I want to begin with a confession. I love projective geometry! I've studied it for over 30 years and I still can't get enough. I'm convinced that this modern geometry (discovered in the Renaissance and re-discovered in the Romantic era), has much to offer we humans as we evolve to higher levels of understanding of ourselves and the world we live in. I'll begin by introducing the subject in a way which may not seem like mathematics (no equations, no variables, no algebra) to some readers. Once I've established the key ideas I'll turn to some themes which hopefully will help the reader understand my enthusiasm for the subject, by connecting it to larger issues in science, society, and human development.
There are in fact some very good web sites devoted to projective geometry and its potential significance for the human future. For example, Nick Thomas's projective geometry site is one such. It gives an overview of projective geometry and how it has begun to be applied to scientific research, using abundant illustrations and non-technical language.
This blog will represent my particular perspective on projective geometry. For example, one of my special interests is creating interactive software for all kinds of geometry. I'd like to use this blog to make available interactive software which I've written over the years for exploring themes in projective geometry. I'd also like to present in understandable form some ideas which form part of my Ph. D. thesis (TU-Berlin, 2011).
For this beginning post, I'd like to close with a couple of examples which give a flavor of the kind of phenomena one meets in projective geometry.
One of the fundamental theorems of projective geometry is Desargues Theorem, which concerns the relationship of two triangles. It states that if the joining lines of corresponding vertices of the two triangles meet in a point, then the intersection points of corresponding sides (considered as infinite lines!) lie on a line. And vice-versa! This interactive applet allows you to play around with this theorem. Pay especial attention to what happens as pairs of lines become parallel. In projective geometry such pairs still have an intersection point, allowing the fluid motion to continue undisturbed.
A second famous theorem of projective geometry is Pascal's Theorem. It begins with 6 points A, B, C, D, E, and F on a conic section. Consider the six (infinite!) joining lines of adjacent points AB, BC, etc. These six lines are arranged in pairs of opposite lines, for example, AB and DE, BC and EF, and CD and FA. Then the theorem asserts that the intersection points of these three pairs of lines lie on a line. This interactive application allows you to explore this configuration.
Note: in this figure point B has a distinguished role: it cannot be moved by the user. In fact, the five other points determine a conic section, and B is constructed from these five points using Pascal's Theorem. Also, by moving the other points one obtains a wide variety of conic sections, including ellipses and hyperbolas, but also parabolas, even a pair of straight lines can be obtained.
Before proceeding: please play with these apps! If they don't work, let me know (firstname.lastname@example.org). Hands-on experience is invaluable in developing a relationship to this geometry.
With a little experience, I think you'll agree that both of these theorems are "different" from the geometry you learned in school. In fact, they illustrate a fundamental quality of projective geometry: the geometric phenomena are much more dynamic and flexible than in ordinary "school" geometry. We can simply note how many different configurations one can arrive at by moving one or the other of the special points of the configurations. Later perhaps we can consider why that is.
This quality of projective geometry is related to its genesis in the birth of perspective painting in 15th century Italy. The human being at this time learned to see the world in a new way, and projective geometry in this sense is the mathematics of this seeing. "School" geometry, more accurately known as euclidean geometry for its great expositor Euclid, can be thought of as the mathematics of touch. Many of the paradoxes and peculiarities of projective geometry can be grasped in terms of this tension between these two fundamental human senses. And the relative "strangeness" of projective geometry can be understood as an expression of its relative youth in comparison to euclidean geometry, rather than any intrinsic deficiency.