# Math problem

David113
Posers and Puzzles 03 Jun '08 12:46
1. 03 Jun '08 12:46
I don't have a solution...

Is there a set of points S in the plane, such that every straight line in the plane contains exactly 2 points of S?
2. 03 Jun '08 15:30
Originally posted by David113
I don't have a solution...

Is there a set of points S in the plane, such that every straight line in the plane contains exactly 2 points of S?
My first instinct is to say no. I think I can prove that no continuous curve works, and clearly no bounded set works. But there could be some bizarre collection of discrete points that does the job - I can't see how to disprove that quite yet.
3. 03 Jun '08 15:411 edit
What about the points that define a circle with a radius of infinity?
Then every line should intersect this circle in exactly two points, at the polar coordinates r1=r2=inf, and phi1=f and phi2=f+pi.
4. forkedknight
Defend the Universe
03 Jun '08 15:41
Originally posted by David113
I don't have a solution...

Is there a set of points S in the plane, such that every straight line in the plane contains exactly 2 points of S?
The closest thing that exists would probably be the Euclidean axes, which would work for all lines except y=0 and x=0
5. 03 Jun '08 16:15
Originally posted by forkedknight
The closest thing that exists would probably be the Euclidean axes, which would work for all lines except y=0 and x=0
And all lines through the origin if that does not count as 2. Me thinks.
6. 03 Jun '08 16:39
What about the points that define a circle with a radius of infinity?

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.
7. Palynka
Upward Spiral
03 Jun '08 16:52
Originally posted by Jirakon
What about the points that define a circle with a radius of infinity?

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.
A circumference is the immediate intuitive answer, but I fail to see what sense it makes to consider these intersections at infinity.

Give me a functional form taking the limit of r to infinity and I'll give you an epsilon that proves that there is at least a line that doesn't intersect on two points.
8. 03 Jun '08 17:19
Originally posted by Jirakon
What about the points that define a circle with a radius of infinity?

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.
Care to mention one of the points in that set then?
9. 03 Jun '08 17:20
Originally posted by Tera
And all lines through the origin if that does not count as 2. Me thinks.
And all lines parallel to either the x- or y- axis.
10. 03 Jun '08 17:54
Originally posted by FabianFnas
What about the points that define a circle with a radius of infinity?
Then every line should intersect this circle in exactly two points, at the polar coordinates r1=r2=inf, and phi1=f and phi2=f+pi.
A point is an ordered pair of two REAL numbers. No infinity.
11. Palynka
Upward Spiral
03 Jun '08 18:013 edits
Originally posted by mtthw
My first instinct is to say no. I think I can prove that no continuous curve works, and clearly no bounded set works. But there could be some bizarre collection of discrete points that does the job - I can't see how to disprove that quite yet.
I don't think it can be discrete points. Imagine if you have a set of discrete points and pick a line that intersects only two points. If you pick one of those points as center and keep rotating that line infinitesimally, then I don't see how it's possible to cover all possibilities with any mesh of discrete points.

Edit - So I'd say no, there is no such set S.

Edit 2 - Which probably means I'm wrong and missing an ingenious answer.
12. wolfgang59
03 Jun '08 19:20
I agree there is no such set.
,,
and I can kinda explain

but a rigorous proof is alluding me.ðŸ™„
13. 03 Jun '08 21:26
Originally posted by Palynka
A circumference is the immediate intuitive answer, but I fail to see what sense it makes to consider these intersections at infinity.
.

if you do include any trans finite numbers in the plane; then ithink that you need to include all trans finite numvbers.

another quwstion is: can you define a plane, and lines on that plane, so that set s exists.
14. 03 Jun '08 21:33
A point is an ordered pair of two REAL numbers. No infinity.

If that's the case, then there is no solution:

Suppose such a set exists in which all points have finite coordinates. Take the point(s) with the highest y-value (y_max). Now imagine the line y = y_max + 1. This line does not intersect any of the points. Therefore no such set exists.
15. 03 Jun '08 21:47
Originally posted by Jirakon
What about the points that define a circle with a radius of infinity?

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.
Well, it was a wild guess, the closest I can get to an answer.

Is it possible to construct a plane where inf really is included? As you can with R, i.e. R* ?