Foundations of Geometry
Instructor: Ward Heilman Office: 239 Hart Hall
Office Hours: Tuesday & Thursday 12:30 to 1:30 PM,
Monday 3:00 to 4:00 PM and by appointment
Office Phone: (508) 531- 2352 email: wheilman@bridgew.edu
Text: A Survey of Classical and Modern Geometries by Arthur Baragar
Useful website: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
We will begin our study of geometry with the classic text The Elements
by Euclid. We will review and prove some theorems (including several of the famous ones) from this book. Along the way we will examine The Elements from a more modern viewpoint and point out some technical difficulties while still admiring the amazing structure, strict logic and overall beauty of this work. We will use the ideas of straightedge (line) and compass (circle) to construct some geometric figures and to determine some constructible and nonconstructible numbers. We will briefly look at some solid geometry and the platonic solids. Then, a more exacting approach to the axiomatic method, axiomatic systems and models will lead to the study of finite geometries. We will conclude the course with discussions of hyperbolic and spherical geometries. An emphasis will be placed on proving things and on analyzing proofs carefully to gradually improve the ability of each student to read, understand and eventually construct her/his own well-organized and correct proofs.
Grading: There will be three in-class tests and a final comprehensive examination. Your grade will be determined by your homework, your scores on the three in-class tests and your final examination.
Homework, Attendance 15% of the final grade
Test 1 Friday, September 28, 2007 20% of the final grade
Test 3 Friday, December 1, 2007 20% of the final grade
Final Examination (comprehensive) Wednesday December 19, 2007 8 AM 25% of the final grade
Homework will be assigned every class. Some will be collected and graded. These problems will be labeled “Hand In”. On the problems that are handed in, you should work independently. Please do not copy or rewrite someone else’s work. One of the most important goals of this course is to develop your ability to solve problems, including problems that may be somewhat different from what you have seen before. This is frustrating but rewarding work. You will not benefit from it unless you put in the time and energy yourself. It will be better for all of us if we admit when we are unable to do some work rather than handing in a version of someone else’s work. Some problems will be assigned for you to do but not to hand in. On those problems that are not collected you may work together, discuss approaches and share solutions to the problems; however, remember that you take the tests on your own. Homework type problems often occur on the tests. Doing and understanding the homework is an essential part of this course. Your success in this course depends on making sure that you understand the information discussed in class, how to do the homework problems, and how to solve problems.
Attendance is always important. In this class it is essential. It is expected that you will attend each and every class. At times I will hand out notes and I will be lecturing and expanding on the topics in those notes and in the text. You are responsible for knowledge, activities and announcements presented in the class. Failure to attend class almost always shows up in poor performance on tests. Please note that, in fairness to all students, tests cannot be made up.
To do well in this course you need to attend each and every class. Furthermore you should be in class and prepared to take the tests on the dates they are scheduled as described in the syllabus or as discussed in class. The syllabus explains that tests cannot be made up (i.e. the instructor will not give a student the same or another test at a different time and place if the student is not in class or not prepared to take the test when it was scheduled) unless there is a justifiable reason why the student was not able to be in the class to take the test at the regularly scheduled time.
Please be sure to be here for each class.
In compliance with Bridgewater State College policy and equal access legislation, I am available to discuss appropriate accommodations that you may require as a student with a disability. Requests for academic accommodations should be made during the drop/add period, unless there are unusual circumstances, so that appropriate arrangements can be made. Students are encouraged to register with the Disability Resources Office in Boyden Hall for disability verification and determination of reasonable academic accommodations.
Bridgewater State College values diversity. It is committed to providing a learning, working and living environment for all members of its community and to assuring that the “college experience” challenges, empowers, supports and prepares its students to live, work and value a world which is increasingly global and diverse. Diversity of socioeconomic, racial, ethnic, religious, gender, sexual orientation, age and disability enriches the institution and its many constituencies.
Bridgewater State College will not tolerate behavior based in bigotry that has the effect of discriminating unlawfully against any member of its community.
Provides opportunities to BSC undergraduates who wish to pursue independent research, scholarship or creative activities under the guidance of a full-time BSC faculty or librarian mentor. Through ATP, students design and develop research projects, learn new research skills, gain a more sophisticated understanding of the nature of academic research, and have opportunities to present their research and creative work at regional and national conferences. The expected outcome of this program is to graduate students with the self-confidence, motivation and ability to conduct independent scholarship and research.
ATP Summer Grant Program. ATP Summer Grants are awarded to BSC undergraduates engaged in an in-depth project involving research or creative work, conducted under the supervision of a BSC faculty or librarian mentor. The 10-week grant program will run between May and August.
Eligibility. ATP Summer Research Grants are open to all BSC full- and part-time undergraduates (six credit hours minimum), who have completed a minimum of 12 credits at BSC by May, and will be enrolled as an undergraduate at BSC through at least the following fall semester. Applicants must maintain good academic standing through the spring semester, and must register for fall courses. Although there is no formal GPA requirement, an overall GPA of 3.0 or higher is recommended. Students graduating in May are not eligible.
Stipend. Students who are awarded an ATP Summer Grant will receive a stipend of $3,200 for the ten-week program, and you may also apply for up to $500 to cover the cost of research-related expenses. Your mentor will receive a $1,250 stipend for work associated with mentoring an ATP summer student.
Application Procedure. Complete each section of the application as instructed. You may go online for an application or additional copies are also available in the Honors Center, located in the Academic Achievement Center, Maxwell Library. If any information is missing or incomplete by the application deadline, your application will not be evaluated. You should identify a mentor at least six weeks before the proposal deadline, as you are expected and strongly advised to consult with your research mentor for guidance in preparing your research proposal and completing this application. However, please note that proposals written by the mentor will be denied funding.Deadline. The complete application, including recommendation letters, must be received by the Office of Undergraduate Research, 212 Tillinghast Hall, in early February. Check the ATP website for exact deadline. Late or incomplete applications will not be reviewed. Award announcements will be made in early March.
If you are interested in this (or other ATP grants) please talk to me. In general I am willing and would be glad to mentor an ATP summer research project. If would like to work with other members of the Mathematics and Computer Science Department please contact them soon to begin discussing a project.
Having made it this far, each person in this class has the ability and habits to do well in this course. Success will depend on working hard, attending each and every class, overcoming frustration, asking questions, and doing the problems. These problems are essential to understanding the ideas and concepts we will be learning. Only certain of problems will be collected and graded but all of the problems assigned are important.
Remember that mathematics is a cumulative subject; people have been working on and modifying it for thousands of years. So if something is not clear to you in 15 minutes, that should not bother you. Don’t worry about it. Relax, take a deep breath, go for a short walk. Then come back to it. Read the notes again, review the given examples, make up small examples and test them, look up the definitions, check the proofs, ask yourself what facts, results or theorems are related to the problem. Write down what exactly your question is, what part you don’t understand. If possible ask someone who might be able to help. You can always ask questions in class and come to my office. Work hard and enjoy.
Preface Read page ix
Chapter 0 Introduction
0.2 3 Basic Terminology
0.3 3-4 Notations, notes on exercises
Chapter 1 Euclidean Geometry
1.1 8-11 Pythagorean Theorem 1.1.1 (Euclid I.47)
Converse Theorem 1.1.2 (Euclid I.48)
1.2 12-14 Axioms 1-5 (Euclid), Definitions of distance,
isometry, congruence, circle, right angle,
axioms 6-8 (existence of isometries)
1.3 15-16 SSS Theorem 1.3.1 (Euclid I.8)
SAS Exercise 1.13 (Euclid I.4)
ASA Exercise 1.14 (Euclid I.26)
1.4 17-20 Equivalence of Parallel Postulates
(Text and Euclid axioms 5)
Angles of V sum to 180 Theorem 1.4.6 and
Exterior R = 2 other interior R’s Corollary 1.4.7
(both parts of Euclid I.32)
1.5 20-21 Pons Asinorum Theorem 1.5.1 (Euclid 1.5)
Converse Exercise1.24 (Euclid I.6)
1.6 21-22 Star Trek Lemma Theorem 1.6.1 (Euclid III.20)
1.7 25-29 Definition 10 Similar triangles, Corollary 1.7.4- basic property of similar V’s
Ratios of corresponding segments
Theorem 1.7.2 and converse Theorem 1.7.3
form Theorem 1.7.1
1.9 34-35 Theorem 1.9.1 medians meet at centroid G (def’n 10)
(2/3 the distance from vertex to side)
1.10 36-39 Theorem 1.10.1 angle bisectors meet at incenter I
incircle, inradius r, excircle, excenter, exradii
Define sine and cosine of an angle
Area Theorems: 1.10.2, Heron’s (1.10.4)
Law of Cosines Theorem 1.10.3
1.11 41-42 Perpendicular bisectors meet at circumcenter O
circumcircle, circumradius R
Theorem 1.11.1 Law of Sines
1.12 44 Theorem 1.12.1 altitudes meet at orthocenter H
Theorem 1.12.2 O, G, and H are collinear- Euler Line
1.13 46-47 Theorem 1.13.1 Nine Point Circle
Complementary triangles, Quadrilaterals
Chapter 2 Geometry in Greek Astronomy (briefly)
2.1 60-62 The moon and the sun
2.2 62 The earth
2.3 64-67 Timeline
Chapter 3 Classical constructions using straightedge and compass
Introduction- Plato and Euclid
The four problems of ancient Greek straightedge and compass constructions
(i) What regular polygons are constructible?
(ii) Can we trisect an arbitrary angle?
(iii) Is it possible to double the volume of a cube?
(iv) Is it possible to square the circle?
3.1 70 The rules
3.2 70-71 Constructing a regular triangle (equilateral), square and hexagon
3.3 71-73 Bisecting an angle, perpendicular bisectors of a line segment, and constructing circles with a
given length (i.e. a collapsible compass is not required)
3.4
74-77 Sums, differences, products and quotients of
constructible lengths are constructible,
is
constructible, constructible numbers form a field.
3.5
77-80 (2
/
5) is constructible thus a regular pentagon is
constructible
Solid Geometry (Compare to Euclid’s first three axioms.)
Axioms
(i) There is a unique plane containing 3 non collinear points.
(ii) If two distinct points are in a plane then the line they determine is entirely in that plane (lines can be extended – they stay in the plane).
(iii) If two planes intersect their intersection consists of more than one point.
Results in systems of linear equations corresponding to the number of intersections of lines and planes in two and three dimensional geometry
and the generalization to hyperplanes (systems of n equations in n unknowns)
Volume:
The volume of any prism is (B
h),
where B is the area of the base and h is the height of the prism.
Special cases:
The volume
of a parallipiped- ((lw)h),
cube- ((ss)
s)
= s
,
cylinder-
((
r
)h).
The volume
of any pyramid is 
of
the volume of the corresponding
prism, i.e. (B
h),
where B is the area of the base and h
is the height of the pyramid.
The volume
of a rectangular pyramid- ((lw)h),
square pyramid-
((ss)
h)
= sh,
cone-
((r)h).
Experiment: Cut a triangular prism into three (not necessarily congruent) pyramids of equal volume. One of the pyramids should have the same base and height as the prism. Thus the volume of the prism is three times the volume of the pyramid with same base and height. Justify your results.
The surface area of the
sphere is 4r.
The volume of a sphere is
((r)h).
For a fixed perimeter, the circle maximizes area.
For a fixed rectangular perimeter, the square maximizes area
For a fixed surface area, the sphere maximizes volume.
For a fixed rectangular surface area, the cube maximizes volume.
For a fixed cylindrical surface area, the cylinder with h=2r maximizes volume.
Chapter 5 Platonic Solids
5.1 97-99 Theorem 5.1.1 Exactly five Platonic solids
5.2 100 Duals of the Platonic solids
5.3 100-102 The Euler Characteristic
Topology: Tilings, Lattices and Patterns. Tiling the plane and other surfaces
The Axiomatic Method and Axiomatic Systems
History: Thales, Pythagoras, Hippocrates, Plato (Socrates), Eudoxus (Aristotle), Euclid, the parallel postulate and alternatives.
Finite Geometries:
Four-line geometry- axioms, theorems and models
(dual) Four point geometry- axioms, theorems and models
Jack and beanstalk geometry (Fe, Fo) - axioms, theorems and models
Properties of axiomatic systems: Consistency, Completeness, and Independence
Fano geometry- axioms, theorems and models
Young geometry- axioms, theorems and models
Chapter 6 Hyperbolic Geometry
Axioms for Hyperbolic Geometry
6.1 115-118 Models- disc without the boundary, trumpet horns
6.2 118-121 Neutral Geometry
Lemma 6.2.1 Perpendiculars are constructible.
Lemma 6.2.2 Angles of
V sum to
180
.
Lemma 6.2.3 Two angles of
V sum to < 180.
Lemma 6.2.4 Given VABC, there is a VA’B’C’
such that C’
C,
and A+B+C = A’+B’+C’.
6.3 121-122 Congruence of Similar Triangles
Theorem 6.3.1 In any
VABC, A+B+C <
180.
Corollary 6.3.2 Angles of
quadrilateral sum to < 360.
Theorem 6.3.3 If
VABC~VA’B’C’
thenVABC
VA’B’C’.
6.4 122-124 Parallel and Ultra Parallel Lines
Theorem
6.4.1 The angle of parallelism,
,
depends only on the length 
PQ
.
Theorem 6.4.2 A line, L, and an ultraparallel line to L have a common perpendicular.
Chapter 10 Spherical Geometry
Trigonometry of angles, degree and radian measure.
A model for Spherical Geometry: Double elliptic geometry on Sphere (of
radius 
)
– points, lines (great circles), lines are not infinite, intersection and the
parallel postulate, distance between points P and Q is
(
POQ),
where the angle is measured in radians. Poles, antipodes
10.1
209-211 Lunas: area of
luna of angle 
=
.
Theorem 10.1.1
ABC=
(A+B+C–).
10.2 209-213 Right Triangles
Theorem
10.2.1 In a right VABC
(right at
C) on the unit sphere, cos c =cosa cos b.
Theorem
10.2.2 In a right VABC
(right at
C) on the unit sphere, sin A= 
and
cos A = 
.
Theorem: There is no perfect planar map of the surface of the sphere.
10.8 224-225 Elliptic Geometry– On the sphere identify antipodal points (collapse them to one point). Every pair of lines intersect exactly once.