Non-Euclidean Geometry
(MATH 6118-001, Spring 2022)

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Last update: Sunday, December 12, 2021


The Greeks believed that only the truth suits divine matters, not error or approximation, and the divine character of some matter made them more demanding in regard to accuracy. (We do exactly the opposite, being deformed by customs and propaganda.) It was because they saw a divine revelation in geometry that they invented rigorous demonstration.

(Simone Weil: Gravity and Grace)


Instructor: Gábor Hetyei
Office: Fretwell 335F, Phone: 704-687-1045, E-mail: ghetyei@uncc.edu
Office hours: Tuesdays 2:30-2:45 pm and 4:30-5:15 pm, Thursdays 2:30-3:30 pm or by appointment (eventual changes will be announced in class).
Text: Roads to Geometry, 3d Edition by Edward C. Wallace and Stephen F. West.
Prerequisite: Consent of the department.
Topics:
  1. Axiomatic Systems: Historical Background (only important highlights)/ Axiomatic Systems and Their Properties (read at home)/ (skip Finite Geometries) / Axioms for Incidence Geometry
  2. Axiom Sets For Geometry: Euclid's Geometry and Euclid's Elements / Modern Euclidean Geometry / Hilbert's Axioms for Euclidean Geometry / Birkhoff's Axioms for Euclidean Geometry / The SMSG Postulates for Euclidean Geometry / Non-Euclidean Geometry
  3. Neutral Geometry: Preliminary Notions / Congruence Conditions / The Place of Parallels / The Saccheri-Lengendre Theorem / The Search for a Rectangle
  4. Euclidean Geometry of the Plane: The Parallel Postulate and Some Implications / Congruence and Area / Similarity / Some Euclidean Results Concerning Circles / Some Euclidean Results Concerning Triangles / More Euclidean Results Concerning Triangles / The Nine-Point Circle / Euclidean Constructions / (skip Laboratory Activities Using Dynamic Geometry Software)
  5. Analytical and Transformational Geometry: We will cover Inversion, the rest of the chapter will be covered if we have time left and there is interest.
  6. Non-Euclidean Geometries: A Return to Neutral Geometry: The Angle of Parallelism / The Hyperbolic Parallel Postulate / Hyperbolic Results Concerning Polygons / Area in Hyperbolic Geometry / Showing Consistency: A Model for Hyperbolic Geometry / Classifying Theorems / Elliptic Geometry: A Geometry with No Parallels? / Geometry in the Real World / (skip Laboratory Activities Using Dynamic Geometry Software)
We will try to cover as much of the book as we can, with a focus on the above indicated topics.
Test dates:
Midterm: Tuesday March 2, 5:30-6:50 pm.
Final Exam: Tuesday, May 10, 5:00 -7:30 pm.
(Double-check in the UNCC Exam schedule !)
The final exam will be cumulative, consisting of two parts. The first part will be mandatory, covering only material taught after the midterm. The second half will contain questions reviewing the material taught before the midterm. This part will be optional, if left unanswered, I will substitute your midterm score.
Homework: Homework will be assigned every week, some to be turned in in writing, some will be presented by a randomly selected student in class. Since we meet only once per week, you should be planning on attending all lectures. If you are unable to come to a class, you have to work out in writing and turn in all homework assigned for oral presentation on that day.
Evaluation: Grades will be based on: 34% for the homework, 33% for the midterm, and 33% for the final (22% for the mandatory part, 11% for the optional part).
Class meeting: Tuesdays 5:30 - 8:15 pm in Denny 215
Homepage: http://webpages.uncc.edu/ghetyei/courses/6118/index.html
Disabilities: UNC Charlotte is committed to access to education. If you have a disability and need academic accommodations, please send me your accommodation letter as early as possible. You are encouraged to meet with me to discuss the accommodations outlined in your letter. For more information on accommodations, contact the Office of Disability Services at 704-687-0040 (Fretwell 230).
Rules of the Classroom: To ensure that your fellow students' right of learning is protected, please observe the following:
  • No conversations during class. Raise your hand if you have any question.
  • If you attend the class via Zoom (currently this is not an option), your microphone should be muted. Use the "chat" feature to ask questions.
  • Students must have proper registration to attend the class.
  • For further rules (banning discrimination and harassment of any kind) see the following University policies: Policy 406, The Code of Student Responsibility, Policy 502, Sexual Harassment Policy and Grievance Procedures; and see the Information Technology Services Standard for Responsible Use
  • As for my behavior, correct me, if I address you incorrectly. If you changed your name recently and it is not reflected on the official roster, please use your Ninernet login ID to identify yourself on exams. This is also a good idea if you have the same name as someone else in class.
Academic Integrity: All students are required to read and abide by the Code of Student Academic Integrity. Violations of the Code of Student Academic Integrity, including plagiarism, will result in disciplinary action as provided in the Code. Definitions and examples of plagiarism are set forth in the Code. The Code is available from the Dean of Students Office or online. In this class, the following special rules apply:
  • You are allowed to collaborate on homework solutions, but you have to put them in writing using your own words. If two or more write-ups are identically worded, all of them will be rejected.
  • Exams will be closed book, without notes. The current plan is to have all tests on campus. Should we be forced to have online exams, expect that I may require to have your web-camera on during the test. In either case verbatim quotes from the textbook, including theorem and page numbers will be considered academic misconduct.
Copyright issues: My lectures and course materials, including presentations, tests, exams, outlines, and similar materials, are protected by copyright. I am the exclusive owner of copyright in those materials I create. I encourage you to take notes and make copies of course materials for your own educational use. However, you may not, nor may you knowingly allow others to reproduce or distribute lecture notes and course materials publicly without my express written consent. This includes providing materials to commercial course material suppliers such as CourseHero and other similar services. Students who publicly distribute or display or help others publicly distribute or display copies or modified copies of an instructor's course materials may be in violation of University Policy 406, The Code of Student Responsibility. Similarly, you own copyright in your original papers and exam essays. If I am interested in posting your answers or papers on the course web site, I will request your written permission.

I wish to especially underscore that under no circumstances should you make homework solutions publicly available.