The one exception to my high school math misery was Geometry. It was the only subject I was able to do fairly well — probably because it had pictures. And so, after two years of recapitulating high school algebra with its numbers and letters and piled-up polynomials, I embarked on first year Geometry in the fall of 2002.
One of my 'friendly scientists' gave me a textbook, simply titled 'Geometry,' dated 1984. Simple geometry hadn't changed since the Hellenistic era, but our world had done quite a bit of changing just since the book's publication. The cover of this book featured a dizzying perspective looking up from the base of one of the World Trade Center towers. Inside the book were color pictures of geometric forms drawn by that new-fangled contraption, the desktop computer with its computer-aided design programs.
One of the authors of this book bears the amazing name of Phares O'Daffer. As of 1999, this gentleman was still with us, professor emeritus of Illinois State University; perhaps by now he has gone on to mathematical heaven. I thank Mr. O'Daffer and his co-authors Stanley R. Clemens and Thomas J Cooney for being my guides through the angles and planes of memory and logic.
I had not done geometry, let alone formal proofs, for 33 years. Yet I soon remembered my two-column proofs from my high school days. I loved proofs then, and I love them now. Proofs are little chapels of order, logic, and rightness in a world of ambiguity and chaos. Statements two and three follow obediently from statement one, living in an ideal world where serene arguments lead to amiable and agreeable conclusions.
And yet there were moments of ambiguity even in the world of proofs. It turned out that there were many ways to prove even the simplest problems, and that the solution cited in the book was not the one I came up with, even though both worked. I had to judge for myself that I had solved it.
I covered literally hundreds of papers with neatly arrayed two-column proofs. Sometimes I copied the diagram, sometimes not; but it was part of my graphic honor that all my signs and writing be clear and readable, as if I were submitting these exercises to some ethereal but persnickety high school teacher. But there was no teacher, only the book and me. Nevertheless, I didn't even abbreviate; I wrote 'parallelogram' rather than 'p-gram;' 'perpendicular' rather than 'perp.' It was a precision for the sake of precision alone, rather like making the covered seams of a garment perfect, or finishing the walls inside hidden spaces the tenant will never see.
Much of the language and logic of this elementary geometry is based on congruence. Two triangles can't be the same, they are inescapably two, occupying different spaces, even in the virtual world of the mind. Thus they are congruent rather than equal. The philosophical design lines had been drawn in the sand more than 2000 years ago. The laws of congruence were the beams that held most of the two-column proofs together in a straightforward classical architecture. I must have 'congrued' hundreds and hundreds of angles, segments, and whole triangles. Vertical angles, made from two lines crossing, were particularly satisfying.
Geometry was of course the basis of architecture, which I have been drawing professionally for more than 15 years. It was fascinating to see the walls and windows and roof-pitches of my commercial work turned into pure abstractions. Sometimes, though, the book would feature well-drawn little scenes of houses and surveyors and ships and engineering projects, to show us how geometry worked in the real world. I had been doing it all along, with my verticals, horizontals, rectangles and perspectives.
There was something physical and kinetic about working with geometric forms. I could almost feel the sharpness of angles in my hands (no doubt due to the feel of the sharp plastic of my draftsman's triangle). But I could also feel the stability of triangles as well as the shear of parallelograms and the solemn rotation of regular polygons. When I worked through the section on the geometry of circles, I actually felt dizzy, as if I were going around in circles! Lines and rays also had their impression of motion, as if they were the trajectories of birds or bullets. In a sense, geometry was like playing a sport, especially baseball, which is the most geometric of all sports.
As I got toward the end of Professor O'Daffer's book, I re-encountered solid geometry, which brought back memories of my childhood schoolroom's perfect, fragile wooden blocks illustrating the different basic shapes. These blocks, predictably, had been abused, broken, or stolen by the heedless students. But here in the cloister of Geometry, the shapes re-assumed their perfection. Inside a drawing of a three-dimensional object on a two-dimensional page is a space that is clearer than the purest water or finest glass, because it is a space of the mind. There, inside the cubes and prisms and pyramids and cylinders, was a frictionless dimension where problems had solutions and nothing broke.
Yet also inside these perfect shapes is an infinity of irrational numbers, square roots of two and three and especially that chaotic creature Pi. I asked an impertinent question: if God is perfect, then why did He create so many irrational numbers? If a circle is the symbol of perfection, why is its circumference related to its diameter, and its area to its radius, by such an irredeemably irrational number? Would not a trinitarian God make Pi a perfect three? These are questions that bemuse mathematicians and annoy philosophers, or vice versa. Somehow, this world exists because of all these endless, messy irrational numbers, not only the perfection of the rational numbers.
It took me almost a year to go through my geometry book. I may be the only person who has done every exercise in the book (omitting those which needed schoolroom playthings which I didn't have). At times I got tired and frustrated with the endless sequences of congruence; I felt as though I were cutting the lawn one angular blade at a time. Finally, one of my friendly mathematicians offered that perhaps I didn't have to write out every single proof perfectly, that if some proofs were similar, it was all right to abbreviate and note them in shorthand rather than lay out the numbers and the columns over and over again. A radical notion, that, but it helped.
At the end of the book, in the summer of 2003, I re-visited algebraic geometry for at least the third time since I had begun this mathematical journey. Relating all those equations to geometric forms made them more learnable and memorable. But all these memories are fugitive, needing review to keep them solid and clear.
Yet my cherished goal of studying physics is still receding from me. It seems I just don't know how to get started. The electrons inside my mind are impatient: when will you learn about us? I need more math, always more math. I am still only in the equivalent of high school. But when will I do physics?
The circle spins, and the angles tell each other's story: my next subject is trigonometry, without which no physics can be done.