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Math 4060 -- Foundations of Geometry
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Class Meets:
TR 3:30 p.m. - 4:50 p.m. in PHYS 115.
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| Office Location: | General Academic Building
(GAB) 405. |
| E-mail: |
wcherry@unt.edu
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Prerequisites
All students should have successfully completed Math 2510
(Real Analysis I) before taking this course. The reason for
this is that this course is a proof intensive course and students
should already be familiar with basic proof writing techniques and
strategies.
Prior or concurrent enrollment in either Math 2520
(Real Analysis II) or Math 3510 (abstract algebra)
is also strongly recommended. This is meant to ensure familiarity with proof
and proof technique. Nothing from real analysis or abstract
algbra will actually be needed.
Students who have not yet taken Math 2520
or Math 3510, but who are confident in their proof-writing ability,
are welcome in the course.
Course Description
Early interest in geometry was almost certainly motivated by
a desire to improve building techniques and to be able to
create interesting shapes for temples, altars, toys, and machines.
Scholars in ancient Greece "abstracted" the study of shapes
into an idealized form. The most widely read text book in history
is a geometry text by Euclid known as the Elements.
Thousand's of years later, much of what
we learn in high school geometry is based on the ideas in Euclid's
text. Euclid's goal was to create a logical foundation for idealized
geometry. He wanted to provide proofs for geometric facts based on
as few axioms as possible. The course will begin with a careful study
of parts of Euclid and an exploration of what he was trying to do and
why. The course will then move on to the contributions of
19th and 20th century scholars who built on Euclid's work. The second half of
the course will look at alternative geometries where the so-called
parallel postulate need not hold.
The course will emphasize the role of proof in geometry, its historical
development, and the philosophical implications that proof had on
the development of scientific thought. Students will begin by reading
Euclid and continually develop their own proof writing skills and
geometric intuition.
Textbooks
Robin Hartshorne, Geometry: Euclid and Beyond,
Springer Verlag, 2000. ISBN 0-387-98650-2.
Euclid's Elements, The Thomas L. Heath translation edited
by Dana Densmore, Green Lion Press, 2002. ISBN 1-888009-19-5.
Euclid is also available online at
http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
Course Requirements
Grading in the course will be based on weekly homework assignments,
an in class mid-term, and an in class final exam.
| Homework | 40% |
| Midterm |
30% |
| Final Exam | 30% |
Digitization of Ancient Copies of Euclid
Click here to see a handwritten copy of
Euclid's Elements made by Stephen the Clerk
in Constantinople in the year 880.
Click here for a Latin translation of Euclid's Elements.
Important Dates
| Midterm: | Thursday, March 13 |
| Final Exam: | Thursday, May 8,
1:30 - 3:30 p.m. |
Course Handouts
Course Syllabus
Article on Rusty Compass Constructions
in Islamic Art
Picture illustrating the Euler line when
the circumcenter and orthocenter fall outside the triangle
Pictures Illustrating the 9-Point Circle
A picture of the Miquel point outside the triangle
Hilbert Axioms up through section 7
Information about the Midterm Exam
Ask for a copy of last year's midterm (not available on the web).
Ask for the solutions to some of the homework from section 6
and 7 (not available on the web).
Hilbert Axioms
Information about the Final Exam
Ask for a copy of last year's final (not available on the web).
Maple Library (Optional)
For those of you interested in and pretty expert with Maple, you may
be interested in my Maple library for drawing ruler/compass type constructions
in Maple. This library is not documented, so you will have to be pretty
expert to figure out how to use this. The Maple procedures are written
in a plain text in the file geom.txt. To use this, you will need
to read this file into Maple with the "read" command. Be sure you either
give a complete path name, e.g. starting with C:, or be sure geom.txt is
in the directory that you start Maple from. I've also included the worksheet
that deomonstrates the proof of the 9 point circle so you can see how it
works. Working with Maple is not required for this course, so you
do not need to play with this stuff except for fun. To save these files,
you want to click on the links with the RIGHT mouse button and choose
"Save Link As".
Maple Library for Ruler and Compass Constructions
Maple worksheet illustrating the
proof of the 9 point circle
Homework Assignments
Homework #1 due Jan 17, Jan 22, or Jan 24
Homework #2 due Jan 24
Homework #3 due Jan 29
Homework #4 due Jan 31
Homework #5 due Feb 5
Homework #6 due Feb 7
Homework #7 and #8 due Feb 12 and Feb 14
Homework #9 and #10 due Feb 19 and Feb 21 (Due date for HW#10 extended until Feb 26)
Homework #11-14 due Feb 28, Mar 4,
Mar 6, and Mar 11. (Due date for HW#13 extended until Mar 11 and HW#14 canceled)
Homework #15 due March 27
Homework #16 and #17 due April 1 and 3.
Homework #18 and #19 due April 8 and 10.
Homework #20 and #21 due April 15 and 17.
Homework #22, #23 and #24 due April 22, 24 and 29.
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Trouble reading or printing links on this page?
The homework assignments and supplemental materials above are
in Adobe PDF (or Acrobat)
format. If you are having trouble reading or printing these hand-outs,
click here.
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