Fundamentals of Modern Geometry – MATH 505

CG • Section 8WK • 11/08/2019 to 04/16/2020 • Modified 02/09/2022

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.

For information regarding prerequisites for this course, please refer to the Academic Course Catalog.

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 the Course Overview.

Discussion: Student Introduction Video

This is a short video introduction of each student to the class and the instructor. The video must not exceed 2 minutes. 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 Demonstration Assignments.

Video Demonstration Assignments (4)

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 practicing correct presentation of mathematical proofs in a simulated classroom setting. Each demonstration must be less than 6 minutes long.

Homework (6)

Homework problems are essential to this course and students will be assigned homework problems to complete throughout the course. One homework problem out of the assigned problems will be submitted for grading in Modules 1-3 and 5-7.

Homework Portfolio Assignments (2)

Homework will be assigned weekly. The first 4 homework assignments will be scanned into a single document and submitted as the Homework Portfolio: Midterm Assignment, before the Quiz: Midterm. The next 3 homework assignments will be scanned into a single document and submitted as the Homework Portfolio: Final Assignment, before the Quiz: Final. Many exam problems will come from the assigned homework.

Quizzes (7)

Each quiz will be timed, open-book/open-notes/open-video, and cover material from the presentations in the assigned modules. The time limit for each quiz is 45 minutes. The quizzes are some combination of true/false, multiple choice, and fill-in-the-blank questions.

Quiz: Midterm Exam

The exam will be timed, handwritten, and open-book/open-notes/open-video, and it will cover the course material for modules 1–4. The time limit is 120 minutes (2 hours). On all written work, the student is expected to write correct mathematics to avoid point deductions.

Quiz: Final Exam

The exam will be timed, handwritten, and open-book/open-notes/open-video, and it will cover the course material for the entire course. The time limit is 120 minutes (2 hours). On all written work, the student is expected to write correct mathematics to avoid point deductions.