Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
3answers
60 views

Neighboring solids in tetrahedral-octahedral honeycomb

In the tetrahedral-octahedral honeycomb, each vertex seems to be incident to 6 octahedra and 8 tetrahedra: Such simple combinatorial fact is probably well-known, or perhaps even obvious. However, ...
1
vote
1answer
20 views

Substitution matrix for the Ammann-Beenker tiling

On the wikipedia page for the Ammann-Beenker tiling, it says the following: I am trying to derive the substitution matrix for and show that its eigenvalues are $(1\pm\sqrt{2})^1$... so far the things ...
6
votes
2answers
120 views

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 \,...
0
votes
0answers
18 views

Proof for replicating tiles

A rep-tile is a shape which can be divided into n smaller shapes that are similar E.g. a square can be reproduced by 4 smaller squares rep-n means that a shape can be created from n smaller copies. ...
2
votes
1answer
39 views

Permutations for tilings / tessellations

I'm an artist, many years past my maths a-level - so apologies for any idiocy up front. At the moment, I'm working with tilings/tessellations of right-angled isosceles triangles. I have two designs, ...
4
votes
0answers
46 views

How does one generalize the dual of a tiling by regular polygons to other tilings?

This question is motivated by some work in curriculum design for outreach math. Normal tilings of the Euclidean plane are a popular topic since people of all ages can play with it, but on the other ...
2
votes
0answers
45 views

Prove that the shortest path between two points in a Delaunay triangulation minimizes angle at each step.

Say we have a set of points $p_{k (k \in K)}\in P$ Poisson distributed in a real coordinate plane $X$ residing in ${\rm I\!R}^{2}$ and with Euclidean distance function $d$ and $K$ is a set of indices. ...
2
votes
1answer
31 views

Self-Similar Polygon Tessellations

It is well-known that the only regular polygons which tessellate the plane (using only one shape) are the triangle, square, and hexagon. However, there are many more tessellations of the plane by ...
3
votes
0answers
51 views

Determine the tiles intercepted by a given function in an irregular tessellation

Consider an irregular tessellation of the plane composed of convex tiles, such as the following one. For each tile, the coordinates of each vertex (black dots) are known up to a finite precision. ...
1
vote
0answers
76 views

What is the best way to tessellate sphere into equal area in any level of detail? HEALPix or Geodesic Grid or another method?

I want to tessellate sphere into a grid in my 3D world map. There was 2 ways that I was consider right now, HEALPix and Geodesic If I use it specifically for world map that could be zoom into any ...
1
vote
1answer
106 views

Moscow Seven Sisters

Fix $n$ points in the plane in generic position, i.e. no three of them on the same line, etc. The number of lines joining two of them is ${n \choose 2}$. The number of regions in which $\ell$ lines ...
0
votes
1answer
35 views

Finding the right grid tessellation, for a chess board with elliptical geometry

I came up with the idea of non-euclidean chess(the chess board I'm working on will have 2D elliptical geometry) but I came across a problem. The question: What kind of grid do I put on it(what shapes ...
2
votes
1answer
180 views

Spiral path on a Penrose tiling

I would like to color a Penrose tiling by following a "spiral path", painting each tile according to a given color sequence. In this picture, I illustrate what I am looking for: The dashed line ...
2
votes
0answers
40 views

One element space tessellations

Disclaimer: i am bioinformatician and programmer, please excuse if my wording and definitions are far from elegant and occasionally imprecise. Intro: I am interested in space tessellations of n ...
0
votes
3answers
61 views

Are there any known plastic constant triangles?

I am trying to determine if there are any known plastic constant triangles. By this I mean specifically triangles for which all sides are powers of the plastic constant, $p\approx1.324717957244746$. ...
0
votes
0answers
20 views

How to recover spatial dimension from a tesselation?

With a tiling of a space, you start with the space and tile it. But if I have a tiling, can I recover the dimension? If you give me a triangle tiling or a square tiling or etc of the plane, how can I ...
1
vote
0answers
27 views

Multiple use of Fermat's Principle , or Simple, Finite Tessellation (Unfolding)

First consider a specific question. Given a triangle and two interior points, a person at Point-1 is required to 'touch' all the 3 sides of the triangle exactly once and return to Point-2. Find the ...
3
votes
2answers
61 views

What determines if a hyperbolic tiling with ideal vertices is regular?

I'd like to say I need this community's help in clearing my mind of the clutter that leads me to this contradiction: As an example, the Wikipedia article for the Infinite-order triangular tiling ...
0
votes
0answers
16 views

Pseudo-Hillbert curve equivalent for hexogonal and triangular tesselations.

Triangles, squares and hexagons can all be used to fill a surface (tessellation). For now let's assume the surface has a limited number of tiles (triangles, squares or hexagons) The goal is to ...
6
votes
1answer
202 views

Name and number of “equilateral tessellations with same angles on all vertexes”

Longer background, shorter questions below: Tessellations of 2D plane consisting of regular polygons are usually described with vertex configurations such as "3.4.6.4" meaning that there are a ...
3
votes
1answer
218 views

How to discretize a sphere?

I would like to discretize a sphere into icosahedra whose vertices are equidistant, i.e., I want to plot $n$ equidistant points on the surface of a sphere. I am familiar with R, Python, and Matlab. ...
5
votes
1answer
57 views

Is there a name or a reference for these aperiodic rhomboidal tilings?

Fill space with unit cubes and then remove all cubes that are not completely within a given half space. An isometric view of the remaining cubes will look like the following image. This is in ...
1
vote
0answers
54 views

Uniform distribution on infinite binary grid

Is there an infinite binary array, so that the distribution of all possible binary $N\times N$ arrays on it will be uniform? In other words, randomly picked $N\times N$ square will be $\sim\mathrm{...
1
vote
3answers
171 views

Is it possible to tile the pane with a semi-regular tesselation given only the vertex type?

I want to write a computer program that tessellates the plane with semi regular tiling, i.e these tilings: https://en.wikipedia.org/wiki/List_of_convex_uniform_tilings If I start with the vertex ...
6
votes
0answers
91 views

A series that converges to π/3 [duplicate]

While surfing on YouTube, I stumbled into this video which gave me a new insight about the well-known series $$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}+ \ldots $$ The idea shown ...
0
votes
0answers
25 views

How to construct an algorithm to move along a discrete grid at constant steps?

Assume we have a tessellation of cubes of length one of a 3D space. We have a starting point $P$ and a direction $\vec V$ of length one. Our goal is to move the point $P$ by direction $\vec V$ such ...
3
votes
1answer
94 views

Do the loops “Snakes” by M.C. Escher correspond to a regular tilling of the hyperbolic plane?

In M.C. Escher's Snakes, you have three snakes going through some loops. I'm more interested in the loops though. In this image, a ring model of the hyperbolic plane is given. It is given by $w=e^{za}...
1
vote
1answer
44 views

Prove that only triangles, quadrilaterals, and hexagons will Tesselate the plane

Prove that only triangles, quadrilaterals, and hexagons will Tesselate the plane So, I have almost completed the proof, I will write it down all the way to the part I'm stranded in. Theorem: only ...
0
votes
0answers
31 views

How many shapes with Euler characteristic 2 of which all faces border the same amount of faces (through edges and vertexes combined)?

I'm trying to create a shape that I can map to both a plane and a sphere. I don't care about distortions at all, so, for example, platonic solids work. However, I need a solution that works for a ...
3
votes
2answers
160 views

Is this Escher artwork a tessellation of the half-plane model of hyperbolic space?

One Escher's prints look like this. A similar one is this. These look suspiciously like Poincaré half-plane models of the hyperbolic plane (there are pieces of artwork by Escher specifically based on ...
2
votes
1answer
85 views

Why do matching rules make a substitution tiling aperiodic?

Wikipedia hath written: Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution ...
6
votes
1answer
79 views

How can group theory explain movement on a hexagonal tiling?

(As a prelude, I have no formal math training other than high school. I am a beginner with group theory and have just recently begun picking it up and seeing its potential uses.) Imagine an infinite ...
4
votes
2answers
76 views

Is there a geometric realization in integer-sided squares of $70^2 =\sum_{j=1}^{24} j^2 $?

I saw this in the NAdigest mailing list, and it was obviously suggested by $70^2 =\sum_{j=1}^{24} j^2 $: From: Gerhard Opfer gerhard.opfer@uni-hamburg.de Date: November 06, 2017 Subject: ...
2
votes
1answer
47 views

Can we tile 3d-space so that every solid has exaclty 4 next neighbours?

I'm curious if this exists, but simply don't know. I'm thinking of euclidian, flat, 3d-space. Preferably filling with one type of solid. Next neighbours means shares a plane with. Here are my thoughts ...
0
votes
1answer
131 views

regular k-gons that can tessellate a plane

What is the sum of all possible values of $k$ for which regular $k-$gons can tessellate a plane? This is one of the NAT problems. While I am familiar with tessellations, I don't quite get what they ...
1
vote
1answer
37 views

Hyperbolic honeycombs

I am puzzling with hyperbolic honeycombs or space filling stacking of hyperbolic regular polyhedrons and it looks all very complex to me . As far as I tried I could not even find a stacking where two ...
2
votes
0answers
40 views

Is there a one-to-one mapping between every planar graph and a Delaunay tessellation of the plane?

For every planar undirected finite graph is there a corresponding Delaunay tessellation of the plane, and vice versa? Has this been proven?
13
votes
2answers
247 views

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 ...
1
vote
0answers
95 views

Does this hinged tiling have a name?

This hinged tessellation is made of rectangles centred on a triangular lattice. Does anyone know if it has a name or has any reference to some paper which studied this configuration or other info such ...
6
votes
2answers
721 views

Description of the order-5 square tiling of the hyperbolic plane as a graph

What is a description of the graph representing the faces of the Order-5 square tiling. Alternatively, this can be seen as the graph of vertices of the its dual, the Order 4 pentagon tiling. ...
1
vote
1answer
57 views

Finding center of tessellating hexagons

Given a hexagon such that opposite angles and radii are equal, how can I find the center point of any number of other hexagons (of the same dimensions) that form a tessellation? In this (very ugly) ...
0
votes
2answers
136 views

Are there higher-dimensional tessellations touching only nearest neighbours?

One property of a hexagonal tiling is that each hexagon only touches its nearest neighbours - in contrast to e.g. a square tiling, where each corner also connects to a second-to-next neighbouring ...
12
votes
1answer
465 views

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. ...
2
votes
1answer
63 views

Why should $\frac{2\pi}{5}$-rotations extend to symmetries of this platonically tesselated surface?

I am reading in The geometry of Klein's Riemann surface, by Karcher & Weber. Section 3 starts off like this: Let's try to construct a genus $2$ surface $M^2$ that is platonically tesselatied$^*$...
10
votes
1answer
272 views

Interesting tiling with a lot of symmetrical shapes

I have such an interesting observation: if I take a square grid and rotate it over itself by atan(3/4) , it forms a structure which has four axes of reflection symmetry: The resulting structure is ...
2
votes
0answers
62 views

Are these three-spoke dovetailing tile tessellations known to geometry? (Animation)

Trispokedovetiles Animation Webpage I've programmed a webpage using Javascript to display an animation which shows a range of different trispokedovetiles, each of which can be specified by a "CIRCLE" ...
2
votes
1answer
142 views

16-cell honeycomb (4D cross-polytope tesselation)

A 4-dimensional cross-polytope (also called 16-cell) is a regular polytope whose vertices are all permutations of $(\pm1,0,0,0)$. It is known that this body tiles the space $\mathbb{R}^4$ by ...
2
votes
1answer
60 views

What is a good algorithm, and framework, to calculate centres of gravity or mass (cog)?

I'd like to take an photograph, subdivide it into a tesselation, either of squares, or (ideally), hexagons, and then find the centre of gravity (or, if you prefer, centre of mass) of each cell of the ...
1
vote
1answer
121 views

Round random points to the nearest vertices of a regular tessellated hexagon

For a simulation I need to be able to take points that are scattered around randomly and move their point to the nearest vertex of a tessellated regular hexagon. That way each point is sitting on a ...
1
vote
0answers
183 views

Extension of Planar Algorithms to Higher-Dimensional Voronoi Diagrams

Voronoi diagrams are not new, and there are many established algorithms (Fortune's, Lloyd's) for generating them (or their duals, the Delaunay triangulation). There are many recent-ish papers too, ...