Q 3b b State the planedual of each of the axioms of this geo

b) State the plane-dual of each of the axioms of this geometry. In each case, state whether or not the plane-dual is true in the same geometry. Explain. 3· The axioms of Fano\'s geometry are as follows: i) There exists at least one line in this geometry. ii) Every line in this geometry has exactly three points on it. i) Not all points of the geometry are on the same line. iv) For any two distinct points, there exists exactly one line on both of them. v) Each two lines have at least one point on both of them. Can the diagram with points A, B, C, D, E, F, and G, below serve as a model for this geometry? Explain your answer. a) 8 Ca b) Prove that in Fano\'s geometry, each point lies on exactly three lines. The Axioms of the finite geometry of Desargues are as follows: 1· There exists at least one point 2. Each point has at least one polar 3. Every line has at most one pole 4.

Solution

3b. Pick a line l (which exists by Axiom 1). Choose any point P not on l (which exists by Axiom 3). Since l has 3 points (by Axiom 2), joining P to each of them gives three distinct lines through P (by Axiom 4 there is a line through P and each of these points. If one of these lines contained two points of l, it would have to be l by Theorem 1.7, but this contradicts the fact that P is not on l.) If there where another line through P, it would not meet l, contradicting Axiom 5, so there are exactly 3 lines through P. This argument takes care of all points not on l, to deal with a point on l, say Q, choose a line through P which does not contain Q and repeat the argument using Q and this line.

Alternate Method:  

(Using Theorem. 1.8) Let P be any point. There are 6 other points in the geometry besides P. P is joined to each by a line (Axiom 4). Since each line contains 3 points (Axiom 2), each of these lines contains two points besides P. Thus, there are at least 3 lines through P. Any other line through P would have to contain two points, one on each of two different lines that we have already constructed through P (otherwise Axiom 4 is violated), but this violates Axiom 4. So, there are exactly three lines through P.