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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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 ...
19
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2answers
927 views

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 honeycomb or some other not ...
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3answers
417 views

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 ...
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3answers
2k views

how to generate tesselation 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 ...
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1answer
1k views

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 ...
8
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1answer
132 views

How to determine whether a polytope is self-tessellating?

I'm looking for polytopes that can be tessellated by a finite number of scaled versions of themselves. I'll use the term component for such a scaled version in the text below. Self-tessellating ...
8
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2answers
353 views

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, ...
7
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1answer
552 views

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the ...
7
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0answers
53 views

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 ...
6
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4answers
952 views

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 ...
5
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1answer
374 views

How is tessellation defined in Mathematics?

Hi. I am a GCSE student and I am interested in Maths. I read few books on maths and learned some mathematical analysis. I know of convergent series but I would like to know how identical sets(not ...
5
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2answers
322 views

Has anyone discovered a convex space-filling 15-faced polyhedron?

I've been looking for extensive surveys regarding space-filling polyhedra, but have only come across Michael Goldbergs "Convex polyhedral space-fillers of more than twelve faces" from 1979, stating ...
5
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3answers
2k views

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 fll the plane without overlaping are the 3,4 and 6 sided ones. I have also heard about penrose tilings but this question ignores ...
5
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4answers
631 views

How many ways to arrange Lego bricks on a Lego board?

Let's say I have a board like this one (though significantly smaller, it's 4x7) and I have two 2x3 bricks. I'd like to know how many ways to arrange the bricks on the board. The bricks should ...
5
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1answer
123 views

Giant Pufferfish skin pattern―how could that be generated

I just started my investigations about tesselations and tilings for some very special kind of design Project. I came over that image: It shows a part of the Giant Pufferfish's skin and I am very ...
5
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0answers
101 views

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 ...
4
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3answers
288 views

Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes

It is know that a square can be dissected into other square such that no two of the squares have the same size. This is the simplest dissection of that kind: Is it also possible to dissect an ...
3
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4answers
4k views

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 ...
3
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2answers
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Employing optimal packing, how many circles (51 mm diameter) can I cut from a rectangle (330 mm×530 mm)

I know that I should use some kind of honeycomb structure but can't work out in which orientation I should arrange it. I've looked at a few websites now and although I have a slightly better idea of ...
3
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1answer
176 views

Triangulation on Euclidean Space

I have a couple of questions about triangulations of the Euclidean space: Is it possible to have an infinite triangulation of the Euclidean space $\mathbb{R}^2$ such that only a finite number of ...
3
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5answers
292 views

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 ...
3
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1answer
377 views

Is there a way to tessellate an area using triangles and minimize/specify the number of unique triangles?

Is it possible to tessellate a planar surface from triangles but with the following constraints: density (average number of triangles) can be varied. a finite set of unique triangles are used for ...
3
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2answers
175 views

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, ...
3
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1answer
271 views

Nets of Geodesic spheres

I would realize the papercraft of a geodesic sphere like this: It is the dual of the one discussed in THIS OTHER QUESTION . Where can I find the printable nets, or the online resources to create ...
3
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1answer
88 views

Not understanding this proof in Grünbaum-Shephard's Tilings and Patterns

I'm reading Grünbaum and Shephard's Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). ...
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0answers
201 views

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?
2
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1answer
258 views

Aperiodic hexagonal tiling?

Is there any known aperiodic tiling of the plane using hexagons? Wang tiles are a known aperiodic tiling using squares. I'm looking for something similar using hexagons.
2
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1answer
246 views

A Voronoi diagram with two dimensional generators on a “warped” plane

Consider this set of two dimensional generators (red polygons top left). A Voronoi diagram of these polygons is shown at bottom left. Now consider the same set of two dimensional generators on some ...
2
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1answer
330 views

Rotation of Tetrehedra for 3d Tessellation

I'm trying to render some $3D$ graphics with a bunch of tetrahedra. I'm trying to figure out how to rotate one tetrahedron such that it will be perfectly face-to-face with another tetrahedron. If this ...
2
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1answer
32 views

Edge length of hyperbolic tesselations

If I have a general uniform tesselation in hyperbolic plane (same configuration of regular polygons at every vertex, but multiple types of polygons allowed), how can I find the edge length and/or ...
2
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1answer
211 views

How do I calculate the unique k-dimensional hypersphere's center from k+1 points?

I'm working with the Bowyer-Watson algorithm to determine the Delaunay tessellation of stochastic points in k-dimensional space. This algorithm assumes that the center of a simplex can be used as the ...
2
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1answer
56 views

Shapes for tiling a circular disk?

Does anyone know of a catalog of sorts for what shapes are allowed for tiling a circular disk? For example, if you are allowed one piece to tile the disk, are all the possibilities essentially ...
2
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0answers
66 views

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 ...
2
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0answers
32 views

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 ...
2
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0answers
28 views

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 ...
2
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0answers
72 views

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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1answer
305 views

Questions about triangles, n-gons, and tessellations of the hyperbolic plane

Why there are infinitely many regular tessellations of the hyperbolic plane? Can there be a triangle made up of three straight lines in the hyperbolic plane? I know it's impossible since the angle ...
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2answers
229 views

Are there formulae to determine close-packing polyhedra?

Is there a formula to determine which polyhedra will tessellate in 3D without any spaces?
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2answers
45 views

how to find spherical coordinates of adjacent vertices surrounding central vertex in A3/D3 lattice

How could you define (using spherical coordinate system) all the adjacent vertices directly connected to a central vertex in a tetrahedral octahedral honeycomb? Alternatively it would be useful to get ...
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1answer
853 views

Points of symmetry of tessellation.

I was given this irregular hexagon: Then I was told to tessellate it: Now, I am being asked to find all the points on the hexagon (first picture) which are points of symmetry of my tessellation ...
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1answer
85 views

Heptagonal tesselations

Are there any tesselations of the Euclidean plane that use only regular polygons such that one of them is a heptagon? If so, what is the tesselation that uses the fewest different types of polygon ...
1
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1answer
41 views

form of groups of motions of tessellations

I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says: The triangle and hexagon tessellations have ...
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1answer
159 views

Mathematical notation to describe tiling shapes?

I stumbled across the following Wikipedia article which contained information on tiling by regular polygons. Underneath each image, it contained a sort of sequence of numbers which appears to be ...
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1answer
111 views

counting edges in tesselations of a torus

Tesselate a torus with finitely many simply connected polygons. Do not allow four or more of them to meet at a point. In counting the edges, don't count a "straight line" as just one edge if it's ...
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1answer
36 views

Maximum number of equilateral triangles in a circle

I am stuck with a question. Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or ...
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0answers
48 views

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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0answers
46 views

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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1answer
45 views

What does “uniqueness of composition” mean here? (Grünbaum-Shephard's Tilings and Patterns)

This is a follow-on from a previous question, in which I paraphrased Statement 10.1.1 of Grünbaum and Shephard's Tilings and Patterns. The original statement is shown below: where Figures 10.1.3 to ...
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0answers
233 views

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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1answer
117 views

What is the dual lattice of Kagome lattice?

We know that the dual lattice of a triangular lattice is the honeycomb lattice. What is the dual lattice of Kagome lattice?