This time, one with a geometric flavor.

Two straight lines on the Euclidean plane

Rcan intersect at only a single point, while the addition of a third such line gives a maximum of three points of intersection. What is the maximum number of intersection points that can be produced by an arbitrary number of straight lines?^{2}

Ought to be fairly easy, I should think. Note that I assume no two lines are parallel, so trivial solutions like "an infinite number" are ruled out.

Each line would intersect every other line once. Therefore, there is one intersection point for every pair of lines. The maximum number of intersection points for n lines is then equal to n!/(2!(n-2)!).

Posted by: Zack | February 01, 2005 at 05:08 AM

The (n+1)th line intersects the n existing lines at most n times by a controversial axiom of Euclid's! Hence by induction the max number of intersections for n lines is 1/2.n(n-1). Actually this seems a little too easy... Have I missed something? Thinking about planes intersecting in R^3 now...

Posted by: Delmore Macnamara | February 01, 2005 at 01:01 PM

How dull! Same answer for all m-dimensional hyperplanes in R^(m+1) so far as I can make out, by same argument.

Posted by: Delmore Macnamara | February 01, 2005 at 01:19 PM

On sphere suppose answer is just n(n-1). How about S^m?

Posted by: Delmore Macnamara | February 01, 2005 at 01:25 PM

Fibonacci series, innit? Same inductive argument as Delmore, but I don't think the addition actually works out to 1/2.n(n-1).

Posted by: dsquared | February 01, 2005 at 01:37 PM

How many times do "great m-spheres" intersect in S^(m+1)? Still trying to visualise this...

Posted by: Delmore Macnamara | February 01, 2005 at 04:04 PM

n order to exclude an infinity of intersection points you only need to rule out colinear lines, not all parallel lines . Leaving non colinear parallel lines gives us the same inequality we were looking for .

Posted by: ogunsiron | February 02, 2005 at 01:37 AM

D'oh! Having switched on brain, now realize that two "great (m-1)spheres" in S^m intersect in precisely one (m-2) spheres". The two points in which great circles on a sphere intersect are "really" a 0-sphere. Probably this was obvious to everyone else. In any case, the 1/2.n(n-1) (if I did my adding up right) formula still holds.

Posted by: Delmore Macnamara | February 02, 2005 at 09:17 AM

Believe formula also holds for RP^m.

For torus S^1 X S^1 seem to get an (uncountable?) infinity of potential intersections because therexist irrational flows...

May be interesting to consider into how many regions n of the relevant (m-1)dimensional subspaces divide R^m, S^M & RP^m. At least one might get some different answers then, e.g. for instance with n=3, m=2, get 7 for plane but 8 for sphere.

Going away now to do some tax avoidance...

Posted by: Delmore Macnamara | February 02, 2005 at 10:06 AM