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

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)