Connecting N points pairwise in volume V, average density of lines?

In summary, the number of segments that intersect a small volume v is given by the sum of the potential energy of each particle in the volume, and the density of line segments drops off near the inside boundary of the volume.
  • #1
Spinnor
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Say we take N random points in a volume V and connect the points pairwise with line-segments. I would like to estimate the number of segments that intersect some small volume v, and where N is large enough so that any small random sample volume v will have many intersections. Little volume v may or may not enclose any points.

Does this get me close? First let us estimate the total length of the line segments, Ʃ. Use an average separation distance D between each pair of points. The distance D is of order one half the length of the volume V, V = L^3, D = L/2.

The total number of segments is N(N-1)/2 so an estimate for the length of line-segments,

Ʃ = D*N(N-1)/2 for large N this is about D*N^2/2
Ʃ ≈ D*N^2/2

Assume this total length is evenly divided into each small volume v. The length in volume v is the fraction [d^3/D^3] times Ʃ,

[d^3/D^3]*Ʃ = [d^3/D^3]*D*N^2/2 = σ

Assume the average length of the line-segments that intersect the little volume v is one-half the length the little volume v, d/2.

Then the average number of line-segments in v, ω, is,

ω = σ/[d/2] = {[d^3/D^3]*D*N^2/2}/[d/2] = d^2*N^2/D^2

using Wolfram calculator,

http://www.wolframalpha.com/

Using our Universe as an example, let d = 1m, N = 10^80, D = [3.5*10^80m^3]^.3333 ≈ 7*10^26

ω = 2*10^106 segments intersecting a volume of 1m^3.

We can ask what must the size of the volume v above be so that on average there will be only one line-segment intersecting it.

Set ω = 1 = d^2*N^2/D^2 now d is unknown and we use N and D above,

d = D/N = 7*10^26/10^80 = 7*10^-54m. If we are too near a point this estimate is bad. If we enclose a point ω jumps by about N

We can also ask how many points must a volume v have so that ω above changes significantly because of the additional line-segments from the enclosed points.

I made many bad estimates but I think I'm within a factor of a billion above?

Thanks for any help!
 
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  • #2
This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?
 
  • #3
Tinyboss said:
This is an interesting problem! I wonder, are the endpoints important, or do they just serve as a way of defining some random lines? Would it serve your purpose equally well to have some number of lines (the infinite kind) randomly distributed throughout your space?

I had something in mind for the endpoints. If you have N charged particles in some volume then the electrstatic potential energy is given by the sum of the potential energy of each pair of charged particles. And for N particles with mass the gravitational potential energy (Newtonian Gravity) is given by a similar pairwise sum. In either case the length of the line segments is important. So if there were some unseen connection between each pair of particles in our universe I was curious just how dense would those connections be. Just an idea. It seems for the numbers I used if there were some type of unseen connections they would have to be very dense. I also had the idea that disturbances of these connections might be gravitons and photons. Spinors would have to pop out of such an idea and I don't know if that is possible.

After some thought I realized that for N random points in a box that the density of line segments would fall off near the inside boundary of the box. Things could be more uniform in some closed space like S^3 but then a pair of points in S^3 gives two (or more) geodesic paths between the points? Maybe you include both paths?

So do you think I'm in the ballpark with those numbers?
 

1. How do you determine the optimal number of points to connect in a given volume?

The optimal number of points to connect in a given volume depends on several factors, including the desired density of lines, the available space, and the purpose of the connections. A general rule of thumb is to try to evenly distribute the points in the volume and increase or decrease the number of points as needed to achieve the desired density.

2. What is the average density of lines that should be achieved when connecting N points in a given volume?

The average density of lines can vary depending on the specific application and the density of points, but a common goal is to achieve a density that allows for efficient communication and connectivity between the points. For example, in a network of sensors, a density of lines that allows for reliable data transmission would be considered optimal.

3. How does the average density of lines affect the overall connectivity in the volume?

The average density of lines directly affects the overall connectivity in the volume. A higher density of lines generally leads to a greater number of connections between points, resulting in a more interconnected and efficient network. On the other hand, a lower density of lines may result in disconnected or isolated points, reducing the overall connectivity.

4. What methods can be used to connect N points in a given volume?

There are various methods that can be used to connect N points in a given volume, including physical cables, wireless connections, and virtual connections through software. The appropriate method will depend on the specific application and the desired density and reliability of the connections.

5. How can the connectivity of a network of N points in a given volume be optimized?

The connectivity of a network of N points in a given volume can be optimized by adjusting the density of lines, utilizing efficient connection methods, and regularly monitoring and maintaining the connections. Additionally, using algorithms and optimization techniques can help determine the most optimal way to connect the points in a given volume.

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