# Tagged Questions

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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### 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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### 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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### Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?

I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that ...
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### How do I draw this picture in squares of discrete $\sqrt{z}$?

From Richard Kenyon's homepage gallery: I want to understand the mathematics of this, and similar/related transformations. ... An explanation in words (1st year uni level maths) would be ideal. I'...
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### Pentomino Tessellation Explanation

I need to explain why this pentomino tessellates in a mathematically coherent way. Here is the pentomino and the tessellation I have made. This pentomino can be translated to form a diagonal ...
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### Area of polygon of hyperbolic disc

Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex. {hyperbolic disc} There exists formula which defines ...
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### Plane tesselation, using stairs $n\times n$, is it possible?

The other day I was constructing new mathematical problems for my pupils and thought of something like this: Given the infinite sequence of "stairs" $n\times n$, constructed from $1\times1$ ...
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### Aperiodic tessellations of the plane

Here are some examples of non-periodic tessellations of the plane. Sir Roger Penrose is the expert in that field. How could someone go about proving that a certain tiling of the infinite plane with ...
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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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### 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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### How to smooth a very narrow quadratic bezier curve with a very low number of points?

I am a software engineer working on a whiteboard application for iOS. One of the features we have is a drawing tool. This tool gathers x,y coordinates and other information like the applied pressure, ...
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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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### Questions about the definition of a periodic pattern

In this article, Doris Schattschneider defines what a repeating (or periodic) pattern is. The definition goes as follows: A periodic pattern in the plane is a design having the following property: ...
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### Tetrahedra require octahedra; 5-cells require…?

It's well known that equilateral triangles tessellate $\Bbb R^2$ but regular tetrahedra do not tessellate $\Bbb R^3$. However, in three dimensions, we can make a a tessellation if we are permitted to ...
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### elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
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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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### Fill a cube with small cubes with different integer side lengths

We are given two (potentially unlimited) sets of cubes, say red cubes (with side $n$) and white cubes (with side $m$), with $m,n \in \mathbb{N} \setminus \{0\}, m \neq n$ (let's assume $n>m$). ...
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### Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
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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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### Is Dodecahedron tesselation somehow possible?

In this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender, ...
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### Find the length of each side of a square containing regular hexagons [duplicate]

I have to find the length of each side of a square such that all the regular hexagons of same length side and radius lying inside the square have centers either inside the square or on the boundaries ...