True proximity in plane and space (Island and star best friends)

[Ultimadark]

”True” proximity in plane and space (”Island and star best friends”)

”True” proximity in plane and space (”Island and star best friends”)

(To read this post in a more pleasant format, click here. My questions are in that document though, you will find them in this post.)

Recently I was on holiday in Greece, on the island of Karpathos. While I was relaxing on the beach, I got to thinking about a certain mathematical problem. The closest island to Karpathos is the island of Kasos, a few kilometers southwest of Karpathos. The problem that first appeared in my mind was “If Kasos is the closest island to Karpathos, what is the probability that Karpathos is the closest island to Kasos?”. (For those of you who have been to Karpathos, or those of you who study the map close enough, you will know that the island of Saria to the north of Karpathos is actually closer. Let’s just ignore that for the moment).

As I continued to think about this problem, I called the phenomenon “true closest islands”, i.e. if Karpathos and Kasos where true closest islands, Kasos would be the closest island to Karpathos and Karpathos would be the closest island to Kasos. Another term, perhaps better suited, would be to call them “Island best friends”. If you study the map below, you will see that Karpathos and Kasos are NOT island best friends. The closest island to Karpathos (the biggest island on the map) is indeed Kasos (the second largest island, southwest of Karpathos), but there is another smaller island that is closer to Kasos than Karpathos is (this island is called Armathia, lies to the northwest of Kasos and is colored green on the map).

Additional explaining of my definition of "A is closest to B"

What my program does, is that for each point A (1 point is considered to be an island) it calculates the distance to all the other points. It checks which distance is shortest, and the island that has the shortest distance to A is what I consider "the island closest to A".

An example: Consider 5 points (islands): A, B, C, D, and E. I want to find out which island is closest to A. I calculate the distance from A-B, A-C, A-D and A-E. I compare the distances, and find that of the distances, the distance A-D is the shortest. Then I consider D to closest to A. If, when doing the same calculations for D, I find that the distance D-A is the shortest, A is also the closest to D, and they are "island best friends". If some other distance, let's say D-E is shorter then D-A, the islands A and D are not "island best friends".

Map showing Karpathos and nearby islands (from Google Maps)

Of course, the way islands are placed in the real world is not random; it depends on the way continental plates are shaped etc. As I got home from my vacation however, I continued to think about the problem of finding the probability that islands are island best friends. I have some experience with a program called Maple, so I decided to use maple to find out about the probability that islands are best friends. I have now created a “program” in Maple that calculates this probability for randomly distributed islands on a rectangular map (grid), which easily allows the user to set the number of islands to place, the grid size and the number of simulations the program will run. Below is a sample of a map created by this program.

A map of a simulation with 200 islands, map size 1000x1000

This particular map consists of 200 point-sized islands placed randomly on a 1000x1000 grid (map). The program then calculated how many of the islands have an “island best friend”. I instructed the program to run this simulation first 100, then 1000, then 10000 times. Below is a histogram showing the percentage of “island best friends” for the 10000 simulations.

Probability distribution, 200 islands, 1000x1000 grid, 10000 simulations

In the case of 200 point-sized islands randomly placed on a 1000x1000 grid, “the most probable probability” is that around 62-64 percent of the islands have an island best friend. Now, one might wonder how this changes when the map size is changed or the number of islands is changed. Here is one example with the same map size as before, 1000x1000, but with a 100 islands instead of 200. Shown is the result of 1000 simulations.

Probability distribution, 100 islands, 1000x1000 grid, 1000 simulations

The peaks are at about the same place, but this histogram is a little bit wider than the one with 200 islands.
You might also wonder what the probability distribution looks like if we move from 2 dimensions to 3 dimensions, and instead randomly place point-sized stars in a cube or cuboid. I modified my program to allow me to do these calculations, i.e. to calculate the percentage of “star best friends”. Below is an example of one simulation of 200 randomly placed stars in a “space map” of size 100x100x100.

A map of a simulation with 200 stars, map size 100x100x100

I have so far not yet have time to do a lot of calculations on the probabilities in space. Below is a distribution that shows the probability that stars will have a best friend, based on a space map of size 100x100x100, 200 stars and 1000 simulations.

Probability distribution, 200 stars, 100x100x100 grid, 1000 simulations

It’s pretty similar to the one made from 10000 simulations, 1000x1000 map with 200 islands showed earlier, except that the histogram has moved a little bit to the left. Also noteworthy is that these two configurations have the same number of possible positions for the stars/islands, namely 1 million.

Questions i have
• Is this the right forum to post this thread? Do you have any tips on websites where I might post this and get some kind of response/input/ideas from other people?
• Do you know if anyone has studied this problem before?
• Is the probability distribution shown in my histograms a normal distribution?

What’s left for other people to do?
• 4 dimensions and higher.
• Further studying of different map sizes and number of islands.
• More efficient algorithms for faster calculations.
• Maps that aren’t rectangular or cuboidal, for example circular and spherical.
• Actual statistics of real-world islands.

About me
I’m a 24 year old Swedish male, studying to be a math and physics teacher.

 

[HallsofIvy]

After all of that, most of which are irrelevant, you have not said what is most important- what is your specific definition of "A is closest to B". Do you mean that "of all islands other than A itself, B is closest" or "of all islands other than B itself, A is closest". Those are very different. Further, because islands are not points, how are you defining the distance between two islands- as the shortest distance between any two on each island or as the distance between geometric "centers" of each island?

 

[Ultimadark]

After all of that, most of which are irrelevant, you have not said what is most important- what is your specific definition of "A is closest to B". Do you mean that "of all islands other than A itself, B is closest" or "of all islands other than B itself, A is closest". Those are very different. Further, because islands are not points, how are you defining the distance between two islands- as the shortest distance between any two on each island or as the distance between geometric "centers" of each island?

First of all, thank you for reading through it (I know the story about Karpathos etc. is irrelevant; I wanted to give some background to why I was doing these calculations).

Thank you for pointing out that I didn't define" A is closest to B", I will try to do it in this reply and then put the explanation in the original post as well:

What my program does, is that for each point A (1 point is considered to be an island) it calculates the distance to all the other points. It checks which distance is shortest, and the island that has the shortest distance to A is what I consider "the island closest to A".

An example: Consider 5 points (islands): A, B, C, D, and E. I want to find out which island is closest to A. I calculate the distance from A-B, A-C, A-D and A-E. I compare the distances, and find that of the distances, the distance A-D is the shortest. Then I consider D to closest to A. If, when doing the same calculations for D, I find that the distance D-A is the shortest, A is also the closest to D, and they are "island best friends". If some other distance, let's say D-E is shorter then D-A, the islands A and D are not "island best friends". I hope this answers your question.

As to treating the islands as points, it's how I do because I figured that would be easiest. My program is perhaps not a calculation of how the probability for real islands is; rather, I call the points "islands" because that's how I started thinking about this problem. The calculations might be more correct in 3D when I consider the points to be stars, since on a cosmological scale I would guess that stars can be treated as points when calculating the distance to other stars.