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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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
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 ...
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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 ...
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 ...
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1answer
110 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
950 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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2answers
228 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
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 ...
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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
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 ...
107
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4answers
9k 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
925 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
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, ...