NON-EUCLIDEAN GEOMETRY
SYLLABUS
COURSE NUMBER:
COURSE TITLE:
COLLEGE OR SCHOOL:
SEMESTER AND YEAR:
SEMESTER HOURS CREDIT:
MATH 7395
Non-Euclidean Geometry
Science and Mathematics
Spring, 2012
3
INSTRUCTOR:
TELEPHONE:
E-MAIL:
WEBSITE:
OFFICE:
OFFICE HOURS:
Charles Koppelman
(678) 797-2051 office
ckoppelm@kennesaw.edu
http://www.science.kennesaw.edu/~ckoppelm
Math & Statistics Building, Room 018
Monday 2:00 – 3:00 and Thursday 2:00 – 4:00
TEXT: Euclidean and Non-Euclidean Geometries, Fourth Edition
Author: Marvin J. Greenberg
COURSE DESCRIPTION: This course examines the development of the axiomatic
basis for non-Euclidean geometry and its relationship to Euclidean geometry, and
analyzes proofs of important theorems in non-Euclidean geometry. Topics include
Hilbert’s axioms, affine and projective planes, neutral geometry, Hilbert planes,
Euclidean planes, and hyperbolic planes. Special emphasis will be given to the nature of
geometric proof and historical attempts to prove the Euclidean parallel postulate.
GRADING: There will be two (2) group problem sets each worth 10% of the final
grade. Satisfactory completion of assigned end-of-chapter exercises is worth 10% of the
final grade. There will be a midterm and a final exam, each worth 30% of the final grade.
Students will also work in groups to complete at least one independent project worth 10%
of the final grade.
ATTENDANCE: Regular class attendance is essential. PowerPoint notes for each class,
as well as all handouts, will be posted both on D2L and on my website. In the event of
absence students are responsible for all material, assignments and announcements made in
class.
TOPICS AND LEARNING OUTCOMES:
I.
Gaps in Euclid’s geometry, incidence geometry, and logic
Students will be able to:
1. apply Euclid’s postulates and identify the gaps that led to the development
of non-Euclidean geometry
2. apply the rules of logic and its notation
3. apply the incidence axioms and prove the propositions that derive from
them.
4. identify finite models of incidence geometry and establish isomorphisms
between models
5. determine whether a finite model is an affine plane, projective plane, or
neither, and prove their conclusion
6. distinguish between consistent and inconsistent axiomatic systems
7. apply the principle of duality for projective planes
8. form the projective completion of an affine plane.
II.
Hilbert’s axioms and neutral geometry
Students will be able to:
1.
2.
3.
4.
5.
apply the betweenness axioms
prove the propositions that derive from the betweenness axioms
apply the propositions to prove Pasch’s theorem and the crossbar theorem
apply the congruence axioms
prove selected propositions and theorems that derive from the congruence
axioms
6. apply Archimedes axiom, Aristotle’s axiom, Dedekind’s axiom and the
other continuity axioms and their corollaries
7. identify the relationship between Hilbert planes, neutral geometry, and
Euclidean planes.
III.
Logical equivalence to Hilbert’s Euclidean parallel postulate
Students will be able to:
1. apply and prove parts of the measurement theorem based on Aristotle’s
axiom and Dedekind’s axiom
2. prove the equivalence of Hilbert’s Euclidean parallel postulate, Euclid V,
and the angle sum theorem
3. prove the equivalence of selected theorems in Euclidean geometry and
Hilbert’s Euclidean parallel postulate.
IV.
Historical attempts to prove the Euclidean parallel postulate
Students will be able to:
1. identify Saccheri and Lambert quadrilaterals and prove related theorems
including the uniformity theorem, Saccheri’s angle theorem, and the
Saccheri-Legendre theorem
2. identify the logical flaws in historical attempts to prove the Euclidean
parallel postulate, including proofs by Proclus, J. Bolyai, Wallis, and
Legendre
3. define the defect of a triangle and prove the additive property of the defect
4. prove that rectangles do not exist in a Hilbert plane satisfying the acute
angle hypothesis
5. prove the universal non-Euclidean theorem
6. prove that non-congruent, similar triangles do not exist in a Hilbert plane
satisfying the acute angle hypothesis.
V.
Proving the fundamental theorems of hyperbolic geometry
Students will be able to:
1. prove theorems associated with parallel lines that admit a common
perpendicular
2. supply key steps and reasons in the proofs of theorems relating to parallel
lines containing limiting parallel rays
3. prove that Hilbert’s Hyperbolic Axiom of parallels holds in a nonEuclidean plane satisfying Dedekind’s axiom (real hyperbolic plane)
4. prove that parallel lines containing limiting parallel rays do not admit a
common perpendicular or congruent alternate interior angles.
WITHDRAWAL FROM THE UNIVERSITY OR FROM INDIVIDUAL COURSES
AND ACADEMIC INTEGRITY
Withdrawal
Students who find that they cannot continue in college for the entire semester after being enrolled,
because of illness or any other reason, need to complete an online form. To completely or
partially withdraw from classes at KSU, a student must withdraw online at www.kennesaw.edu,
under Owl Express, Student Services.
The date the withdrawal is submitted online will be considered the official KSU withdrawal date
which will be used in the calculation of any tuition refund or refund to Federal student aid and/or
HOPE scholarship programs. It is advisable to print the final page of the withdrawal for your
records. Withdrawals submitted online prior to midnight on the last day to withdraw without
academic penalty will receive a “W” grade. Withdrawals after midnight will receive a “WF”.
Failure to complete the online withdrawal process will produce no withdrawal from classes. Call
the Registrar’s Office at 770-423-6200 during business hours if assistance is needed.
Students may, by means of the same online withdrawal and with the approval of the university
Dean, withdraw from individual courses while retaining other courses on their schedules. This
option may be exercised up until March 12, 2014.
This is the date to withdraw without academic penalty for Spring Term, 2014 classes. Failure to
withdraw by the date above will mean that the student has elected to receive the final grade(s)
earned in the course(s). The only exception to those withdrawal regulations will be for those
instances that involve unusual and fully documented circumstances.
Academic Integrity
Every KSU student is responsible for upholding the provisions of the Statement of Student Rights
and Responsibilities, as published in the Undergraduate and Graduate Catalogs. Section II of the
Student Code of Conduct addresses the University's policy on academic honesty, including
provisions regarding plagiarism and cheating, unauthorized access to University materials,
misrepresentation/falsification of University records or academic work, malicious removal,
retention, or destruction of library materials, malicious/intentional misuse of computer facilities
and/or services, and misuse of student identification cards. Incidents of alleged academic
misconduct will be handled through the established procedures of the University Judiciary
Program, which includes either an "informal" resolution by a faculty member, resulting in a
grade adjustment, or a formal hearing procedure, which may subject a student to the Code of
Conduct's minimal one semester suspension requirement.