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.

Tessellation is the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher, who was inspired by studying the Moorish use of symmetry in the Alhambra tiles during a visit in 1922. Tessellations are seen throughout art history, from ancient architecture to modern art.

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Can someone explain the math behind tessellation?

Tessellation is fascinating to me, and I've always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would've gone about calculating tessellating shapes. In my spare time, I'm playing a lot with a series of…
ocodo
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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 the other 15 cases (not covered immediately by…
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how to generate tessellation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's "Circle Limit" drawings, which use a Poincare disk…
LarsH
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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 investigation of tilings of rectangles by 1D animals, it is…
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doubly periodic functions as tessellations (other than parallelograms)

I think of a snapshot of a single period of a doubly periodic function as one parallelogram-shaped tile in a tessellation. Could a function have a period that repeats like a honeycomb or some other not rectangular tessellation?
futurebird
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Looking for references about a tessellation of a regular polygon by rhombuses.

A regular polygon with an even number of vertices can be tessellated by rhombii (or lozenges), all with the same sidelength, with angles in arithmetic progression as can be seen on figures 1 to 3. Fig. 1 Fig. 2 Fig. 3 I had already seen this kind…
Jean Marie
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Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right direction for an algorithm to calculate, and ideally…
James Coote
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Is it possible to uniquely number faces of a hexagonal grid with consecutive numbers?

You have a grid of regular hexagons. The aim of the game is to have each hex contain the numbers 1-6 on its edges. Each edge must also be connected to another edge that has a value one higher and one lower than the value of itself, with 6's wrapping…
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Is a regular tessellation $\{p,q\}$ always possible on some closed surface $S$?

Suppose that we are given positive integers $p$, $q$, $V$, $E$, $F$, a closed surface $S$ and the Euler characteristic $\chi(S)$ of that surface. Suppose that we also know that the following relations hold:…
gebruiker
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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 discusses this. Maybe it is too trivial, but I am…
Artium
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What is the most frequent number of edges of Voronoi cells of a large set of random points?

Consider a large set of points with coordinates that are uniformly distributed within a unit-length segment. Consider a Voronoi diagram built on these points. If we consider only non-infinite cells, what would be (if any) the typical (most frequent)…
mbaitoff
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Why are triangles, squares and hexagons the only polygons with which it is possible to tile a plane?

Ok so I have heard that the only regular polygons which can completely fill the plane without overlapping are the 3,4 and 6 sided ones. I have also heard about Penrose tilings but this question ignores them. How can one prove that there isn't…
Asinomás
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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 size at least $N/7$ such that every pair of points…
Pedro
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What are the conditions for a polygon to be tessellated?

Upon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate spheroid that we call a planet. I'm most interested…
Nick
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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 \,\, exactly \,\, 17 \,\, wallpaper \,\, groups…
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