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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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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1answer
38 views

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 ...
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34 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 ...
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0answers
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Which functions preserve convexity of the cells of a polyhedral tessellation of $\mathbb{R}^n$?

Let $\mathcal{X}_1, \ldots, \mathcal{X}_m \subseteq \mathbb{R}^n$ be disjoint sets with $\bigcup_{i=1}^m \mathcal{X}_i = \mathbb{R}^n$. Furthermore, let each $\mathcal{X}_i$, $i = 1, \ldots, m$, be an ...
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29 views

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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3answers
75 views

Can I fix a tiled floor with only one wrong tile left? [closed]

If I have a $n\times m$ rectangular floor completely tessellated with $2\times 2 $ and $1 \times 4$ tiles and it now happens that I accidentally break one of those (no matter which one)- Can I then ...
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0answers
35 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
22 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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1answer
55 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
114 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 use ...
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3answers
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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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1answer
60 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 ...
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1answer
77 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?
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4answers
2k 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 ...
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1answer
90 views

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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4answers
461 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 ...
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0answers
36 views

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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1answer
41 views

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 ...
2
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1answer
207 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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1answer
37 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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2answers
229 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 ...
2
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1answer
184 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 ...
5
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1answer
98 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 ...
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3answers
1k 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 ...
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1answer
121 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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3answers
249 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 ...
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0answers
50 views

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 ...
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1answer
401 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 ...
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0answers
167 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?
5
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1answer
303 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 ...
13
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1answer
982 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 ...
2
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1answer
224 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.
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1answer
49 views

Largest online database of quasicrystal

This paper gives a broken link and Google was not of any help either yielding introductory materials on the subject. Anyone knows if the database (if it exists) has been moved to a new server and if ...
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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 ...
2
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0answers
68 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
224 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 ...
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1answer
345 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 ...
2
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1answer
173 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 ...
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1answer
109 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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4answers
588 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 ...
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1answer
185 views

Are there formulae to determine close-packing polyhedra?

Is there a formula to determine which polyhedra will tessellate in 3D without any spaces?
3
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1answer
154 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 ...
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2answers
2k views

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 ...
2
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1answer
272 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 ...
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3answers
7k views

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 ...
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2answers
815 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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2answers
269 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, ...