SYLLABUS

Axiomatic
Geometry (Math 340/Section 001)

Fall
2017

Professor Shields

Office: 4 Greenway Room 302

Office Hours: Tuesday 11-12, Thursday 1:45-2:45 or by
appointment.

Office Phone: 953-5919

E-mail: shieldss@cofc.edu

Website: http://shieldss.people.cofc.edu/

*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)

Homework 10%

Computer projects 10%

Final Exam (weighed as one test grade)
20%

92% guarantees A

82% guarantees B

72% guarantees C

62% guarantees D

Exam dates:

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*

HILBERT'S AXIOMS

INCIDENCE:

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.

BETWEENNESS:

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.

CONGRUENCE:

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.