All Questions
Tagged with plane-geometry co.combinatorics
8 questions
10
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How can we find n points on a plane so that as many pairs of points as possible have the same distance?
There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
- 159
6
votes
1
answer
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Convex planar regions with all area bisectors having equal length
Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries.
An area bisector (perimeter bisector) of a planar convex region is a chord ...
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13
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3
answers
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Koebe–Andreev–Thurston theorem - where can I find a proof?
Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
- 2,178
6
votes
1
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Bichromatic pencils
A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...
- 18.9k
3
votes
1
answer
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Planar subsets with many pairs of points on distance $1$ [duplicate]
Let $X$ be a subset of $\mathbb R^2$ consisting of $n$ distinct points. Let $d_1(X)$ be the number of pairs of points of $X$ on distance $1$ from each other. Define
$$d_1(n)=\sup_{X\subset \mathbb R^2|...
- 14.3k
28
votes
6
answers
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How fast are a ruler and compass?
This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...
- 513
24
votes
3
answers
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What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?
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 ...
- 4,922
18
votes
2
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Arrangements of points in the plane
Let $p_1,\ldots,p_n$ be a collection of distinct points in $\mathbb{R}^2$, no three of which lie on a line. For each $p_i$, let $\omega_i(p_1,\ldots,p_n)$ be the following ordered list (well-defined ...
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