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This was the homework from last time: Complete reading Chapter 2

This was the homework from last time: Complete reading Chapter 2 Exercise: 15a and revisit exercise 10. Major Exercises: 4 a, b, 7 Begin thinking about the Problem Set (which you will have time to complete in class next time). Definition

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This was the homework from last time: Complete reading Chapter 2

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  1. This was the homework from last time: Complete reading Chapter 2 Exercise: 15a and revisit exercise 10. Major Exercises: 4 a, b, 7 Begin thinking about the Problem Set (which you will have time to complete in class next time).

  2. Definition A projective plane is a model of incidence geometry having the elliptic parallel property (any two lines meet) and such that every line has at least three distinct points lying on it. (strengthened Incidence Axiom 2) Definition An affine plane is a model of incidence geometry having the Euclidean parallel property: For every line l and for every point P that does not lie on l, there exists a unique line mthrough P that is parallel to l. I-1: For every point P and every point Q not equal to P, there exists a unique line lincident with P and Q. I-2: For every line lthere exist at least two distinct points incident withl. I-3: There exist three distinct points with the property that no line is incident withall three of them.

  3. Exercise 15 a: Four distinct points, no three of which are collinear, are said to be a quadrangle. Let P be a model of incidence geometry for which every line has at least three distinct points lying on it. Show that a quadrangle exists in P. I-3 1. There exist three distinct non-collinear points A, B, and C. I-1 and definition of non-collinear 2. There exists a unique line m incident with A and B, that does not contain C. by hypothesis 3. There exists a point D on m such that D ≠ A or B. m D A 4. There exists a unique line l incident with D and C.   B I-1  5. There exist a point E on l such that E ≠ D or C. by hypothesis   Points E and C are not on line m, because if they were, l and m would both have two points in common (D and either E or C) which is not possible. Similarly, A and B are not on line l. Therefore, points A, B, C, and E are four points, no three of which are collinear. Therefore, A, B, C, and E are a quadrangle. E C l I-1

  4. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane.

  5. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane. Let P be a point in A. There exists a line l not containing P (Prop 2.4) a)

  6. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane. Let P be a point in A. There exists a line l not containing P (Prop 2.4) Then l has exactly n points. (exercise 14d) a)

  7. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane. • Let P be a point in A. There exists a line l not containing P (Prop 2.4) • Then l has exactly n points. (exercise 14d) • There exist exactly n unique lines l1, l2, l3, . . . , ln , through P and each of these n points, for a total on n lines. (I-1) a)

  8. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane. • Let P be a point in A. There exists a line l not containing P (Prop 2.4) • Then l has exactly n points. (exercise 14d) • There exist exactly n unique lines l1, l2, l3, . . . , ln , through P and each of these n points, for a total on n lines. (I-1) • There exists a unique line m through P parallel to l. (definition of affine plane - Euclidean postulate) a)

  9. Major Exercise 4: Let A be a finite affine plane so that, according to exercise 14d, all lines in A have the same number of points lying on them: let n be this number, with n  2. Show the following: Each point in A has n + 1 lines passing through it. The total number of points in A is n2. The total number of lines in A is n(n + 1). The number n is called the order of the finite affine plane. • Let P be a point in A. There exists a line l not containing P (Prop 2.4) • Then l has exactly n points. (exercise 14d) • There exist exactly n unique lines l1, l2, l3, . . . , ln , through P and each of these n points, for a total on n lines. (I-1) • There exists a unique line m through P parallel to l. (definition of affine plane - Euclidean postulate) • Since m andl have no points in common, m is not equal to any ofl1, l2, l3, . . . , ln. Therefore there are n + 1 lines in A passing thorough P. a) b) Each of the n lines in step 3 contains n - 1 points other than P, for a total of n(n -1) = n2 – n points. However, line m contains n additional points (including P). Thus there are (n2 – n) + n = n2points in A. c) There are exactly n + 1 lines for each of the n points on line m (including point P), (step 5). Thus there are n(n + 1) lines in A.

  10. Major Exercise 7: Some authors characterize projective planes by three axioms: Axiom I-1, the elliptic parallel property, and the existence of a quadrangle. Show that a model of those axioms is a projective plane under our definition, and conversely. We already know by our definition of projective plane, that axiom I-1 and the elliptic parallel property hold, and that a quadrangle exists (by exercise 15a). Now begin with a model having the three given axioms and prove it is a projective plane by our definition.

  11. Major Exercise 7: Some authors characterize projective planes by three axioms: Axiom I-1, the elliptic parallel property, and the existence of a quadrangle. Show that a model of those axioms is a projective plane under our definition, and conversely. We already know by our definition of projective plane, that axiom I-1 and the elliptic parallel property hold, and that a quadrangle exists (by exercise 15a). Now begin with a model having the three given axioms and prove it is a projective plane by our definition. Axiom I-1 and the elliptic parallel property are given.

  12. Major Exercise 7: Some authors characterize projective planes by three axioms: Axiom I-1, the elliptic parallel property, and the existence of a quadrangle. Show that a model of those axioms is a projective plane under our definition, and conversely. We already know by our definition of projective plane, that axiom I-1 and the elliptic parallel property hold, and that a quadrangle exists (by exercise 15a). Now begin with a model having the three given axioms and prove it is a projective plane by our definition. Axiom I-1 and the elliptic parallel property are given. Axiom I-3 is guaranteed by the existence of the quadrangle.

  13. Major Exercise 7: Some authors characterize projective planes by three axioms: Axiom I-1, the elliptic parallel property, and the existence of a quadrangle. Show that a model of those axioms is a projective plane under our definition, and conversely. We already know by our definition of projective plane, that axiom I-1 and the elliptic parallel property hold, and that a quadrangle exists (by exercise 15a). Now begin with a model having the three given axioms and prove it is a projective plane by our definition. Axiom I-1 and the elliptic parallel property are given. Axiom I-3 is guaranteed by the existence of the quadrangle. All that remains it to verify Strengthened Axiom I-2 (every line has at least three points).

  14. Axiom I-1 and the elliptic parallel property are given. Axiom I-3 is guaranteed by the existence of the quadrangle. All that remains it to verify Strengthened Axiom I-2 (every line has at least three points). Proof: Let the given quadrangle be composed of the distinct points (call them vertices) P, Q, R, and S. Letl be the line containing P and Q, and m be the line containing R and S. l and mmeet at a point O by the elliptic parallel property. Point O is not one of the four vertices of the quadrangle, since if it were, either PQ or RS would contain three of the vertices, violating the definition of quadrangle. So l and m each have at least three points on them. P  S  Q  R 

  15. Axiom I-1 and the elliptic parallel property are given. Axiom I-3 is guaranteed by the existence of the quadrangle. All that remains it to verify Strengthened Axiom I-2 (every line has at least three points). Proof: Let the given quadrangle be composed of the distinct points (call them vertices) P, Q, R, and S. Letl be the line containing P and Q, and m be the line containing R and S. l and mmeet at a point O by the elliptic parallel property. Point O is not one of the four vertices of the quadrangle, since if it were, either PQ or RS would contain three of the vertices, violating the definition of quadrangle. So l and m each have at least three points on them. Let k be an arbitrary line not equal to l or m and not containing any of the vertices. Then by Axiom I-1, lines l, m, andtheline through P and R each must intersect k (elliptic parallel property) at distinct points. Hence k has at least three points. P  S  Q  R  If k contains one of the vertices, say R, use the line through S and Q instead of the line through P and R.

  16. 10 a. Any one-to-one correspondence will work, as long as it is set up so that if P P and Q Q, we make sure PP corresponds to QQ. This is rather obvious since axiom I-1 requires all three pairs of points of each model to have unique lines through them, and axiom I-3 requires all three points of each model to be non-collinear.

  17. 10 b. No, all four point models of incidence geometry are not isomorphic.    

  18. 9b. Points are lines, lines are planes (all passing through point O. 9d. Points are pairs of antipodal points, lines are great circles (all on a given sphere). 10c. Show that the models in 9b and 9d are isomorphic. P  R Q  Q  R  P Every plane through point O intersects the sphere in a unique great circle with two unique antipodal points. Let the “point” {P, P} in 9d correspond to the point in 9b represented by unique line PP. It is easy to prove that this correspondence is an isomorphism.

  19. We proved that the model below is an affine plane. What would have to be added to the model to change it from an affine plane to a projective plane? M H A  E T

  20. Is this model isomorphic to M C F A H A G    C B E T D E B

  21. The model on the right is called the projective completion of the affine plane on the left. It is the smallest possible projective plane. M C F A H A G T D E B

  22. True or False: Euclidean geometry is an affine plane. What is its projective completion? In order to form the projective completion for Euclidean Geometry we must: Add points at which every pair of parallel lines meet. 2. Add a line that contains all of these points.

  23. True or False: Euclidean geometry is an affine plane. What is its projective completion? Define a relation “” (read “is equivalent to”) between lines in an affine plane as follows: For any two lines l and m, lm if l = m or lm

  24. Consider the set of all lines “equivalent” to a given line l. We call this set an equivalence class and we denote it by [l]. Note: If l m, then [l] = [m] Let’s “add” all equivalence classes to the Euclidean (affine) plane as POINTS and agree that all lines in [l] meet at point [l]. Let’s call these new points “points at infinity” and call all other points “ordinary” points. Do we now have a projective plane? Define l to be the “line” containing all points at infinity. Do we now have a projective plane?

  25. Principle of Duality: Let P be a projective plane. Define the dual interpretation P* of P to have as its points the lines of P, as its lines the points of P, and as its incidence the same incidence relation. Then P* is a projective plane. Major Exercise 3 • Let P be a finite projective plane so that all lines in P have the same number of points lying on them (as per exercise 14c). Call this number n + 1, with n  2. Show: • Each point in P has n + 1 lines passing through it. • The total number of points in P is n2 + n + 1. • The total number of lines in Pis n2 + n + 1.   By the Principle of Duality

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