Let no one ignorant of geometry enter here.

1. Let no one ignorant of geometry enter here. Inscription above the gate of Plato’s Academy – 4th century B.C.

2. Non-Euclidean Geometry Charles Koppelman • Office: Mathematics and Statistics Building, Room 018 • Phone: 678-797-2051 • Email: ckoppelm@kennesaw.edu • Website: www.science.kennesaw.edu/~ckoppelm • Office Hours: Wednesday 2:30 – 4:00 • Thursday 11:00 – 12:00 • Other times by appointment

3. From the syllabus GRADING: There will be two (2) group problem sets each worth 10% of the final grade. Satisfactory completion of assigned end-of-chapter exercises is worth 10% of the final grade. There will be a midterm and a final exam, each worth 35% of the final grade. Graduate students will also work in groups to complete at least one independent project. ATTENDANCE: Regular class attendance is essential. PowerPoint notes for each class, as well as all handouts, will be posted both on D2L and on my website. In the event of absence students are responsible for all material, assignments and announcements made in class. * Dates tentative

4. Topics: • Students will analyze Euclid’s postulates and identify the gaps that led to the development of non-Euclidean geometry. • Students will apply the axioms of incidence geometry to finite affine and projective planes. • Students will apply Hilbert’s axioms to develop the theorems of neutral geometry. • Students will prove the equivalence of the angle sum theorem and Hilbert’s Euclidean axiom of parallelism. • Students will identify the logical flaws in attempts through history to prove the Euclidean parallel postulate. • Students will develop fundamental theorems of hyperbolic geometry by applying one negation of Hilbert’s Euclidean axiom of parallelism. • 7. Students will examine the independence of the parallel postulate and the • Beltrami – Klein Model. (Coverage of this topic is dependent on time • constraints)

5. Euclidean and Non-Euclidean Geometries Fourth Edition Marvin Jay Greenberg From the Preface “This book presents the discovery of non-Euclidean geometry and the subsequent reformulation of the foundations of Euclidean geometry as a suspense story.”

6. Theorem: Any point inside a circle is on the circle.  A  Given: Point A is inside circle O. Prove: Point A is on circle O. . T

7. . T

8. Construct Ray OA intersecting the circle at point P. - - T

9. Choose point T on ray OA so that (OA)(OT) = r2 T

10. Construct the perpendicular bisector of segment AT. (OA)(OT) = r2 T

11. Construct the perpendicular bisector of segment AT. (OA)(OT) = r2 T S

12. OA = OM – AM (OA)(OT) = r2 OT = OM + MT = OM + AM (OA)(OT) = r2 = (OM – AM)(OM + AM) r2 = OM2 – AM2 OR2 – RM2 (AR2 – RM2) r2 = – from ORM from ARM A r2 = OR2 – AR2 r2 r AR = 0 T Since the distance from A to R is 0, A and R must be the same point and, therefore, A is on the circle. S

13. Where is the flaw in the proof?

14. The recommended prerequisite for this course is either MATH 3395 (Geometry) or MATH 7714 (Geometry from Multiple Perspectives). The major part of the content of these two courses is the study of Euclidean Geometry The Introduction section of the textbook says that after completing chapters 1 – 5: Your Euclidean conditioning should be shaken enough so that in Chapter 6 we can explore “a strange new universe,” one in which triangles have the “wrong” angle sums, rectangles do not exist,and parallel lines may diverge or converge asymptotically.

15. The Introduction of the textbook also states: “Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. “ Only impractical dreamers spent two thousand years wondering about proving Euclid’s parallel postulate, and if they hadn’t done so, there would be no spaceships exploring the galaxy today. Albert Einstein stated that without this new conception of geometry, he would not have been able to develop the theory of relativity.

16. Let no one ignorant of geometry enter here. Inscription above the gate of Plato’s Academy – 4th century B.C. Plato and Einstein weren’t the only two great minds of history to recognize how important geometry was to our intellectual development. http://www.youtube.com/watch_popup?v=DUCZXn9RZ9s&vq=medium

17. From Chapter 1 of the textbook: Two requirements that must be met for us to agree that a proof is correct. Requirement 1. Acceptance of certain statements called axioms or postulates without further justification. Requirement 2. Agreement on how and when one statement “follows logically” from another, i.e., agreement on certain rules of logic.

18. From Chapter 1 of the textbook Euclid’s monumental achievement was to single out a few simple postulates, statements that were acceptable to his peers without further justification, and then to deduce from them all conclusions known at the time in elementary geometry…. One reason the Elements is such a beautiful work is that so much has been deduced from so little.

19. From Chapter 1 of the textbook: Two requirements that must be met for us to agree that a proof is correct. Requirement 1. Acceptance of certain statements called axioms or postulates without further justification. Requirement 2. Agreement on how and when one statement “follows logically” from another, i.e., agreement on certain rules of logic. Requirement 0. Mutual understanding of the meaning of the words and symbols used in the discourse.

20. Requirement 0. Mutual understanding of the meaning [definitions] of the words and symbols used in the discourse. Definitions cannot be circular For example – you would not define a diagonal of a quadrilateral as a segment whose endpoints are opposite vertices and then define opposite vertices of a quadrilateral as the endpoints of a diagonal. Note: Dictionaries are always circular. Sometimes a proof must be given in order for a definition to be acceptable. For example – If we define  as the number equal to we are assuming that this series converges. That would have to first be proven.

21. Of course, this requirement that definitions not be circular implies that some terms must remain undefined. Undefined terms point line we will not use the adjective “straight”

22. Why not use adjectives like “straight”? “… in later chapters we will provide alternative interpretations of some of the undefined terms that may startle you… the flexibility to interpret the undefined terms in a manner not originally intended often leads to some very important new mathematics. That is the modern point of view.” “You could do what a blind person must do. Having no image for our undefined terms, just reason carefully about these terms using only the properties we will assume about them in our axioms.”

23. Of course, this requirement that definitions not be circular implies that some terms must remain undefined. Undefined terms point line we will not use the adjective “straight”

24. Of course, this requirement that definitions not be circular implies that some terms must remain undefined. Undefined terms/relations point line This resembles the language of sets. However, we are not saying that a line is a set of points. If we want to talk about the points that lie on a line l, we will use the symbol {l}. lie on (as in “point P lies on line l ”) l passes through P P is incident with l (P Il) l contains P

25. Of course, this requirement that definitions not be circular implies that some terms must remain undefined. Undefined terms/relations point line lie on (as in “point P lies on line l ”) between(as in “point C is between points A and C”)

26. Of course, this requirement that definitions not be circular implies that some terms must remain undefined. Undefined terms/relations point line lie on (as in “point P lies on line l”) between(as in “point C is between points A and C”) congruent Other candidates for undefined terms: set, equal, unique, distinct,member

27. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. l  Q  Definition Given distinct points A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line AB and are between A and B. The two given points A and B are called the endpoints of segment AB. PQ or l P

28. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.  Q  Definition Given distinctpoints A and B. The segment AB is the set whose members are the points A and B and all points C that lie on line AB and are between A and B. The two given points A and B are called the endpoints of segment AB. P

29. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE.

30. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Definition Given distinct points O and A. The set of all points P such that segment OP is congruent to segment OA is called the circle with O as center and OA as radius. For each point P in that set, we say that P lies on the circle and OP is called a radius of the circle. We will only be considering two dimensions.

31. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Euclid’s Postulate III For every point O and every point A not equal to O, there exists a circle with center O and radius OA. A   O

32. Definition The ray AB is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on AB such that B is between A and C. The ray AB is said to emanate from the vertex A and to be part of line AB.   A        B C C C C C C C

33. Definition The ray AB is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on AB such that B is between A and C. The ray AB is said to emanate from the vertex A and to be part of line AB. Definition Rays AB and AC are opposite if they are distinct, if they emanate from the same point A and if they are part of the same line AB (or AC).

34. Definition The ray AB is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on AB such that B is between A and C. The ray AB is said to emanate from the vertex A and to be part of line AB. Definition Rays AB and AC are opposite if they are distinct, if they emanate from the same point A and if they are part of the same line AB (or AC). Definition An angle with vertex A is a point together with two distinct non-opposite rays AB and AC (called the sides of the angle) emanating from A. Note: If we denote the two rays in the definition above as ray r = AB and ray s = AC, we say that r and s are coterminal and we can denote the angle as (r, s)

35. Definition The ray AB is the following set of points lying on the line AB: those points that belong to the segment AB and all points C on AB such that B is between A and C. The ray AB is said to emanate from the vertex A and to be part of line AB. Definition Rays AB and AC are opposite if they are distinct, if they emanate from the same point A and if they are part of the same line AB (or AC). Definition An angle with vertex A is a point together with two distinct non-opposite rays AB and AC (called the sides of the angle) emanating from A. Definition If two angles DAB and CAD have a common side AD and the other two sides AB and AC form opposite rays, the angles are supplements of each other, or supplementary angles. Definition An angle BAD is a right angle if it has a supplementary angle to which it is congruent.

36. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Euclid’s Postulate III For every point O and every point A not equal to O, there exists a circle with center O and radius OA. Euclid’s Postulate IV All right angles are congruent to one another.

37. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Euclid’s Postulate III For every point O and every point A not equal to O, there exists a circle with center O and radius OA. Euclid’s Postulate IV All right angles are congruent to one another. Definition Two lines landmare parallel if they do not intersect, i.e., if no point lies on both of them. We denote this by lm.

38. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Euclid’s Postulate III For every point O and every point A not equal to O, there exists a circle with center O and radius OA. m P Euclid’s Postulate IV All right angles are congruent to one another.  l The Euclidean Parallel Postulate (Euclid V) For every line l and for every point P that does not lie on l, there exists a unique line m through P that is parallel to l.

39. From Chapter 1 of the textbook “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption.”

40. From Chapter 1 of the textbook “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption.”

41. Euclid’s Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q. Euclid’s Postulate II For every segment AB and for every segment CD there exists a unique point E on line AB such that B is between A and E and segment CD is congruent to segment BE. Euclid’s Postulate III For every point O and every point A not equal to O, there exists a circle with center O and radius OA. Euclid’s Postulate IV All right angles are congruent to one another. The Euclidean Parallel Postulate For every line l and for every point P that does not lie on l, there exists a unique line m through P that is parallel to l.

42. From Chapter 1 of the textbook “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption.”

43. From Chapter 1 of the textbook “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption. For about two thousand years, mathematicians tried to derive it from the other four postulates or to replace it with another postulate, one more self-evident. All attempts to derive it from the first four postulates turned out to be unsuccessful because the so-called proofs always entailed a hidden assumption that was unjustifiable.”

44. From Chapter 1 of the textbook “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption. For about two thousand years, mathematicians tried to derive it from the other four postulates or to replace it with another postulate, one more self-evident. All attempts to derive it from the first four postulates turned out to be unsuccessful because the so-called proofs always entailed a hidden assumption that was unjustifiable.”

45. “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption. For about two thousand years, mathematicians tried to derive it from the other four postulates or to replace it with another postulate, one more self-evident. All attempts to derive it from the first four postulates turned out to be unsuccessful because the so-called proofs always entailed a hidden assumption that was unjustifiable. Many of these “hidden assumptions” derived from misusing diagrams. Diagrams are dangerous because they often mask hidden assumptions and cause them to go undetected.

46. “Remember that an axiom was originally supposed to be so simple and obvious that no educated person could doubt its validity. From the very beginning, however, the parallel postulate was attacked as insufficiently plausible to qualify as an unproved assumption. For about two thousand years, mathematicians tried to derive it from the other four postulates or to replace it with another postulate, one more self-evident. Many of these “hidden assumptions” derived from misusing diagrams. Diagrams are dangerous because they often mask hidden assumptions and cause them to go undetected, as was the case in the “proof” that every point inside a circle is on the circle.

47. Theorem: Every right angle is obtuse.

48. Things that you should remember: from high school geometry: Two triangles can be proven congruent by SAS.

49. Things that you should remember: from high school geometry: Two triangles can be proven congruent by SAS. Two triangles can be proven congruent by SSS.

50. Things that you should remember: from high school geometry: Two triangles can be proven congruent by SAS. Two triangles can be proven congruent by SSS. Base angles of an isosceles triangle are congruent (Isosceles triangle theorem).