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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 vertices have degree less or equal than 6?

  • If not, is it possible to have a triangulation where the average degree is greater or equal than 7? Here by average degree I mean the limit in $r$ of the average degree of all the points in the ball of center the origin and radius $r$.

Thanks!



Jim below answered my question with a nice example! Now I have a follow up related question:

  • Consider a density in the Euclidean space and randomly deploy points accordingly to this density. Now generate the corresponding Delaunay triangulation. Does there exists a density whose average degree is greater or equal than 7 almost surely?
  • 0
    Your related question is not going to work for a finite number of "randomly deployed points"2011-10-14
  • 0
    No, of course. The number of deployed points will be infinite.2011-10-14

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