Geometry MAT346
and
Noneuclidean Geometries MAT636

The masters students in this class must do all the work assigned to the undergraduates including exams and homework. In addition, masters students must complete the final project. Undergraduates who wish to complete this final project as well may do so for extra credit. Any student who has registered for the course as MAT636 will be considered a masters student.

Grading Policy for Undergrads: 4 Proving Projects 10% each... 1 Midterm Exam: 25%... Classwork: 5%... Final Exam: 30%...

Grading Policy for Masters Students: Final Project: 25% 4 Proving Projects 5% each... 1 Midterm Exam: 20%...Classwork: 5%... Final Exam: 30%...

One of the primary goals of this course for all students is to become comfortable with proving mathematical facts. For each lesson a homework assignment will be posted on this page and starred problems will be presented in class by volunteers at the following meeting. Any students with questions are encouraged to come to the professor's office hours. These assignments will not be handed in and should be kept with the class notes for studying purposes. There will be four "proving projects" which consist of difficult proofs similar to the ones on the homework assignements. The two exams will have easier problems including at least one proof each.

The masters students will also complete a final project consisting of a set of geometry lessons for high, junior high or elementary school students. These final projects will be posted on the webpage Explore Geometry so they should be very neat and publishable. Students who create webpages for their final projects will receive extra credit. The professor can provide sample templates to make it easier for the students to turn their projects into webpages.

Pages 399-402 has a list of axioms which should be brought to exams and refered to on the homework. Please make a xerox copy for your reference.

Syllabus: Exact assignments will be posted as the course progresses. Note that while many topics appear similar to high school geometry/trigonometry we will be examining everything axiomatically and exploring the corresponding theorems in the noneuclidean spaces S and H .

Sept 3: Euclidean Space (E) versus the Sphere (S) and Hyperbolic Space (H) ( 1.1, Example 2 on p 344 and "The Upper Half-plane Model" p351-2)
HW: Buy a compass, ruler, small ball and thin rubber bands.
Sept 5: Axioms of Incidence and Proof by Contradiction (1.2 pages 1- 4)
HW: page 14/problem 1. Then write out the complete proof in 2 columns with explanations next to each step, finally rewrite the proof in paragraph form. Do E, S and H all satisfy the axioms of incidence?

Sept 10: Axioms of Betweenness and Proofs with Cases (1.2 pages 4-8)
HW: Page 14/problem 2*, 3*, 5*
Sept 12: Line Separation and Pasche's Theorem (1.2 pages 8-9)
HW: Page 14/ problems 4, 6*, Do E, S and H all satisfy the axioms of betweenness?

Sept 17: no class monday schedule (I speak at Rutgers)
Sept 19: Angles and "if and only if" (1.2: pages 9-14)
HW: Page 14/ problems 7*, 8*, explore the definition of angle on H.
Proving Project I (due Oct 1): page 14/ problem 4 and Proof by Induction Handout problem 4. 1.2/problem 9 is false and is no longer on the proving project. Do proofs in 2 columns and then rewrite in paragraph form. See the proof webpage .

Sept 24: Proof by Induction and Countable Unions and Intersections (Handout)
HW: Do problems on handout.
Sept 26: Convex Sets (1.3)
HW: 1.3/3abc, Draw convex sets on hyperbolic space.

Oct 1: Lengths and Degrees 1.4 (proving project I due before class)
HW:1.3/5,6 (try by induction), 1.4/ 1,2
Oct 3: Triangles 1.5
HW 1.5/1 abc, 2abc, 7, Try to draw a rhombus in hyperbolic space!
Proving Project II (due October 15): 1.3/3abc, Verify that the rational plane satisfies all the axioms up to the principles of continuity that is Axioms I1-I3, B1-B4, S1 and S3

Oct 8: Triangles and the Sphere 1.5
1.5/ 4, 6,7, finish reading 1.5, study the sphere's axioms
Oct 10: Circles 1.6
Weekend Review: Work on the proving project!

Oct 15: Sups and Infs 1.7
Oct 17: Continuity Principle 1.7
Weekend Review: Get started on the Final Project!

Oct 22: Review for the Midterm. (proving project II due before class)
Oct 24: Constructions 1.8
Weekend Review: Review all concepts for the midterm.

Oct 29: Midterm Exam.
Oct 31: Metric Spaces 7.2
Proving Project II resubmission is due Nov 5.
Masters students should choose a topic for the final project which could involve Euclidean, hyperbolic, spherical or another geometry and ask students to study triangles or balls, or parallel lines, the actual project must include a typed up lesson with worked out examples and homework and must be handed in on a disk including any graphics on that disk (photos). Be sure not to have faces of your students in any photos as the projects will be posted on the web.

Nov 5: Rectifiable curves and Length spaces handout
Proving Project 3: Do any three *ed problems on the handout due Nov 19. Any further problems which are completed will count as extra credit towards your final grade.
Nov 7: Euclid's Fifth Postulate 2.1-2.2
Weekend Review: work on the proving project

Nov 12: In class project workshop. (Nov 13 is withdrawal deadline)
Nov 14: Euclidean Triangles 2.2-2.3,
Weekend Review:

Nov 19: Similar Triangles 2.3 (proving project III is due before class)
Homework: p86/4,
Nov 21: Euclidean Circles 2.4, HW p95/1,3
Weekend Review: Proving Project IV (due Dec 10, no resubmission): p86/4, p95/1,3, Use proof by induction to make the proof of Lemma 8.2.2 rigorous.

Nov 26: Hyperbolic Geometry 8.1-8.2
Nov 28: Thanksgiving
Weekend Review: Work on the Final Project

Dec 3: Hyperbolic Parallel Postulate 8.3,
(Proving Project III resubmition and Extra Credit due, no extensions).
Dec 5: Elliptic Geometry
Weekend Review: for final

Dec 10: Spherical Gauss Bonnet 7.4 if time
Dec 12: Review for final, Final Projects due today!!!! (Proving Project IV due, no resubmition or extensions)

Topics on the final:

There will be simple proofs: Practice proving page 14 exercise 1 directly from the axioms, also proving that a space is a metric space like proving that the real line such that d(x,y)=|x-y| is a metric space,

Also tested: Supremums and the completeness axiom of the reals should be understood, basic geometry as in 2.2-2.4 must be known on Euclidean space so that you can use them to figure out angles and lengths using the notations we've been using, understand the definition of opposite sides. Knowing how to use Gauss Bonnet both for the sphere and hyperbolic space to find angles and areas. Drawing on both Euclidean and Hyperbolic space (constructing triangles, parallel lines, making diagrams to match a proof). knowing how the axioms of neutral geometry fail on various spaces like the sphere and the three point space.

It is crucial that you understand the differences between hyperbolic and euclidean space: both satisfy neutral geometry and have triangles whose angle sum is less than or equal to 180, understand how the sum of the angles of a triangle is related to Playfair's postulate, understand how the angle sums of traingles relate to the angle sums of quadrilaterals, understand how quadrilaterals relate to the existence of similar triangles of different sizes, understand how in Hyperbolic space similar triangles are congruent and the proof of this fact, understand how the existence of one triangle with an angle sum = 180 implies all triangles have an angle sum equal to 180 (how we made large triangles using 45 45 90 and small triangles by taking slices of triangles,

Final Exam is Tuesday Dec 17 6:15-8:16 pm. Do not bring any notes to the final. Axiom sheets will be distributed as well as a few key theorems and definitions.