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Results tagged with discrete-geometry
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user 3401
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
0
votes
Path connected coloured sets on the squared paper
Edit: This argument is wrong, as pointed out in the comments. I'll leave it up as a warning to others.
I am going to respond to the question under the assumption that this is about infinite paper (as …
- 4,922
4
votes
What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly r...
A colleague, Nicolas Broutin, who is not on MO, pointed me to a recent "preliminary report" of Gábor Pete, about joint work with Christophe Garban and Oded Schramm, on the scaling limit of the MST. In …
- 4,922
24
votes
3
answers
4k
views
What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly r...
Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay triangu …
- 4,922
2
votes
Cutting convex sets
There is a generalization of this question, where "area" and "circumference" are replaced by arbitrary "nice" measures (for the purpose of this answer, say absolutely continuous measures) $\mu$ and $\ …
- 4,922
16
votes
2
answers
1k
views
Are Penrose tilings universal? Do aperiodic universal tilings exist?
Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile …
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