youngs, pappus, desargue Finite geometry.pptx

youngs, pappus, desargue

1. Finite geometry Young’s, Pappus, Desargues Kenny d. peria I Maed math

2. Finite geometry Finite Geometry is any geometric system that has only a finite number of points. It followed the axiomatic systems in the late 1800s. It was developed while attempting to prove the properties of consistency, independence, and completeness of an axiomatic system. Geometers wanted models that fulfilled specific axioms. Often the models found had finitely many points which contributed to the name of this branch of geometry. When it is confined to a plane, all finite geometries are either projective plane geometry (has no parallel lines) or affine plane geometry (has parallel lines).

3. Young’s Finite geometry John Wesley Young ● Mathematics professor at Dartmouth College ● Introduced the axioms of projective geometry ● Was a proponent of Euclidean geometry and held it to be substantially "more convenient to employ" than non-Euclidean geometry.

4. Young’s Finite geometry Axioms of Young’s Geometry 1. There exists at least one line. 2. Every line of the geometry has exactly three points on it. 3. Not all points of the geometry are on the same line. 4. For two distinct points, there exists exactly one line on both of them. 5. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line.

5. Young’s Finite geometry Theorem of Young’s Geometry 1. For every point, there is a line not on that point. 2. For every point, there are exactly four lines on that point. 3. Each line is parallel to exactly 2 lines 4. There are exactly 12 lines. 5. There are exactly 9 points

6. Young’s Finite geometry Theorem 1: For every point, there is a line not on that point. To prove: Let p be a point and by axiom 2 let L be the line P (point) is not on L (line) thus we say that it satisfies the theorem 1.

7. Young’s Finite geometry Theorem 2: For every point, there are exactly four lines on that point. To prove: ● Let C be a point, then by Theorem1 there exist a line L2 set. C∉L2. ● By Axiom 2, L1 contains 3 points, A, B and C. ● By Axiom 4, there are three lines L1, L2 and L3 (the red lines). from intersecting D with the points A, B and C ● By Axiom 5, there is a line L2 contains D but not any of the point A, B and C (L2 is parallel to L1). Thus, there are at least 4 lines through D To prove exactly 4: ● L2 is parallel to L1. ● By Axiom 5, there is no fifth line on D that is parallel to L1. ● By Axiom 4, there is no fifth line on D that intersect L1. Thus, there are exactly four lines through D.

8. Young’s Finite geometry Theorem 3. In Young’s geometry, each line is parallel to exactly two lines. To prove: To prove: (At least two lines) ● By Axiom 1, Let L be a line. ● By Axiom 2, L contains 3 points p1, p2 and p3 ● By Theorem 2, there are three other lines (L1, L2 and L3) on p1, ● By Axiom 2, L1 contains two other points, s2 and s1 ● By Axiom 5, there are two lines (orange lines) r1, r2 parallel to L (violet line), since s2, s1∉L) Thus, there are at least 2 lines parallel to L. To prove: (Exactly 2 lines) ● Assume that there is a line r3 parallel to L, ● By Axiom 5, L is the unique line that is parallel to r3 through the point p1 ● Since p1 is on L1, then L1 must intersect m3 (Axiom4 C!) Thus, there are only two lines parallel to l.

9. Young’s Finite geometry Theorem 4. In Young's geometry, there are exactly 12 lines. To prove: (At least 12 lines) ● By Axiom 1 there is a line l, which is on three points p1, p2, and ● p3, by Axiom 3. ● By Theorem 2, on each of these points there are exactly three other lines. No two of these additional lines can be the same, since none may be on two points of l. ● So far, we have 10 lines, and there can be no other line which intersects l. ● By Theorem 3, there are exactly two lines parallel to l. Thus, become 12 lines. To prove: (Exactly 12 lines) ● By Theorem2, 3, any 13th line must either intersect or be parallel to l , but this is impossible Therefore, there are exactly 12 lines.

10. Pappus’ Finite geometry Pappus of Alexandria (340 A.D) ▪ Philosopher ▪ lived about the time of the Emperor Theodosius the Elder [379 AD - 395 AD] ▪ the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.

11. Pappus’ Finite geometry Axioms of Pappus Geometry 1. There exists at least one line. 2. Every line of the geometry has exactly three points on it. 3. Not all points of the geometry are on the same line. 4. There is exactly one line through a point, not on a line parallel to the given line. 5. If P is on a point not on a line, there exists exactly one point P’ on the line such that no lines join P and P’. 6. With the exception of Axiom 5, if P and Q are distinct points, then exactly one line contains both of them

12. Pappus’ Finite geometry Theorem 1.9: Theorem of Pappus If A, B, and C are three distinct points on one line and A’, B’, and C’ are three different distinct points on a second line, then the intersections of line AC’ and line CA’, line AB’ and line BA’, and line BC’ and line CB’ are collinear. If points A, B, And C are on one line and A’, B’, and C’ are on the other line, then the points of intersections of the lines AB’ and BA’, AC’ and CA’, and BC’ and CB’ lie on the common line called the Pappus Line of the configuration.

13. Pappus’ Finite geometry Theorem 1.9: Theorem of Pappus If A, B, and C are three distinct points on one line and A’, B’, and C’ are three different distinct points on a second line, then the intersections of line AC’ and line CA’, line AB’ and line BA’, and line BC’ and line CB’ are collinear. If points A, B, And C are on one line and A’, B’, and C’ are on the other line, then the points of intersections of the lines AB’ and BA’, AC’ and CA’, and BC’ and CB’ lie on the common line called the Pappus Line of the configuration. Theorem 1.10 Each point in the geometry of Pappus lies on exactly three lines.

14. Pappus’ Finite geometry Theorem 1.10 Each point in the geometry of Pappus lies on exactly three lines. Sample Proof in Euclidean Geometry Point/Line Duality If you have any diagram of points and lines, you can replace every point with coordinates (a, b, c) with the line coordinates (a, b, c) and vice-versa, and you still have a valid diagram. If you do this in Pappus Theorem, you will get another version of Pappus’ theorem, called the “Dual” version.

15. Pappus’ Finite geometry Pappus’ Theorem: Dual Formulation Pick any two points. Through it, draw blue lines, green lines, and red lines. Find the intersection of the line of different colors. Draw the lines that connect the two blue-green crossings, green-red crossings, and red-blue crossings.

16. Pappus’ Finite geometry