Math 4520, Classical Geometries
Fall 2017
Monday, Wednesday, Friday in Malott 205
10:10 to 11:00 AM

Instructor: Bob Connelly (
Office: 565 Malott
Office hours: Monday, Wednesday, Friday 1:30-2:15 in Malott 565
Last updated:  November 27, 2017

Prerequisites: A good course in linear algebra such as Math 2210 and an interest in geometry.

Exams:  We will have a take-home exam, a project, and a final take-home exam in addition to weekly homework when there is not  take-home exam due.
Projects: Everyone will have project that will be due before the last day of classes, December 1, 2017.  Here are some ideas to get started, but before you start, come to see me so we can discuss it.
Take-home exam: (handed out in class).

In class prelim, October 23, covering all chapters up to and including Chapter 10.  Bring a ruler.

Reading and Homework:

1.  Chapter 1:  Due Friday, August 25.  Do Problems 1, 5, 6, 7.  (extended due date: Monday, August 28.)
2.  Chapter 2:  Due Friday, September 1,  Do Problems 3, 4, 5, 6, 8, 9
3.  Chapter 3:  Due Friday, September 8,  Do Problems 4, 7, 9, 10, 11
4.  Chapter 4:  Due Friday, September 15, Do Problems 3, 4
5.  Chapter 5:  Due Friday, September 22, Do Problems 2, 3, 9, 10
6. Chapter 6:  Due Friday, September 29,  Do Problems 2, 3, 4, 6, 11
7. Chapter 7: Due Friday, October 6, Do Problems 1, 3, 4, 6, 7
8. Chapert 8: Due Friday, October 13, Do Problem 9, and find 3 4-by-4 orthogonal latin squares.
9. Chapter 9: This tells about orthogonal latin squares.  Hand in Exercise 3,  Friday, October 13
10. Chapter 10:  This is about projection and projectivities.  For Friday.  October 20, Do problems 5, 6, 7, 8, 10.
11. Chapter 11:  This about polarities and duality.  For Friday, October 27. Do problems 2, 3, 4, 5.
12. Chapter 12:  The cross ratio.  Do problems 1, 2, 6  for Friday, November 3.
13. Chapter 13:  The Complex field and circles: Do problems 3, 6, 7 for Friday, November 10.
14. Chapter 14:  The Hyperbolic Plane.  Do problems 1, 2, 4, 5 for Friday November 17.

For those who do not present or hand in a project, do problems 10, 11, 12,  of Chapter 14 to be handed in by December 7, 2017.

Quickie homework due Wednesday Sept 20 in class:  Hand in the dual figure of the of the hyperboloid lemma done in classHere is the figure with key points labeled.  No description needed.  Just hand in your dual figure with the corresponding lines indicated and your name.

Quickie Homework 2:  This is picture of a circle in a square in one-point perspective.  Drawing 4 lines or less, construct a fifth point on the image of the circle.  Just hand in a copy of the figure with the lines indicated and the point on the image of the circle (an ellipse) indicated.  Due Wednesday, September 27.

Here is a reference that can has a lot of very good information about projective geometry and projective planes.  It is

Perspectives on Projective Geometry
A Guided Tour Through Real and Complex Geometry
Authors:  Jurgen Richter-Gebert
ISBN: 978-3-642-17285-4 (Print) 978-3-642-17286-1 (Online)

This is available through the Cornell Library, which you can access as a student.  We will have some problems and readings from this text later.  Take a look, please.

The "Real Projective Plane", by H. S. M. Coxeter is also a good source of information about the real projective plane.

These notes by Chris Zeeman are a great source of ideas and material for projects and exercises on perspective principles.

It is highly recommended that you use LateX (using TeXshop to run it, for example) to hand in your homework.  Here is a sample LateX document that you can use to get started.  It has a lot of definitions that are helpful for our class, and it shows how to import figures and pictures (in pdf format for example).  In order to typset the file above you need to have the following figures in the same directory.  fig1.pdf, Hogarth.pdf

It is also useful to have the software Cinderella to draw accurate pictures conistant with the principles of perspective and projective geometry.  For examle here is a Cinderlla document that you can run to locate the center of vision in 3-point perspective.


The following is a preface to the course, and following that are chapters of some notes that will be used as a text and provide some exercises.

This is an introduction to geometry and how it has been moved by a combination of classical Greek influences as well as a more modern desire simply to understand the world as it is seen.

The influence of Euclid's Elements was pervasive in western thought, especially mathematics. The idea is that you should explicitly state what is assumed. One stated one's axioms. But the idea was to minimize such unproved statements. Euclid created a small list of such axioms, but he delayed using the fifth until he was really forced. Did he really need that fifth axiom after all? Could it be proven from his other axioms?

Meanwhile, during the Renaissance, artists began to seriously ask just how should one draw a picture that accurately shows what we see? The realization of just how this should be done came almost as a revelation. This led to the development of perspective drawing, and that led to projective geometry. The difference between what is drawn in "correct" perspective and what is not, is striking. The principles perspective are simple, but some of the consequences are not what one might expect.

In a way, projective geometry is an example where one can apply the insight obtained from a simple set of axioms, unlike the situation of Euclid, where there were a very large number of hidden, subtle, complicated, axioms. Projective geometry has just three simple axioms, two are just mirror duals of each other, and the third hardly counts. We will use these to show how they provide a great perspective into the nature of geometry, even Euclidean geometry.  Projective geometry provides a toy axiomatic system, without a lot of fuss or mess, yet still delivering what is needed.

What do the practical questions of perspective drawing and projective geometry have to do with the esoteric question of whether Euclid's fifth axiom can be deduced from the others? Projective geometry gives one an excellant point of view on how to think think about geometry in general and Euclidean geometry in particular. Indeed, the answer to the question about the fifth axiom, led to the realization that the hyperbolic plane exists. A very natural way to understand the hyperbolic plane (especially with respect to projective geometry) is to think of it as a geometry, but with a distance measured in terms of an indefinite inner product. This in turn is the basis of special relativity.

Along the way in this story, one sees many other interesting and important subjects in mathematics. For example we consider, finite fields related to finite projective planes, orthogonal latin squares used in statistics, the duality of points and lines, the geometry of circles and complex numbers, and the strong analogy between spherical geometry and hyperbolic geometry.

The following picture shows some of the concepts coming from perspective drawing that are based on some ideas of projective geometry. For example, parallel lines project to lines all through the same point, or they project to a collection of parallel lines again. The properties of the projection of a cube, depending on how the cube sits with respect to the viewing plane, lead to some unexpected properties, as well as some very particular constraints, on how one is to draw a cube in realistic perspective.

The following picture shows the Minkowskii/Lorenz model of the hyperbolic plane, which is very naturally seen as a hyperboloid of revolution with two sheets. The diagram shows how the projection, viewed along the y-axis, of this surface from the "south pole" onto the horizontal plane gives a pleasant "model" for the hyperbolic plane. This is the model that was promoted by Henri Poincare.

Link to the Math Department.