Introduction to the Entire Course

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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.