Modern Geometry Fall 2011

1. Modern GeometryFall 2011 Portfolio By: Alexandria Croom Professor: Dr. David James

2. Description of project • As a senior at Howard University, I enrolled in a directed reading course with Dr. David James. This project is a way to document my progress in the course. This document is a work in progress that will continued to be completed throughout the semester. I would like to especially thank Dr. James for being willing to teach this course. His enthusiasm has brought much joy to the learning process.

3. How was modern geometry Developed? • Check out this video!

4. Table of contents • The Geometries • Euclidean • Hyperbolic • Spherical • Axioms • Distance • Incidence • Lines • Real Ray • Theorems & Proofs • Unique Middle Theorem • Triangle Inequality

5. The geometries Examples of five systems

6. Euclidean Geometry • The Euclidean plane points and lines includes: • Coordinates - set of points is an ordered pair • Equation of lines • Nonvertical y= mx+ b • Vertical x= a • Distance Formula • e(AB)=

7. Hyperbolic Geometry • Consists of all points inside (but not on) the unit circle in the Euclidean plane. That is all (x, y) • Lines in H are the chords of the circle • Distance in the Hyperbolic Plane is calculated: • Where M, N are endpoints of chords, and A, B are points on a line

8. Spherical Geometry • S(r): notation for spherical plane surface of a sphere of radius r • S- set of all (x, y, z) with • Definition: Great circles- intersection of sphere with plane that cuts it in half • How is distance defined in S? • The distance between two points is the length of the minor arc of the great circle (line) through A and B

9. Axioms A description of one or more math structures consisting of terms, definitions, assumptions, theorems, and propositions

10. Axioms of Distance • For all points P and Q • Positivity: PQ • Definiteness: iff • Symmetry:

11. Axioms of incidence • In a system • There are at least two different lines • Each line contains at least two different points • Each two different points lie in at least one line • Each two different points P, Q, with lie in at most one line

12. Three axioms for the line • Definition: Point B lies between points A and C (written A-B-C) provided that • A, B, C are different collinear points, and • AB+BC = AC Unique Middle Theorem follows (see theorems and proofs) • Betweenness of Points Axiom • If A, B, and C are different collinear points and if then there exists a betweenness relation among A, B, and C. • Triangle inequality theorem follows (see theorems & proofs)

13. Three axioms for the line • Quadrichotomy Axiom for Points • If A, B, C, X are distinct, collinear points, and if A-B-C, then at least one of following must hold: • X-A-B, A-X-B, B-X-C, OR B-C-X

14. Three axioms for the line • Nontriviality Axiom • For any point A on a line there exists a point n with

15. Real Ray Axiom • Definition: Let A, B, be points with The ray is the set • For any ray and any real number with there is a point X in with • Leads to Theorem 8.2 ( see theorems & proofs)

16. Theorems & Proofs Key theorems and postulates for working in modern geometry

17. Unique middle theorem • If A-B-C, then both B-A-C and A-C-B are false. • Proof: • By definition of betweenness, A, B, C are distinct collinear point and AB + BC = AC. • Assume B-A-C. Then, BA + AC = BC BA + AB + BC = BC 2AB + BC= BC (by symmetry axiom AB =BA) 2AB=0 Thus, AB= 0. This implies that 0 + BC = AC By positive definiteness this would make A=B, contradicting the assumption that they are distinct points Therefore, B-A-C is false A-C-B is likewise false.

18. Triangle inequality • If A, B, C are distinct and collinear then AB + BC ≥ AC

19. This is still a work in progress! • Special Thanks to : • Dr. David James • And • The Howard University Mathematics Department