Question: As in Problem, we are going to create a geometric structure from a finite number of points in the plane. Suppose that we have seven points a,b,c,d,e, f,g and that we consider a set of points to be a line if it contains exactly three points. We will select some lines, no two of which are parallel (and therefore, every pair of which intersect).
(a) How many possible lines are there?
(b) How many lines are parallel to the line abc?
(c) How many lines intersect the line abc? A pair of lines should only intersect at one point (otherwise they are in some sense curved).
(d) We need to make sure that none of the lines that intersect abc intersect each other at more than one point. How many lines intersect abc and intersect each other at no more than one point?
(e) Suppose we want a geometric structure with no parallel lines (because they remind us of bottomless pits). Can we select lines as in part
(d) so that there are no pairs of parallel lines?
(f) Now suppose further that we want exactly three lines to intersect at a point. Which lines can we choose for our structure?
(g) Try to draw the points and lines of this geometric structure. You will need to draw at least one line in an unusual way in order to succeed. (This geometric structure is called the Fano plane and has many interesting properties-for example, the roles of the points and the lines can be switched.)
Problem: We are going to construct a geometric structure from a set of lines using a finite number of points in the plane. Suppose that we have four points a,b,c,d and that we consider a set of points to be a line if it contains exactly two points.
(a) How many possible lines are there?
(b) How many lines are parallel to the line ab?
(c) How many lines intersect the line ab?
(d) Suppose we want a geometric structure with no parallel lines (because they remind us of bottom less pits-see Burkhard Polster's paper "YEA WHY TRY HER RAW WET HAT"). We might start with the line ab and all lines that intersect it; however, we would need to be sure that no two of these lines are parallel. How many lines can our geometric structure have if no two are parallel? And, which lines can we choose for our structure?
(e) Now suppose further that we want exactly two lines to intersect at a point. Which lines can we choose for our structure?
(f) Try to draw the points and lines of this geometric structure.