In this post I want to share some details from my biography related to my love affair with projective geometry.

My career has revolved around two poles: computer graphics on the one hand, and projective geometry on the other. These two subjects have quite a bit in common, as perhaps I'll be able to make clear through this blog. But in other respects they have opposite natures, at least in the form they appear in my life. To understand why, I need to say more about my interest in projective geometry.

That was first awoken by reading the book "Projective Geometry: Creative Polarities in Space and Time", by Olive Whicher, published by the Rudolf Steiner Press in 1979. This occurred after I received my B. A. in Mathematics from UNC-CH in 1978. I had already been bitten by the computer graphics bug, had taken a computer graphics course, written my first hidden surface removal algorithm (on punch cards) and produced pictures using the IBM mainframe in the basement of Phillips Hall (which houses the math department of UNC-CH and at that time also the computer center of the CS Department). I had also taken a job doing graphics programming at RTI, a research institute in the so-called Research Triangle not far from Chapel Hill, after my experiences as a teaching assistant for beginning calculus classes showed me I had little skill as a teacher. I found I was a very good programmer, however. At the same time, I was missing the sense of meaning in the mathematics I was doing, and the book from Olive Whicher filled this hole superbly.

Through a series of coincidences, immediately after reading the book I was able to attend a 3-week summer school in which Olive Whicher (coming all the way from England!) taught a course in this subject. That was at the Rudolf Steiner Institute, held that summer in Natick, MA. Through this institute, I came into contact for the first time with anthroposophy (established in the early part of the 20th century by said Rudolf Steiner). In the course with Olive Whicher I was exposed to a new way to consider the role of mathematics in the world and in the human being. Projective geometry plays a key role in this renewal of mathematics. The approach resonated deeply with my own strivings, as yet unsatisfied, and I realized I had found something important for the rest of my life. Olive Whicher had been a co-worker of George Adams for 28 years, until his death in 1963. George Adams (1894-1963) was a student of Rudolf Steiner's who devoted his considerable intelligence and energy to working out indications Rudolf Steiner had given in his lectures regarding the renewal of natural science, based on the thought-forms of projective geometry.

On the other hand, I had a growing interest in representing mathematics using the newly developing medium of computer graphics. As a graduate student in the math department of the Univ. of North Carolina at Chapel Hill from 1978 to 1983, I had access to one of the best computer graphics labs in the world (through the Computer Science Dept) and I developed into quite a "hacker", finally earning my master's degree with a project which implemented the euclidean wallpaper groups on one of the first color "frame buffers". These skills led to jobs on the West Coast after graduation, and from 1984-1987 I had the opportunity to work at Lucas Films (spun off as Pixar, Inc. during my stint there), meeting the best and brightest in the computer graphics world.

The world of mathematics however exerted a stronger influence and in 1987 I was given the opportunity to be the technical director of The Geometry Center, a large NSF project at the Univ. of Minnesota, which focused on bringing the power of modern computer graphics to research mathematics. In this capacity I had the good fortune to co-direct a mathematical animation "Not Knot" which broke new ground in visualisation of difficult but fascinating mathematical concepts in the area of non-euclidean geometry. It was a natural continuation of my master's project to three dimensions.

During this phase of my development by interest in anthroposophy grew slowly. I began to understand more how the ideas about projective geometry I had learned from Olive Whicher (and in a later course in 1981 from Lawrence Edwards) connected to the larger body of ideas which is anthroposophy. There the focus was on the development in the human being of faculties of independent thinking; the dangers of mechanical assistance (as in computers) was a theme which I confronted when I shared my enthusiasm for computer graphics with other anthroposophists. A growing recognition that I had done enough programming led me to train in Mannheim, Germany to become a Waldorf high school math teacher, a profession which I practiced at Green Meadow Waldorf School in Spring Valley, N. Y., for five years (from 1998 to 2003) under the mentorship of the physicist Steven Edelglass.

The last swing of the pendulum took me back to programming and academia as I returned to the Technical University Berlin in 2003 where I obtained my Ph. D. in September 2011 under the guidance of Prof. Ulrich Pinkall on a topic inspired by George Adams, a treatment of rigid body motion in euclidean and non-euclidean spaces, all based on projective geometry. The result prepares me and others interested in this theme to develop the indications of Rudolf Steiner, initially taken up by George Adams, further using a fully modern formulation. It is this task to which I am now trying to devote my energies. Also with my return to Berlin in 2003, I picked up the graphics programming which I had been so glad to leave in 1997 when I entered Waldorf teaching, so that I am now prepared to generate movies and applications illustrating the ideas which I want to spread. All that remains is, alas, that I get to work and so what I have set out to do.

Hi Charles,

ReplyDeleteany chance to get the basic motivation and ideas of your thesis explained here, sparing the need of digging through the formalism ?

/Christian