All Questions
Tagged with tiling gt.geometric-topology
10 questions
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Tiling the hyperbolic plane by non-regular quadrilaterals
We add a bit to Which polygons tessellate the hyperbolic plane?.
Question: Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the ...
- 5,979
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Regarding fundamental domain of 2 genus surface
Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 ...
- 613
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Which polygons tessellate the hyperbolic plane?
The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
It is well known that in Euclidean geometry, all triangles and all ...
- 5,979
4
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1
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169
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Absolute and relative tilings of the hyperbolic plane
In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it).
The ...
- 9,720
4
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Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?
This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
- 12.4k
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Order question about pentagonal tiling type 9 and type 10
People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see Pentagonal tiling.
These 15 types of pentagonal was named by finding date except type 9 and type ...
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5
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2
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408
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Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
- 6,353
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Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
- 6,353
2
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1
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295
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Aperiodic tiling of compact space by small number of basic tiles
Suppose we have compact space, like sphere or torus in particular dimension $d$.
Is it possible to construct aperiodic tiling in such setting? It seems obvious, answer is yes, because we may just ...
- 1,626
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What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?
I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of Weaire–...
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