Fundamentals of Modern Geometry – MATH 505

CG • Section 8WK • 11/08/2019 to 04/16/2020 • Modified 07/28/2020

Course Description

A treatment of the foundations of modern Euclidean geometry and an introduction to non-Euclidean geometries with emphasis on hyperbolic geometry. The course focuses on demonstrating and explaining geometric concepts through axiomatic methods.

Prerequisites

MATH 132 and MATH 250

Rationale

This course provides an introduction to modern axiomatic geometry with an emphasis on presentation of proofs. The geometry of Euclid is approached through Hilbert’s Axioms of incidence, betweenness, congruence, continuity, and parallelism. A great deal of emphasis is placed on the historical development of the Parallel Postulate and the discovery of non-Euclidean geometry.

Measurable Learning Outcomes

Upon successful completion of this course, the student will be able to:

  1. Distinguish between undefined terms, definitions, axioms and theorems.
  2. Demonstrate the logical structure of a mathematical argument by way of a geometric proof.
  3. Explain why a false argument is not valid.
  4. Outline a proof, indicating the basic steps used when provided with a valid proof.
  5. Prove classical theorems (of Thales, of Menelaus, of Pappus, of Desargues, Euler etc.).
  6. Demonstrate through proof differences that assuming the parallel postulate makes in a geometric system.

Course Assignment

Textbook readings and videos

Course Requirements Checklist

After reading the Course Syllabus and Student Expectations, the student will complete the related checklist found in Module/Week 1.

Introduction Video

This is a short video introduction of each student to the class and the instructor. The student must make sure that he or she can be seen and heard in the video. This assignment also acts as a trial run for the required Video Demonstrations.

Video Demonstrations

Each demonstration is a concise, precise, well-practiced presentation of the proof of some given proposition in geometry. They serve as a way of demonstrating knowledge and practing correct presentation of mathematical proofs in a simulated classroom setting.

Homework Portfolios (2)

Homework will be assigned weekly. The first 4 homework assignments will be scanned into a single document and sent as a homework portfolio before the Midterm Exam. The next 4 homework assignments will be scanned into a single document and sent as a homework portfolio before the Final Exam. Many exam problems will come from the assigned homework.

Quizzes (7)

Each quiz will be timed, handwritten, and open-book/open-notes/open-video, and will cover material from the videos in the assigned modules/weeks. The time limit for each quiz is 30 minutes. The quizzes are some combination of true/false, multiple choice, and fill-in-the-blank questions.

Midterm Exam

The exam will be timed, hand-written, and open-book/open-notes/open-video, and will cover the Reading & Study material for Modules/Weeks 1–4. The time limit is 90 minutes (1 hour, 30 minutes). On all written work, the student is expected to write correct mathematics to avoid point deductions.

Final Exam

The exam will be timed, hand-written, and open-book/open-notes/open-video, and will cover the Reading & Study material for Modules/Weeks 5–8. The time limit is 120 minutes (2 hours). On all written work, the student is expected to write correct mathematics to avoid point deductions.