Axiomatic Geometry (Math 340/Section 001)
Office: 4 Greenway Room 302
Office Hours: Tuesday 11-12, Thursday 1:45-2:45 or by appointment.
Office Phone: 953-5919
Undergraduate Mathematics Program Student Learning Outcomes: Students are expected to display a thorough understanding of the topics covered. In particular, upon completion of the course, students will be able to
1. Using algebra, geometry, calculus and other track-appropriate sub-disciplines of mathematics, students model phenomena in mathematical terms.
2. Using algebra, geometry, calculus and other track-appropriate sub-disciplines of mathematics, students derive correct answers to challenging questions by applying the models from Learning Outcome 1.
3. Write complete, grammatically and logically correct arguments to prove their conclusions.
These outcomes will be assessed on the final exam.
Course Description: This course is on the axiomatic foundations of Euclidean and hyperbolic geometry. Students should have a reasonable amount of experience in proving theorems and in the use of logical arguments involved in some area of mathematics. Topics covered in this course include axiomatic systems and finite geometries, axioms of Euclidean and non-Euclidean geometry, the Poincare model, the hyperbolic plane, the independence of the parallel postulate and classical constructions. Prerequisites: MATH 203, 220, 295, or permission of instructor.
Text: Euclidean and Non-Euclidean Geometry by Marvin Jay Greenberg.
(Note: The text is quite readable and should be read.) We will cover chapters 1-6, and parts of 7 and 9, Chapter 8 is a reading assignment. We will also be using The Geometer's Sketchpad software available in the Computer Lab.
Course Perspective: Euclid's Elements (300 B.C.) proposed 5 axioms from which all of the geometry of the plane can be developed. They can be translated as follows:
I. A line may be drawn from any one point to any other.
II. A line segment may be produced to any length in a line.
III. A circle may be described with any center and any radius.
IV. All right angles are equal.
V. If a line meets two other lines so as to make the interior angles on one side together less than two right angles, then the two lines will meet on that side.
From the beginning, the fifth axiom was controversial. It is certainly more complex than the others, and many felt it could be proved from the others. In fact, Euclid delayed using it in the Elements as long as possible. For the next 2000 years, mathematicians around the world worked to prove that axiom V follows from the other axioms. Their futile attempts finally led to the discovery of another perfectly logical and consistent geometry, based on the truth of axioms I-IV and the negation of V. In this new world, many parallels exist through a given point to a given line, rectangles do not exist and magnification is impossible! More astonishing perhaps is that this new geometry can be useful in the real world. We will examine how Euclidean geometry evolves from axioms I-V, and also explore the world of non-Euclidean geometry based on the negation of axiom V.
Grades will be determined on the following basis:
3 tests (20% each)
Computer projects 10%
Final Exam (weighed as one test grade) 20%
92% guarantees A
82% guarantees B
72% guarantees C
62% guarantees D
Thursday September 21
Tuesday October 31
Tuesday November 21
The last day to withdraw from the course with a W is October 26.
The final exam will be cumulative. It will be on Tuesday, December 12 from 12-3
There will be no exemptions from the final exam. It will be comprehensive and must be taken for a pass in the course.
Warning: All of the test dates are subject to change; if you miss class, it’s your responsibility to find out if any of these tests have been rescheduled for another day.
Attendance: Students are responsible for all material presented in class, so it is in your best interest to attend. Help during office hours is available only to those who either attended the class in which the material was presented or whose absence is excused by the Absence Memo Office at 67 George Street.
Students with disabilities: The College will make reasonable accommodations for persons with documented disabilities. Students should apply at the Center for Disability Services/SNAP, located on the first floor of the Lightsey Center, Suite 104. Students approved for accommodations are responsible for notifying the instructor as soon as possible and for contacting the instructor at least one week before any accommodation is needed.
Academic Integrity Statement: The Honor Code at the College of Charleston specifically forbids cheating, attempted cheating, and plagiarism. Cases of suspected academic dishonesty will be reported directly to the Dean of Students. A student found responsible for academic dishonesty will receive a XF in the course, indicating failure of the course due to academic dishonesty. This grade will appear on the student’s transcript for two years after which the student may petition for the X to be expunged. The student may also be placed on disciplinary probation, suspended (temporary removal) or expelled (permanent removal) from the College by the Honor Board.
It is important for students to remember that unauthorized collaborations—working together without permission—is a form of cheating. Unless a professor specifies that students can work together on an assignment and/or test, no collaboration is permitted. Other forms of cheating include possessing or using an unauthorized study aid (such as a PDA), copying from another’s exam, fabricating data, and giving unauthorized assistance
1-1: For every point P and every point Q not equal to P, there exists a unique line l passing through P and Q.
1-2: For every line l, there exist at least two distinct points that are incident with l.
1-3: There exist three distinct points with the property that no line is incident with all three of them.
B-1: If A*B*C, then A, B and C are three distinct points all lying on the same line, and C*B*A.
B-2: Given any two distinct points B and D, there exist points A, C and E lying on BD such that A*B*D, B*C*D and B*D*E.
B-3: If A, B and C are three distinct points lying on the same line, then one and only one of the points is between the other two.
B-4: For every line l and for any three points A, B and C not lying on l:
(i) if A and B are on the same side of l, and B and C are on the same side of l, then A and C are on the same side of l,
(ii) if A and B are on opposite sides of l, and B and C are on opposite sides of l, then A and C are on the same side of l.
C-1: If A and B are distinct points and if A' is any point, then for each ray r emanating from A', there is a unique point B' on r such that B' ≠ A' and AB is congruent to A'B'.
C-2: If AB is congruent to CD and AB is congruent to EF, then CD is congruent to EF. Moreover, every segment is congruent to itself.
C-3: If A*B*C, A'*B'*C', AB is congruent to A'B' and BC is congruent to B'C', then AC is congruent to A'C',
C-4: Given any angle <BAC and given any ray A'B' emanating from a point A',
then there is a unique ray A'C' on a given side of line A'B' such that <B'A'C' is congruent to <BAC.
C-5: If <A is congruent to <B and <A is congruent to <C, then <B is congruent to <C , Moreover, every angle is congruent to itself.
C-6 (SAS): If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.
CONTINUITY= DEDEKIND'S AXIOM: Suppose that the set of all points on a line l is the union Ll U L2 of two nonempty subsets such that no point of L1 is between two points of L2 and vice versa. Then there is a unique point O lying on l such that one of these subsets is equal to a ray of l emanating from O and the other subset is equal to its complement.
HILBERT'S PARALLEL POSTULATE FOR EUCLIDEAN GEOMETRY:
For every line l and every point P not lying on l, there is at most one line m through P such that m is parallel to l.
HYPERBOLIC PARALLEL AXIOM: There is a line l and a point P not on l such that at least two distinct lines parallel to l pass through P.