Unsolvable?? problem..

[Duplicity]

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 more than 10° (degrees) to any other?
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).
Any angle below 180° has to be greater than 1.

 

[Steven G]

If there was no degree restrictions, then what would be the answer to your question? Now consider the given restriction.

 

[Duplicity]

I think if there were no degree restrictions, it would be infinite numbers.

 

[Steven G]

I think if there were no degree restrictions, it would be infinite numbers.

Correct, now consider the given restriction. What would you say now?

 

[Dr.Peterson]

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 more than 10° (degrees) to any other?
Prove your result.

Forget the AI. What do you think the answer should be?

Try changing the problem to 2 dimensions, and answer it. It's easier to picture it that way. Then you can think about what, if anything, changes in 3D.

I'm wondering if you are misinterpreting the question, or didn't write what you mean. Where did the question come from?

 

[MaxWong]

There are infinitely many.
Proof: Suppose the lines all go through the point $\mathbf x \in \mathbb R^3$. Pick any unit vector $\mathbf v \in \mathbb R^3$. Construct a plane $\Pi$ that passes through the point $\mathbf{x} + \mathbf{v}$ and has normal vector $\mathbf v \in \mathbb R^3$. On the plane $\Pi$, construct a circle centred at $\mathbf{x} + \mathbf{v}$ and with radius $\tan 5^\circ \approx 0.0875$. Pick any point $\mathbf c \in \Pi$ such that $\mathbf c$ lies inside the circle. Let $L_{\mathbf c}$ be the line that passes through $\mathbf x$ and $\mathbf c$. For any points $\mathbf c_1, \mathbf c_2$ in the circle, the angle between $L_{\mathbf c_1}$ and $L_{\mathbf c_2}$ must be less than 10 degrees.

 

[Duplicity]

Forget the AI. What do you think the answer should be?

Try changing the problem to 2 dimensions, and answer it. It's easier to picture it that way. Then you can think about what, if anything, changes in 3D.

I'm wondering if you are misinterpreting the question, or didn't write what you mean. Where did the question come from?

I’ve done that before posting here. It is 18 lines only.

The question just came from inner-me.

I thought of a sphere, and lines that go through its origin. How should they be placed, to make min 10 degree angles?

Then I realized, we don’t need a sphere. (It might make life easier though)

I have a few solutiona in mind, which will take long time to solve for me, but still I would not be sure if one of them will be a final answer.

Let me know please if there is a misundrstood point in my q

 

[Duplicity]

2

Forget the AI. What do you think the answer should be?

Try changing the problem to 2 dimensions, and answer it. It's easier to picture it that way. Then you can think about what, if anything, changes in 3D.

I'm wondering if you are misinterpreting the question, or didn't write what you mean. Where did the question come from?

Easy in 2D

 

[Duplicity]

There are infinitely many.
Proof: Suppose the lines all go through the point $\mathbf x \in \mathbb R^3$. Pick any unit vector $\mathbf v \in \mathbb R^3$. Construct a plane $\Pi$ that passes through the point $\mathbf{x} + \mathbf{v}$ and has normal vector $\mathbf v \in \mathbb R^3$. On the plane $\Pi$, construct a circle centred at $\mathbf{x} + \mathbf{v}$ and with radius $\tan 5^\circ \approx 0.0875$. Pick any point $\mathbf c \in \Pi$ such that $\mathbf c$ lies inside the circle. Let $L_{\mathbf c}$ be the line that passes through $\mathbf x$ and $\mathbf c$. For any points $\mathbf c_1, \mathbf c_2$ in the circle, the angle between $L_{\mathbf c_1}$ and $L_{\mathbf c_2}$ must be less than 10 degrees.

I’m sorry but this is above my understanding. Can you put it in simple drawing.

Eithwr you took my question wrong, or I am not able to follow.

 

[The Highlander]

Easy in 2D

I would imagine that (taking the centre as a 'hinge' point) each of the lines could be moved up (and down) by 10° until it was vertical in order to extend the concept into 3 dimensions. Once they were vertical they would all be coincident and would, therefore need to be replaced by a single line at that stage.
That would result in (18² - 17 = 307) lines in total (if my analysis is correct. )

 

[khansaheb]

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

Imagine a 3-D cone with sidewalls at 10°. Also imagine we need to fill the cone with straws - very very thin straws. Any of those straws will meet with any other straw at the apex of the cone at an angle ≤ 10°. As you can visualize, there is no set upper number of straws (lines). Thinner the straws - more of those you can pack inside the cone.

 

[Dr.Peterson]

Here is why I said,

I'm wondering if you are misinterpreting the question, or didn't write what you mean. Where did the question come from?

You initially asked,

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

That means that every angle between any two lines is less than or equal to 10°.

This is easy: draw them all within a 10 degree sector. That's what people have said when their answer is infinite.

But now you say,

The question just came from inner-me.

I thought of a sphere, and lines that go through its origin. How should they be placed, to make min[imum] 10 degree angles?

That appears to mean that you want every angle between two lines to be greater than or equal to 10°.
That's a very different question, and in 2D is what you illustrated.

So then you said,

Either you took my question wrong, or I am not able to follow.

That is, you expressed your question incorrectly (which is not uncommon when someone makes up his own question), and we all interpreted differently than your intent. It's easy to get this sort of wording wrong!

Given what you apparently mean to ask, the AI is at least saying things that make a little sense, though I don't think it works. Your question is sort of like circle packing on a sphere, which can't be done just in terms of area, any more than circle packing on a plane can.

I'd have to think more about what can be done; but at least I now have an idea of what you intended to ask, and maybe the rest of us do, too. Post #10 appears to be an attempt at that question, but I can't tell whether his thoughts are correct.

 

[The Highlander]

Here is why I said,

You initially asked,

That means that every angle between any two lines is less than or equal to 10°.

This is easy: draw them all within a 10 degree sector. That's what people have said when their answer is infinite.

But now you say,

That appears to mean that you want every angle between two lines to be greater than or equal to 10°.
That's a very different question, and in 2D is what you illustrated.

So then you said,

That is, you expressed your question incorrectly (which is not uncommon when someone makes up his own question), and we all interpreted differently than your intent. It's easy to get this sort of wording wrong!

Given what you apparently mean to ask, the AI is at least saying things that make a little sense, though I don't think it works. Your question is sort of like circle packing on a sphere, which can't be done just in terms of area, any more than circle packing on a plane can.

I'd have to think more about what can be done; but at least I now have an idea of what you intended to ask, and maybe the rest of us do, too. Post #10 appears to be an attempt at that question, but I can't tell whether his thoughts are correct.

If you can't tell whether my "thoughts are correct" then they probably aren't; I didn't give it a great deal of thought.

I just rotated one line upwards in 10° increments until it was vertical, thereby creating 18 lines in a vertical plane (analogous to the 18 lines in the horizontal plane shown in the picture).

I then assumed that this could be repeated for each of the remaining 17 lines in the picture to create a 'sphere-full' of lines but there would then be 18 coincident vertical lines (hence the minus 17).

But I didn't use any clever maths to reach that result and I certainly didn't make any attempt to prove it.

 

[Duplicity]

I would imagine that (taking the centre as a 'hinge' point) each of the lines View attachment 37887could be moved up (and down) by 10° until it was vertical in order to extend the concept into 3 dimensions. Once they were vertical they would all be coincident and would, therefore need to be replaced by a single line at that stage.
That would result in (18² - 17 = 307) lines in total (if my analysis is correct. )

I think rotating this drawing 17 more times in vertical axis, makes all lines to be apart ar least 10 degrees. Which is 324 lines. So that needs to be the least in answers. But my feeling is that a more clever idea could prove it can have much more than 324.

 

[Duplicity]

I would imagine that (taking the centre as a 'hinge' point) each of the lines View attachment 37887could be moved up (and down) by 10° until it was vertical in order to extend the concept into 3 dimensions. Once they were vertical they would all be coincident and would, therefore need to be replaced by a single line at that stage.
That would result in (18² - 17 = 307) lines in total (if my analysis is correct. )

I think rotating this drawing 17 more times in vertical axis

Here is why I said,

You initially asked,

That means that every angle between any two lines is less than or equal to 10°.

This is easy: draw them all within a 10 degree sector. That's what people have said when their answer is infinite.

But now you say,

That appears to mean that you want every angle between two lines to be greater than or equal to 10°.
That's a very different question, and in 2D is what you illustrated.

So then you said,

That is, you expressed your question incorrectly (which is not uncommon when someone makes up his own question), and we all interpreted differently than your intent. It's easy to get this sort of wording wrong!

Given what you apparently mean to ask, the AI is at least saying things that make a little sense, though I don't think it works. Your question is sort of like circle packing on a sphere, which can't be done just in terms of area, any more than circle packing on a plane can.

I'd have to think more about what can be done; but at least I now have an idea of what you intended to ask, and maybe the rest of us do, too. Post #10 appears to be an attempt at that question, but I can't tell whether his thoughts are correct.

you are absolutely right. That was my typo. Actually my original question to AI was:

Consider a sphere in 3D.
How many lines can you draw, which intersect on the center of the sphere, but the angle between them maximum 10 degrees.
Please explain your thinking steps.
Do not count overlapping lines

Here is why I said,

You initially asked,

That means that every angle between any two lines is less than or equal to 10°.

This is easy: draw them all within a 10 degree sector. That's what people have said when their answer is infinite.

But now you say,

That appears to mean that you want every angle between two lines to be greater than or equal to 10°.
That's a very different question, and in 2D is what you illustrated.

So then you said,

That is, you expressed your question incorrectly (which is not uncommon when someone makes up his own question), and we all interpreted differently than your intent. It's easy to get this sort of wording wrong!

Given what you apparently mean to ask, the AI is at least saying things that make a little sense, though I don't think it works. Your question is sort of like circle packing on a sphere, which can't be done just in terms of area, any more than circle packing on a plane can.

I'd have to think more about what can be done; but at least I now have an idea of what you intended to ask, and maybe the rest of us do, too. Post #10 appears to be an attempt at that question, but I can't tell whether his thoughts are correct.

I am terribly sorry, that was a typo. I meant the min angle to be 10 degrees.

I opened a new thread to make it clear:

Unsolvable? problem. (with corrected the typo)

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°...

www.freemathhelp.com

And my original question to claude.ai was:
Consider a sphere in 3D.
How many lines can you draw, which intersect on the center of the sphere, but the angle between them maximum 10 degrees.
Please explain your thinking steps.
Do not count overlapping lines

Sorry again.

 

[Duplicity]

There are infinitely many.
Proof: Suppose the lines all go through the point $\mathbf x \in \mathbb R^3$. Pick any unit vector $\mathbf v \in \mathbb R^3$. Construct a plane $\Pi$ that passes through the point $\mathbf{x} + \mathbf{v}$ and has normal vector $\mathbf v \in \mathbb R^3$. On the plane $\Pi$, construct a circle centred at $\mathbf{x} + \mathbf{v}$ and with radius $\tan 5^\circ \approx 0.0875$. Pick any point $\mathbf c \in \Pi$ such that $\mathbf c$ lies inside the circle. Let $L_{\mathbf c}$ be the line that passes through $\mathbf x$ and $\mathbf c$. For any points $\mathbf c_1, \mathbf c_2$ in the circle, the angle between $L_{\mathbf c_1}$ and $L_{\mathbf c_2}$ must be less than 10 degrees.

Correct!. Please take a look at the corrected one, as my intention was to find the number of lines with at least 10 degrees of angle.

Unsolvable? problem. (with corrected the typo)

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°...

www.freemathhelp.com

 

[Duplicity]

If you can't tell whether my "thoughts are correct" then they probably aren't; I didn't give it a great deal of thought.

I just rotated one line upwards in 10° increments until it was vertical, thereby creating 18 lines in a vertical plane (analogous to the 18 lines in the horizontal plane shown in the picture).

I then assumed that this could be repeated for each of the remaining 17 lines in the picture to create a 'sphere-full' of lines but there would then be 18 coincident vertical lines (hence the minus 17).

But I didn't use any clever maths to reach that result and I certainly didn't make any attempt to prove it.

There was a typo in the question which made it funny, and I corrected that in the new thread:

Unsolvable? problem. (with corrected the typo)

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°...

www.freemathhelp.com

sorry for taking your time