Introduction to the Entire Course
22aug10

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The Warp and Weft of this Course

This course is constructed like a piece of woven cloth, with
a warp and weft. The orderly warp of parallel strings
represents the progression of three kinds of plane geometry,
the affine, projective and transformational. This corresponds
to selections from chapter 1, 2 and 4 in Philippe Tondeur’s
textbook, and chapter 3 of the Topological Picturebook.
The weft of the course is a thread that weaves back and forth
through the warp. It is organized historically according to the
methods typical to each era.

Three Geometries of the Plane

Affine Geometry
In the affine geometry of the plane we subtract something from
Eucid's geometry. We do not use the concept of distance and angle.
Thus the notion circle and perpendicular do not occur in
affine geometry. But parallel, concurrent, proportions
do make sense here.
The method we use in our exposition of affine geometry is
Cartesian because we use vectors.
Mastery of this first module of the course will be evaluated
by the homework, quiz and the midterm.
Projective Geometry
Only twice in our history does art significantly precede mathematics.
From the experimental and visual theory of linear perspective derives
a projective geometry. In the projective geometry of the plane  we change
something in Euclid’s geometry. We add ideal points and lines, and
thereby changing the axioms to make them more "democratic". Just
as any two points are joined by a line, now any two lines cross at
a point. We give up the concept of parallelism to gain greater
generality of our theorems. This material is taken from the author’s
Topological Picturebook and expanded into lessons on perspective
drawings.
Mastery of this second module of the course will be evaluated by
the "daily" drawing exercise, and a project proposed and completed
by the student.
Transformational Geometry
In his Erlanger Program, Felix Klein established the principle that
all geometry derives from the study of transformations. We will study
transformations that only change the scale (dilatations) and those
that preserve distance (isometries). Excerpts from Chapter 2 of
Tondeur’s text and most of Chapter 4 pertain to this module.
This third module of the course will be evaluated by homework, a
take-home test, and the final.
Final Examination and Grade
A comprehensive, 3 hour in-class final is the culmination of this course.
It can only be taken at the announced time. Do not make arrangements
to be absent from the tests or the final. Your grade will reflect my
evaluation of your mastery of the course. It is
based on a great variety of ways you can demonstrate this mastery. Each
contributes a proportion to the grade. A running estimate of your final
mastery based on work observed to date can be made on request, but this
had at least initially, a great margin of error.

Five Themes in Geometry

The weft of the course is a historical thread of five themes that weaves
its way throught he presentation of the course. You should keep track of
which theme a particular part of a lesson fits into in your journal, for
example. Some are pretty obvious. Others may lead an insightful discussion
just which theme it best illustrates.
The Physical Origins of Greek Geometry
From the dawn of agrarian civilization, in the Nile, Euphrates,
Indus valleys, and in China, the earth had to be measured for
purposes of ownership, taxation and inheritance of fields. Seasons
had to be anticipated, and planting calenders constructed based on
astronomical observations. This knowledge was collected by
Greek mathematicians, beginning with Thales and Pythagoras 25 centuries
ago. It reached its height ca -300 with Euclid’s axiomatization of
mathematics in his book, The Elements. Archimedes (-200)
anticipated the calculus as it applies to physics.  It ends with Pappus
of Alexandria at about +300.
Renaissance Perspective and 3-dimensional Drawing
Only twice in the history of humanity did art preceed mathematics.
In the Renaissance, artists discovered and formalized the rules of
linear perspective which only later became projective geometry.
Their practices also laid the foundation for non-Euclidean geometry
in two senses: with analytical geometry the methods of proof that
Euclid enshrined are abandoned. With transformational geometry the
Euclid’s postulates themselves are altered or denied.
The Industrial Origins of Cartesian Geometry
In the 17th century, Descartes and Fermat created what we now call
analytical geometry. Instead of the axiomatic method of the Greeks,
Cartesian geometry is based on the properties of numbers, and uses
algebra as its main method of discovery and proof. The contemporary
form of this is vector geometry. It was further developed by
Newton and Leibniz for their invention of the Calculus. Thus,
Rartesian geometry is the foundation of all present day science
and technology.
Klein’s Erlangen Program to Unify Geometry
In the 19th century, the problem of making geometry, and mathematics
in general, a logically rigorous science was take up in earnest. At
the beginning of the century, the philosopher Kant still taught that
the geometry of Euclid is the only kind of geometry that there is
because any other kind is inconceivable by the human mind. By the
end of the century, mathematicians had not only shown that many
geometries which violated Euclid’s axioms, not only
"could be thought of"
but that they were equi-consistent with Euclidean geometry and with
all other branches of mathematics. This subject is treated in MA402.
Felix Klein proposes that the various kinds of geometry could be based
on their groups of isometries. That is, the collection of transformations
which preserves the properties of interest to the particular geometry, can made into
the foundation of that geometry. For instance, a precise definition of how to measure the
distance between two points suffices for building up the entire edifice of
plane geometry.

The Geometry in Computer Graphics
In the last decades of the 20th century, the artistic and technological
demands of computer graphics has led to corresponding changes in
mathematical teaching and research, much as perspective did in the Renaissance.
This course, MA403, is increasingly a witness to this evolution as it
pertains to college geometry.