Maximum Number of Intersections of n Distinct Lines

Date: 10/07/98 at 23:41:03
From: Sally
Subject: Algebra 2

Find a pattern for the maximum number of intersections of n lines, 
where n is greater than or equal to 2.

I've tried drawing pictures but in geometry class I was taught that 
the number of intersections of a line is infinite, because the line 
never ends.  Was I informed wrong when I was in geometry?  I'm really 
confused with this whole problem.  Please help.  Thanks a lot.

Date: 10/08/98 at 12:55:21
From: Doctor Rick
Subject: Re: Algebra 2

Hi, Sally. I'm not exactly sure what you are thinking of when you say 
the number of intersections of a line is infinite, but in at least one 
sense, you are right, and the problem needs to be restated.

Let's just look at two lines. There are three possibilities according 
to Euclidean geometry:

  1. The lines are parallel, and they do not intersect.

  2. The lines intersect in exactly one point.

  3. The lines are the same line, so they intersect at every point 
     along the line.

(These possibilities arise from Euclid's Parallel Postulate; other 
kinds of geometry have been invented in which this is not true. For 
instance, two "straight lines" on the surface of the earth - or a 
sphere - will meet in exactly two points, on opposite sides of the 
earth.)

It's that third case that causes trouble. Is that what you are 
thinking about? The problem should really have said, "Find a pattern 
for the maximum number of intersections of n _distinct_ lines, where n 
is greater than or equal to 2." Then the maximum number is not 
infinite, and it's much more interesting. 

To get you started, think about a triangle - extend its line segments 
into lines, and you have 3 lines intersecting in 3 points. If you do 
the same with a square, you get two pairs of parallel lines; you can 
get more intersections if they are not parallel. See what you can do. 
Have fun with it!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/