Mathematics 380 :  Visual Mathematics
Fall 2005

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:

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
    To develop visual intuition in mathematics and see several examples of where this intuition is applicable. 

    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. 

    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.

    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).

    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.  


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)