Geometry Autobiography.

Walter Whiteley Updated September 2004

Like many of you, I took my last 'formal' geometry course back in high school. (In my case, Ontario Grade 13, Analytic Geometry in the early 60's. Perhaps you took Algebra and Geometry in OAC?) I also had a formal 'Euclidean Geometry' course in grade 10, complete with 'ruler and compass' constructions and proofs. Such courses are not as common these days.
As an undergraduate math major (physics minor) I saw a few pictures (in calculus, analysis, topology ... ) but there were no courses in 'geometry'. An indication of the 'algebraic' slant of my undergraduate training is that text for the year long linear algebra course had exactly one picture in it! As a result of this training (including how the high school geometry course was taught) I had lots a questions about logic, about proofs etc., but did not even think about geometry, or visual reasoning.
I went to graduate school to study logic - and took many logic courses (maybe ten?) and no geometry courses. (I had some questions - like: Why was `projective geometry' considered a 'dead' subject? What makes a part of mathematics important or something to be left behind?) However, after three years of graduate work, I lost the questions I had about 'proofs' (the professors did not want to talk about my issues) and met a professor who was interested in amost everything. In particular, he had some interesting questions about the form and the content of theorems in analytic geometry. (He also had reasons why he thought these quesions were important.) I did my thesis on aspects of the foundations of analytic geometry -- and had to buy a Schaum's Outline of Projective Geometry to get started! (Actually, my thesis was on a branch of mathematics that the sociologists were then describing as the example of dead mathematics! They were wrong, as predictors of the future, but right as people describing the previous four decades.)

After a year teaching at a small university, I knew I wanted to teach, but I was not sure I wanted to do research. (I still had lots of questions - but not ones many other people shared. Who can keep working on questions for which nobody else will care about the answers?) So I went to teach in a Quebec CEGEP (equivalent to final year high school and first-year university in Ontario). For 20 years, with occasional breaks to 'visit at universities and teach first-year courses', I taught at the college: precalculus, calculus, linear algebra ... . I found I was using more and more pictures: images to visualize and explain how I thought about the math. Often my head was f illed with an 'animated movie' that was difficult to show to my students.
I also met an architectural engineer from the Université de Montreal who had lots of interesting geometry questions. I could answer a few of these, from the work I had done for my thesis - and I joined a group to work on some of the other questions. Essentially the questions concerned the relationship between shapes in 3-space which are flexible when you expect them to be shaky - and pictures of four dimensional objects. Three things made this experience important:

there were other people to talk to, to share the questions, and the excitement of conjectures and answers;
there were tangible 'applications' - I could build actual models, and explore the ideas in examples in my hands and eyes;
there was lots of pretty geometry: I used everthing I knew and had to learn lots more.

I still work with these people, as well as a number of other people around the world whom I met through this geometry. (One of my collaborators is someone I first worked with as a fellow graduate student over 35 years ago.)
I started to do research again - and have continued ever since. After a while, there were Quebec government grants which reduced my teaching hours and gave me more time for research. I broadened my research into some other areas of 'discrete applied geometry': areas related to Computer Aided Design, Geometric Modeling, Robotics, etc.. I also worked with people in Montréal who are doing research of spatial visualization, pedagogy of geometry, etc. (including the same architectural engineer who taught 'descriptive geometry' for over 30 years).

Just over twelve years ago, I was offered to chance to come to York, to teach mathematics (including geometry - for my first time) and to continue my research. I have taught this course, and variations on it. a number of times now. I have lots of mathematics I would like to explore with you, as well as some pedagogical ideas I am working with. I hope to learn more geometry (and more pedagogy) this year and anticipate you will have ideas and questions that I will learn from.
In my research, I have learned lots of geometry. In my travels doing research, I have met lots of people outside of mathematics who have questions about geometry - but know even less than I do. [People in engineering, computer science, design, robotics, biology, philosophy, medicine, psychology, biology, etc.] I still have lots of geometry questions of my own (more questions every year).
In my research, I have used the computer for geometry in a variety of ways. I used the standard `drawing' programs (on the Mac) to illustrate my ideas and to help my thinking as I presented the constructions. I learned some of the program Maple to help with the algebraic manipulation (analytic geometry) and in the process learned some of its plotting features. More recently, I have been using programs such as The Geometer's Sketchpad to investigate geometry problems, and to explore some geometry with undergraduate mathematics students. Because of my research, I visited Berkeley California in 1966 and worked for three weeks with the developers of The Geometer's Sketchpad (and some Spherical models I will show you next class). I am very interested in the use of such tools, and manipulatives (which I use in all my geometry courses), so that people can explore the ideas, make conjectures, and 'own' the material with {\it their} questions, rather than just mine. (However I don't leave my own questions at the classroom door.)
I am pleased that I managed to anticipate changes in geometry teaching - as Geometers Sketchpad is now required for all grade 9 classes (and encouraged for all years) in the new Ontario Curriculum. We can have a class discussion about the good and bad ways to use such softeware. To me, these programs give an opportunity to change what we 'see' in geometry (and in other mathematics) in important ways. If you are interested in more of my thoughts on this, there are two related papers on my main web page which you can download. We will talk more about 'seeing mathematically' during the course.
Throughout the years, dating back to my high school days I have wondered about 'proofs', how we decide which mathematics is 'true' and which is 'important'? I have continued to read on the 'philosophy of mathematics' and reflect on how I (and others around me) do mathematics. I am interested in an `inquiry' approach to this course because I think it will be good for us, and because I should practice the approach I `preach' for teaching high school students. In particular, I have lots of ideas and questions about the potential for diagrams, objects and pictures inside mathematical proofs.
I believe that mathematics, and mathematics education, are undergoing a 'paradigm shift' about the essential role of diagrams in and visual tools in learning, doing and communicating mathematics. At some point, we should have a discussion (debate) about what makes a good 'proof' in geometry, in algebra etc. Again, I talk some about this in the papers on my web site - talks I gave to audiences of mathematics educators.

Since I insist you offer some of your geometry questions, let me close with a few of my questions.

Here are some examples of questions that I have been working on. (I have been thinking about it for about 15 years, and worked on this with undergraduate students over three summers. The Federal government does give research grants for such work through NSERC!)

Remember how you check whether two triangles are congruent?: SSS: three sides equal;; SAS: two sides and the enclosed angle equal;; ASA: two angles and one side equal;; SSA: two sides and the non enclosed angle equal - was ambiguous - but, for example, if both triangles were acute this was enough;
We never needed four facts - only three, and we never used three angles. (Three angles only guarantees similarity - it is the same as having only two angles equal.)

Question: What data should you use to check that two sets of four points (quadilaterals?) are congruent?
How many facts? Which patterns of angles and lengths?

Here is a second, easier geometry problem I play with:

Question: How many fingers do you need on a 'Robot hand' in order to be able to grip and contral a solid object, if you assume the fingers are like 'claws' and there is no friction?
Note that five is the wrong answer!

Here is a third question that I worked on this last summer, with three undergraduate students (one of whom is in our class, and one of whom was in a first year class with me last year). There is a large poster we made this summer, outside of my office door.

Question: What portions of a protein are rigid, and how does it flex? Does a mathematical model of a protein as a structural framework explain how it folds and how it functions? (E.g. Mad Cow Disease is caused by a protein - prion - unfolding and folding into a different shape which becomes a rigid plaque.)

Question: How much of my own reasoning in mathematics is based on my senses (visual and kinesthetic) and my experiences with these senses? How do people learn to work with 3-D symmetry? Do we need to work with 3-D objects in order to later understand computer animations and then to reason 'in our minds eye'?

If you are interested in learning more about the kinds of applied geometry I do, or about summer research next summer, talk to me.

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