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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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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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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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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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 ...
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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 ...
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Tessellations of Flower like Functions of the form k + Sin[mx]/n

When, if ever, does the shape of the polar plot of k + Sin[mx]/n from $0\leq x\leq 2 \pi$, where k,m,n $\in$ N form a monotiling in the Euclidean Plane? If some values of k,m, and n allow for this ...
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hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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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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861 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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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 ...
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Questions about Centroidal Voronoi Tessellation

I am reading the paper "Centroidal Voronoi Tessellations:Applications and Algorithms" by Du, and I need some help understanding the following parts of the proofs. On pg 659 of the paper, Du writes as ...
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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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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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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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1answer
49 views

Hypersphere - Pattern matching using Centroid, Radius and Diameter

I have a hyper-sphere formed with set of $n$-dimensional data points. I could calculate centroid ($X_0$), radius($R$) and diameter($D$). Using these $X_0, R, D$, how I can find whether the a given ...
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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 ...
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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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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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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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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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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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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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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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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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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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86 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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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?
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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 ...
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143 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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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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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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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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309 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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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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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 ...
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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 ...
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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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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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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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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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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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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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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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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 ...
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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 ...
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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.
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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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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 ...