Holy Cross Mathematics and Computer Science
MATH 392, section 3 -- Geometry Through History, Spring 2016
Note: Information on this page, especially the
course schedule, is subject to change. Any changes will be announced here
and in class.
Syllabus and Schedule
Examples, Class Notes, Etc.
- Overview of Greek history and mathematics (.ppt) -- class
on January 27
- Another way to see the proof of Book I, Proposition 47 of Euclid's Elements
- Slides on Elements, Book I (.ppt) -- classes on February 1 and 3.
- The Elements
of Euclid in English translation, with mathematical commentary (from David Joyce's web site at
Clark University ``across town'')
-
The Elements
of Euclid in English translation, with Greek text and facsimile of a Greek manuscript from about 888 CE in Bodleian Library at Oxford University (U.K.) (from the Clay Mathematics Institute)
- The Elements of Euclid,
in English translation by Sir Thomas L.Heath and linked Greek text (Perseus Project)
- Note: the mathematical
contents of these on-line versions of Euclid are essentially equivalent.
I highly recommend you at least take a look at the second one, though, because
you can see what a hand-written copy of this text in the original
Greek from the Middle Ages looks like. (Note that Euclid lived
about 300 BCE -- that is, about 1200 years before this manuscript was produced!)
- Solutions/Lecture Notes from class on Monday, Feb. 8.
- Solutions/Lecture Notes from class on Friday, Feb. 12.
- Proof that parallelism is symmetric from class, Wednesday, Feb. 17.
- Additional properties of parallelism from class, Wednesday, Feb. 17.
- Properties of the hyperbolic plane from class, Friday, Feb. 19 and
Monday Feb. 22
- The Bolyai-Lobachevsky theorem from classes Monday, Wednesday, Friday, Feb. 22, 24, 26.
- Final step in the proof of Bolyai-Lobachevsky
from class Monday, Feb. 29.
- Discussion on Descartes and coordinate geometry, class on
Friday, March 4.
- Solutions for the midterm exam
- Geodesics on a surface of revolution, class on Wednesday, April 20.
Assignments
- Small group discussion day on Euclid, Book I, Proposition 47;
writeups due no later than start of class on Monday, Febrary 1.
- Problem Set 1: Problems 2.8, 2.9, 2.13, 2.14 from McCleary -- due: in class, Friday, February 5. (Note: You may write out these proofs using either the
"statement/reason" 2-column format you may have used in high-school geometry, or as continuous
text in complete sentences. Either way, though, provide a reason for every claim you make.
As the directions indicate for 2.8, note that you can use Postulates I - IV and any
results proved in Book I prior to Proposition 9, but nothing else(!).)
- Problem Set 2, due: in class Friday, February 12.
- Problem Set 3, due: in class Monday, February 29.
- Information on Final Projects
- Problem Set 4: Problems 5.4, 5.5, 5.7, 6.2, 6.3, 6.5, 6.6 (Note for 6.2:
This means that you are to show that each of the properties (1),(2),(3),(4) is
equivalent to saying the curve is a straight line. One strategy -- show they are
all equivalent, and then show that any one of them is equivalent to the curve
being a line) due: in class Friday, April 1.
- Problem Set 5: Problems 7.4, 7.8, 7.9 (Note for 7.4, 7.8, 7.9: The "component
functions of the metric" and the "metric coefficients" are the same things -- the
functions E,F,G appearing in the first fundamental form.) due: in class, Friday
April 8.
- Problem Set 6: Problems 8.2, 8.4 adding: ``and Gaussian curvature''
to the question (Suggestion for 8.4: Use 8.2 and equations:
z2 = 1 - x2 for the cylinder,
and z = x2 - y2 for the saddle.
The completely general ellipsoid makes for a relatively hard calculation.
If you can't get that to work out, try this particular ellipsoid
x2/a2 + y2 + z2 = 1 and experiment with different
parametrizations. For instance, you could "doctor up" the spherical coordinates parametrization of a sphere, or recognize that this particular sort of ellipsoid can be generated in other ways too.), 9.2, 9.3. due: in class, Friday April 15.
- Small group discussion day on the Poincaré upper half plane as an abstract surface, writeups due no later than Friday, May 6.
Announcements
- Don't put off working on the final project for this course; recall that you
will need to submit an annotated bibliography of sources by April 18, and that
I will be meeting with all the project groups during the week of April 18 for
progress reports.
Related Links and Other Information
- Note: The Greek phrase appearing next to the heading of this page is one traditional rendering of
the reported inscription over the entrance to Plato's Academy in Athens
(founded about 387 BCE).
It means (roughly)
Let no one ignorant of geometry enter. This is a reflection of the foundational
role of geometry in Plato's ideas about knowledge and education.
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Last modified: April 21, 2016