Unsolvable? problem. (with corrected the typo) "How many lines can you draw in a 3D space, which intersect on a single point, and none of them has..."

Duplicity

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This just stuck to my mind, and I tried several AI's with no satisfactory answers.

The problem is very short.

How many lines can you draw in a 3D space, which intersect on a single point, and none of them has an angle less than 10° (degrees) to any other? (Angle between any two lines are 10° or more.)

Prove your result.


Thoughts, hints and info:

  • Claude.ai was the best performing among others. It tried a way of steradians. It calculated a vector and its 10° circular area that fills the surface of a containing sphere. By dividing the spheres area to that area the result was simplified to (4𝜋) / ( 2𝜋.(1-cos(5°)) ). The result than becomes 525.58, and it concluded that it must be 525, as you can have only integer number of lines.
  • This makes a lot of sense, on the other hand, it somehow accepts the area of the cirle such that, non-euclidian circles fill all the area of a sphere. (which is not true)
  • All other AI's failed deeply ( I have not tried Chat GPT-4o)
With different approches, lots of answers could be found, but the proof is nearly impossible.

More hints:
If the problem was about to find 180° (instead of 10°) , the answer is 1 ( overlapping lines cannot be counted).
 
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As I previously suggested, this can be turned into a question about packing of equal circles on a sphere. That is generally a problem for computer searches rather than exact mathematical solution. Here is an abstract of an article I found, as an example:


It is also related to maximizing the minimum distance between points on a sphere (though there you are given a fixed number of points and have to find the minimum angle):


As it says there,

Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24. There are conjectured solutions for many other cases, including those in higher dimensions.​
 
As I previously suggested, this can be turned into a question about packing of equal circles on a sphere. That is generally a problem for computer searches rather than exact mathematical solution. Here is an abstract of an article I found, as an example:


It is also related to maximizing the minimum distance between points on a sphere (though there you are given a fixed number of points and have to find the minimum angle):


As it says there,

Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24. There are conjectured solutions for many other cases, including those in higher dimensions.​

Thank you Dr. Peterson.

What I understand is that the solution has not been found yet, (except for special degrees), and this question may stay unsolved for decades, before someone puts a proof.

Maybe putting on a simulation on a computer can find the exact result, but the proof is somewhat hard.

Someone can prove that the number of lines must be >= some specific number. i.e 324

What I am looking is, for now, who can find the greatest amount that can be proved.


🙋‍♂️
 
Thank you Dr. Peterson.

What I understand is that the solution has not been found yet, (except for special degrees), and this question may stay unsolved for decades, before someone puts a proof.

Maybe putting on a simulation on a computer can find the exact result, but the proof is somewhat hard.

Someone can prove that the number of lines must be >= some specific number. i.e 324

What I am looking is, for now, who can find the greatest amount that can be proved.


🙋‍♂️
I'm not sure you understand what proof really means (or for that matter, exact -- which we couldn't know without a proof!). What proofs we have in this area, as I understand it, are computer algorithms, not just "simulations", but somehow breaking down all possible arrangements and determining which gives the greatest number.

But one thing we can do is to find an upper bound -- a number we know can't be exceeded. One simple approach is to divide the surface area of the sphere by the area of a spherical cap with an "angular diameter" of 10 degrees. Try doing that yourself (since we're here to help people learn to solve problems themselves, not just to give answers). The actual number has to be less than that, because there will be space between the circles. This is, in fact, what your AI found -- not the least upper bound, which you want, but an easy upper bound.

By the way, you again said it backward:
Someone can prove that the number of lines must be >= some specific number. i.e 324
That's easy: the number must be at least 1!

In any case, the point of saying that it's difficult is that you'd need a really good reason in order to get someone else to do the work. Why is this important for you?
 
Dear Dr. Peterson,

Thank you for your help on this.

I am sorry, I have not introduced myself, and why I am willing to solve this.

I am an electronics engineer 49yo, running my own company focused on lighting business. Math is somehow my hobby.
I have no given task about this question. Also this problem is not related to my business. I just found it as a good exercise to keep my brain fresh :)

For the upper bound, I could reach as follows:

4π / ( 2π(1 - cos(5°)) )

This gives calculation solves to ~= 525.58

As you mentioned, because of the spaces in between, this can be accepted as the upper bound.

I have also calculated a lower bound, which is creating 2d plane and having 18 lines that intersect on a single point with 10° angles.
Putting that plane on 3D, and rotating it 10°'s for 18 times, we can also say the lower bound is 18*18 = 324.

I am quite sure :

324 <= L <= 525

I will try to find a larger number for lower bound (324), and a smaller number for the upper bound (525), to approach the L.

(I am thinking of finding the minumum space left on a sphere, when filled with circles on its surface.)

Thank you for all the support.

Best,
 
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