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shurtados
shurtados
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Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even more, I think it is not possible to triangulate the euclidean plane with triangles that are $C$-bilipschitz with a unit triangulation for some constant $C$, because that would imply there is a bilipschitz map from $\mathbb{H^2}$ to the euclidean $\mathbb{R^2}$ and this is not possible, is that correct?

So my question is: is it possible to triangulate the euclidean plane with any kind of euclidean triangles such that at every vertex has degree 7. I've been trying to do it by hand but independent of the way I tryalways a lot of little triangles start appearing and I'm not certain if I can continue triangulating in such a way that the whole plane is covered.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even more, I think it is not possible to triangulate the euclidean plane with triangles that are $C$-bilipschitz with a unit triangulation for some constant $C$, because that would imply there is a bilipschitz map from $\mathbb{H^2}$ to the euclidean $\mathbb{R^2}$ and this is not possible, is that correct?

So my question is: is it possible to triangulate the euclidean plane with any kind of euclidean triangles such that at every vertex has degree 7. I've been trying to do it by hand but independent of the way I try a lot of little triangles start appearing and I'm not certain if I can continue triangulating in such a way that the whole plane is covered.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even more, I think it is not possible to triangulate the euclidean plane with triangles that are $C$-bilipschitz with a unit triangulation for some constant $C$, because that would imply there is a bilipschitz map from $\mathbb{H^2}$ to the euclidean $\mathbb{R^2}$ and this is not possible, is that correct?

So my question is: is it possible to triangulate the euclidean plane with any kind of euclidean triangles such that at every vertex has degree 7. I've been trying to do it by hand but always a lot of little triangles start appearing and I'm not certain if I can continue triangulating in such a way that the whole plane is covered.

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shurtados
shurtados
  • 1.1k
  • 6
  • 13

Euclidean triangulation of the plane with degree 7 at each vertex.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even more, I think it is not possible to triangulate the euclidean plane with triangles that are $C$-bilipschitz with a unit triangulation for some constant $C$, because that would imply there is a bilipschitz map from $\mathbb{H^2}$ to the euclidean $\mathbb{R^2}$ and this is not possible, is that correct?

So my question is: is it possible to triangulate the euclidean plane with any kind of euclidean triangles such that at every vertex has degree 7. I've been trying to do it by hand but independent of the way I try a lot of little triangles start appearing and I'm not certain if I can continue triangulating in such a way that the whole plane is covered.