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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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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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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Topological surface covered by hexagons and heptagons

I've found an interesting exercice that I don't know how to approach. It goes like this. We have a topological space which is Hausdorff, compact, connected and locally homeomorphic to ...
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Isohedral polygons that tile space in Voronoi tessellations

Besides the three uniform tilings of space in $\mathbb{R}^2$, are there other regular or semiregular tilings, by one or more types of isohedral polygon, that are also Voronoi tessellations of some ...
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Photo mosaics and the assignment problem

I have been having fun making mosaics from thumbnails. E.g. Until now, thumbnails have been uniform size and my aim has been to use every thumbnail, which makes it very straightforward to use the ...
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What is the optimal tiling of a regular n-gon in the plane?

I want to tile the plane with equal-sized regular polygons of $n$ sides. Obviously for some $n$, the tiles will be able to tessellate and cover the whole plane (e.g triangles, squares, hexagons) I ...
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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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prove that using 10 coins is enough to cover any 10 points in a plane [duplicate]

I recently got this puzzle, cant seem to prove it, i have manage to prove that for large enough N there is no solution, but not for 10. the puzzle asks to prove that using 10 coins is enough to cover ...
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Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
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Kelvin problem with restriction to only one type of polyhedron

If we only allow one type of polyhedron, what would be the answer to the Kelvin problem? Would the Kelvin conjecture be true?
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1answer
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What is the answer to the Kelvin problem if restriced our selves to isohedral polyhedra?

A few days ago I asked this question. It turned out to be a known unsolved problem in mathematics: The Kelvin problem. Now I'd like to slightly change the question: What is the optimal way to ...
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What is the analogon of the hexagonal grid in 3-dimensional space? Rhombic dodecahedral honeycomb?

Conjecture: The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb. Reasoning: "(The rhombic dodecahedral honeycomb) is ...
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Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like the square and the ...
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1answer
61 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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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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1answer
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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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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
55 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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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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1answer
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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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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
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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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1answer
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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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1answer
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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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365 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 ...