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?