MAT 3271: GeometryProfessor: Charles Delman Professor’s Office: M3351Phone: 581-6274 (O) 348-7786 (H, before 9pm, please) Office Hours: MW - 1-2 pm, 3-4 pm1. Course ContentEuclid and the origins of axiomatic geometry; the axiomatic method: undefined terms, definitions,axioms, theorems, logic and proof; incidence geometry; models of axiomatic systems; Hilbert’s axioms;neutral geometry; modern Euclidean geometry; introduction to hyperbolic geometry, it’s historical de-velopment, and its impact. (Note: this course is the first half of a two course sequence, continued byMAT 3272 .) Text: Euclidean and Non-Euclidean Geometries (3rdedition), by Marvin Jay Greenberg2. Objectives(1) The student will independently write clear, logically sound definitions and proofs.(2) The student will be able to discern the errors of reasoning in an incorrect proof and the unstatedassumptions made in a non-rigorous argument.(3) The student will present ideas to the class in a clear and organized fashion.(4) The student will listen attentively to the presentations of others and, in a polite, respectful andconstructive manner, point out errors, raise questions, and offer s uggestions for correction orimprovement.(5) The student will be able to verify that the axioms of a system hold in a model of that system,construct models of simple axiomatic systems, and demonstrate an isomorphism between twoisomorphic models.(6) Without reference to external sources, the student will be able:• to define the major concepts of geometry which have bee n covered and explain their purpose;• to state and prove the major theorems of geometry which have been covered and discusstheir consequences.The purpose of this course is to study geometry critically, rigorously, and from a historical perspec tive.We will begin with the geometry presented by Euclid (the kind you learned in high school), examininingits axioms and exposing Euclid’s hidden assumptions. We will make these assumptions explicit andput Euclidean geometry on a modern rigorous foundation using the axiom system developed by Hilbert.Then we will discuss the independence of the postulate on parallel lines from the other axioms and beginthe study of hyperbolic geometry, which is the system which results if the Euclidean Parallel Postulateis replaced by it’s negations.Euclidean hyperbolicThe development of modern geometry centers on the long controversy surrounding Euclid’s fifthpostulate, which states that given any line l and any point P not on l, there is exactly one line throughP which is parallel to l. We will call this the Euclidean Parallel Postulate. Although the EuclideanParallel Postulate may seem obvious to you, we will see that there is absolutely no more validity inassuming it than in assuming its opposite, namely that there exists a line l and a point P not on lsuch that at least two distinct lines through P are parallel to l. We will call this opposite assumptionthe Hyperbolic Parallel Postulate, and the geometric system which results from assuming it is calledhyperbolic geometry. Hyperbolic geometry has some surprising properties; for example, all similartriangles are congruent - there is no scaling! It also has revolutionary applications.1I hope that, in addition to the concrete objectives stated above, you will come to appreciate thatgeometry, and mathematics as a whole, is a developing science rather than a rigid set of rules handeddown from the past, and that its history has not been without heated controversies. I also hope you willdevelop an awareness of the aesthetic and mysterious qualities of mathematics and enjoy the beautifuland often surprising discoveries which have made the history of geometry so colorful.3. RequirementsClass participation: You are expected to be in class every day and to be prepared to presentyour progress toward a solution of any of the assigned problems. Participation will count toward thehomework portion of your grade (see “Grading,” below), based on the quality of your presentations andcontributions to the discussion. Effort counts.Homework: Written homework problems will be regularly assigned and graded (with comments).In-term Exams: There will be two exams during the class term (in addition to the final), which maybe partly given as take-homes. Make-up exams will be given only under extraordinary circumstances orin case of serious emergency; prior permission to miss an exam must be obtained from the professor ifat all possible.Final exam: The final exam will be comprehensive. It may be partly given as a take-home.4. GradingI do not grade on a “curve”. Under no circumstances will your grade directly depend on how howyour fellow students do. If you do a good job of learning the material, you will receive a good grade,regardless of how well the other members of the class perform. Don’t forget that the reverse is also true:if you do a poor job of learning the material, you will receive a poor grade, regardless of how poorlyeveryone else doe s.I will assign letter (rather than numerical) grades, based on the objectives stated above and standardsclarified in class. Grades correspond to my judgement of quality as follows:• A Excellent. The work exhibits mastery of nearly all important ideas , including those which aresubtle or difficult, much insight and originality, highly coherent organization and fine expositorystyle. Errors and omissions, if any, are minor.• B Good. The work exhibits substantial understanding of most important ideas, including somewhich are subtle or difficult, some insight and originality, cohere nt organization and correctusage, grammer and spelling. There are some substantive errors or omis sions.• C Fair. The work exhibits basic understanding of many fundamental ideas, although not thosewhich are subtle or difficult, and demonstrates some coherence. The presentation is readable,with at most minor errors of usage, grammer or spelling. There are many substantive errors oromissions.• D Poor. The work exhibits some understanding of a few fundamental ideas, but not those whichare subtle or difficult, and it fails to demonstrate coherence. Usage, grammar and spelling aremostly correct. There are a great many subtantive errors or omissions.• F No credit. The work exhibits too few of the positive qualities noted above to be worthy ofcredit.Each requirement will count toward your final grade according to the scheme below (possibly subjectto slight modification):Homework