rectangles that are translated copy of the same rectangle, then each point is either inside a tile, or on a segment common to two adjacent tiles, or a corner common ...
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Isohedral polygons that tile space in Voronoi tessellations

Besides the three uniform tilings of space in $\mathbb{R}^2$, are there other regular or semiregular tilings, by one or more types of isohedral polygon, that are also Voronoi tessellations of some ...
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Photo mosaics and the assignment problem

I have been having fun making mosaics from thumbnails. E.g. Until now, thumbnails have been uniform size and my aim has been to use every thumbnail, which makes it very straightforward to use the ...
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Kelvin problem with restriction to only one type of polyhedron

If we only allow one type of polyhedron, what would be the answer to the Kelvin problem? Would the Kelvin conjecture be true?
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Regular polygon tessellation

A roof if made up of regular tiles, much like a chessboard. The tiles overlap at the base and cannot be the same colour as the ones either to the left, right, up or down to tile. If another colour ...
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Does there exist a tessellation of a 3-D object such that no two objects touch by corner or edge only?

A hex grid is special in 2D geometry because it can tile itself in a way such that any two hexagons that touch a corner must also touch by an edge. This makes it useful for game planning because the ...
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what is the theoretical solution to Voronoi domain distribution for 2D random point sets

Consider a large set of random points in 2D plane generated by Poisson process. And consider only the finite Voronoi domains generated using these points. Is there a theoretical solution to the ...