Professor: Jeff Johannes
Section 1
MWF 11:30a-12:20p Sturgis 103
Office: South
326A
Telephone: 245-5403
Office Hours: Monday 12:30 - 1:20p, Tuesday 8 - 9p,
Wednesday 12:30 - 1:20p, Thursday 3-4, 8 - 9p and by appointment or
visit
Email Address: Johannes@Geneseo.edu
Web-page:
http://www.geneseo.edu/~johannes
Reference books on reserve in the library
Beyond the Third
Dimension: Geometry, Computer Graphics and Higher Dimensions, Banchoff
QA 691.B26 1990
A Survey of
Classical and Modern Geometries, Baragar QA445.B318 2001
Experiencing
Geometry: Euclidean and non-Euclidean with history, Henderson
QA453.H497 2005
Projective
Geometry and Modern Algebra, Kadison, Kromann QA 471.K224 1996
Mathematics in
Western Culture, Kline QA21.K52 1953
Mathematical
Thought from Ancient to Modern Times, Kline QA21.K516 1972
Projective
Geometry and its Applications to Computer Graphics, Penna,
Patterson QA 471 P395 1986
Geometry,
Relativity, and the Fourth Dimension, Rucker QA 699.R8
1977
Mathematics and
its History, Stillwell QA21.S84 1989
The Shape of
Space, Weeks QA 612.2.W44
2002
Perspective in
Perspective, Wright N 7430.5 W7 1983
Purposes
To develop visual intuition in mathematics and see
several examples of where this intuition is applicable.
Overview
The goals of this course are to consider the way in
which we view the world around us and to consider how we may develop
our vision into a reliable mathematical guide. We will begin by
considering how the world we see is different from the world in
abstract existence. This will lead us to consider projective
geometry. There we will consider consequences of the line at
infinity and classical results such as theorems of Desargues and
Pascal.
Next we will consider the fact that abstract reality
still isn’t very Euclidean. We will begin by exploring the
geometry of the Earth – a sphere. We will discuss different rules
for distance, and how lines and triangles behave differently. We
will compute area formulas and some spherical trigonometry.
Does the search for application of Euclidean
geometry take us to the three-dimensional space in which we live?
Perhaps, but perhaps not. We will consider various options for
three-dimensional manifolds representing the universe.
Finally, is the three-dimensions we see the only
possible reality? Perhaps there is more. We will discuss
how to visualise higher dimensions and some of the ramifications of
higher dimensional geometry.
Grading
Your grade in this course will be based on four
problem sets, two in-class exams, and a project (written and
presented). Each of those aspects will be worth at least a
quarter of your grade and each component of each aspect will be equally
weighted. The remaining quarter will be determined by each
student individually. You may distribute that quarter as you see
fit among the announced course aspects or propose a new course aspect
for the remaining quarter. All grading systems must be proposed
by September 9.
Problem Sets
Problem sets will consist of questions related to
each topic area. They will be due the day after we have completed
the topic area. Before these papers are handed in, I
strongly suggest discussing them with me and others outside of
class. These discussions will be graded on a ten point
decile scale based on completeness, accuracy, and writing.
These problems will be evaluated as follows.
0 Missing
3 Question copied, nothing written
6 Something written that appears that it was only
written to take up space
7 Substantially incomplete. Something written,
but does not really answer the main questions. Major errors. Very
poor writing
8 Mostly complete. maybe a few minor errors
9 Complete, no errors, some personal insight,
well-written
10 Wonderful
No late problem sets will be accepted.
Projects
Each student is responsible for completing a
project. Your project will include research on a topic in visual
mathematics. You must include reference to at least five
resources, two of which must be non-internet resources. You also
will
be responsible for a twenty minute presentation of your
project. Selecting the topic by the deadline will be worth
5%, the summary of research (annotated bibliography) will be worth 15%,
the draft will be worth
20%, the presentation will be worth 30%, and the final paper will be
worth 30%.
In-class Exams
There will be two in-class exams. The exam
will be
graded on a scale approximately given by
100 – 80% A
79 – 60% B
59 – 40% C
39 – 20% D
below 20% F
For your interpretive convenience, I will also give you an exam grade
converted into the decile scale. The exam will be challenging and
will require thought and creativity. It will not include filler
questions
(hence the full usage of the grading scale).
Feedback
Occasionally you will be given anonymous feedback
forms.
Please use them to share any thoughts or concerns for how the course is
running.
Remember, the sooner you tell me your concerns, the more I can do about
them.
I have also created a web-site
which accepts anonymous comments. If we have not yet
discussed
this in class, please encourage me to create a class code. This
site
may also be accessed via our course page on
a
link entitled anonymous
feedback. Of course, you are always welcome to approach me
outside
of class to discuss these issues as well.
Religious Holidays
It is my policy to give students who miss class
because of observance of religious holidays the opportunity to make up
missed work. You are responsible for notifying me no later than
September 9 of plans to observe the holiday.
Schedule
August 29 Introduction
31 Projective
September 2 Projective
7
Projective
9 Projective
12 Projective
14
Projective
16
Projective
19 Sphere (Projective PS due)
21
Sphere
23
Sphere
26
Sphere (Project Topic Due)
28
Sphere
30 Sphere
October 3 Sphere
5 Sphere
7 Space (Sphere PS due #1)
12
Space (Sphere PS due #2)
14 Exam (Projective and Spherical)
17
Space (Project Research Summary Due))
19 Space
21
Space
24
Space
26
Space
28 Space
Seaway meeting presentations
31 four (Space PS due)
November 2
four
4 four
7 four (Project Draft Due)
9 four
11
four
14
four
16
four
18 Exam
21 closure (4d PS due)
28 Project presentations
30
Project presentations
December2 Project
presentations
5 Project
presentations
7
Project
presentations
9
Project presentations
12 Project presentations (Project Paper Due)
December 16 Project presentations (6)