Sunday, October 13, 2013

Me and Projective Geometry

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.

Friday, April 20, 2012

Perfect partnerships

The previous post led to the creation of projective geometry by extending "normal" geometry by an ideal plane filled with ideal points and lines.  In this post I want to explore the consequences of this extension in one particular direction, revealing a startling symmetry permeating projective geometry.

It is characteristic of mathematics that it begins with simple elements and combines them to create compound objects.  In geometry, these simplest elements are points, lines, and planes.  These elements in turn have incidence relations to one another: points can lie on lies, planes can contain or pass through points (or lines), etc.

Consider the pairs of incidence statements from planar Euclidean geometry:
A. Every two points have a unique joining line.
B. Every two lines have a unique intersection point.
The truth of Statement A is immediately obvious.  Statement B, on the other hand, is only "almost always" true:   if the two lines are parallel, then they by definition do not intersect.

It turns out that in planar projective geometry, statement B is also always true, since two parallel lines have an ideal point in common.  In fact, in projective geometry, every true statement has a partner which is also true.  This partner is called its dual, and it is obtained from the original statement by exchanging a set of words and phrases with their dual partners.  

For example, statement B can be obtained from statement A by replacing the blue text: by switching "point" and "line", and by switching "joining" and "intersection". These word-pairs are said to be dual in planar projective geometry. Other such pairs include the incidence properties   "contain" and "lie on" (or "passes through").  This set of words and phrases can be thought of as a dictionary for translating any statement into its dual.  One can also dualize configurations of geometric elements, without regard for the truth content.  For example, "3 points and their joining lines" is dual to "3 lines and their intersection points". Finally, notice that the dual of the dual is the original statement, so the two statements (or configurations) are really like a pair of twins.

It's natural to ask, Why should the principle of duality apply to all of projective geometry?  The simplest way to understand this is to observe that the axioms of projective geometry all exhibit this property: statements A and B are examples of two such axioms.  All the other statements in projective geometry can be derived from the axioms by logical necessity; any statement derived from a set of statements exhibiting duality will also exhibit duality, since I can apply the dictionary of duality to any proof to obtain a valid proof of the dual statement.

Duality for planar (2-dimensional) projective geometry is slightly different than for spatial (3-dimensional) projective geometry, which just means that the dictionary of duality for the two cases is slightly different.   Here we first focus on the 2-dimensional case, then discuss and give  an example of 3D duality.

We focus here on the existence of  perfect partnerships in projective geometry, arising as a result of duality.  Start with a point P and a line m which are not incident. Consider the set of all the lines passing through P (called a line pencil centered at P), and the set of all points lying on m (called the point range of m).  See figure on the left. Then, the perfect partnership is established by associated to every line through P, its intersection point with m; and conversely, every point on m is associated to its joining line with P.  It's clear that this partnership is perfect only because the red line in the figure parallel to m intersects m in an ideal point.  In euclidean geometry the partnership is not perfect.

A perfect partnership of this form (between the elements incident with two simple forms which are themselves not incident) is called a perspectivity.  By chaining together such perspectivities that share a common element, one can construct further partnerships.  For example, chaining together two such perspectivities that share a common line pencil establishes a perfect partnership between the points of the two point ranges.   A future post on this blog will take up this theme further.

In 3D duality, points and planes are dual; lines are self-dual.  To see why this is so, consider the statement:  "Two points have a unique joining line."  The spatial dual reads "Two planes have a unique intersection ____."  Clearly the only reasonable choice for the missing term is line.  Hence, a line is self-dual.  To be precise, "a line and all the points it contains" is dual to "a line and all the planes it lies in."  The former is called a point range (as above); the latter is called a plane pencil.

We close today's post by considering a simple example of 3D duality.

Begin with the cube, one of the five Platonic solids, a regular polyhedron consisting of 6 faces, 8 edges, and 8 vertices (image on left).  In order to simplify the procedure, we simplify by thinking of the faces as infinite planes, and the edges as complete lines.  (Dualizing the finite faces and edges is a more difficult task which we'll postpone for later, see exercise below.)   The dual polyhedron will then have 6  vertices, 12 edges, and 8 faces.  By considering the other Platonic solids, one sees that the octahedron satisfies these conditions (image on right).  In order to make sure that this is really the dual of the cube, attempt to translate descriptions of one solid into their dual form, and see whether they in fact are true.  Only when the two figures are dual in all their detailed incidence properties is one justified in calling them dual partners.
 3 faces and 3 edges meet at each vertex of the cube, and 4 faces and 4 edges meet at each vertex of the octahedron.
translates to:
3 vertices and 3 edges lie on each face of the octahedron, and 4 vertices and 4 edges lie on each face of the cube.
It's simple to verify that the second statement indeed is true.  Let's try something a bit more difficult:
The three joining lines of pairs of opposite vertices of the octahedron intersect in a point, the center point of the octahedron.
This translates as:
The three intersection lines of pairs of opposite planes of the cube lie in a plane, the center plane of the cube.
Verify that the dual statement makes sense: pairs of opposite faces of the cube lie in parallel planes, whose intersection line is therefore an ideal line.  Hence all three lie in the ideal plane.  Furthermore,  duality implies that this plane should be considered the center of the cube!  As Dorothy said, "We're not in Kansas anymore."   Rather than trying to explain how to think about this middle plane, we leave the reader to ponder it.  Those who are interested in further exploration in this direction  are invited to try the following exercise.

Exercises. 1. Devise a reasonable definition to decide when a point is inside the octahedron.  (Hint: start by defining that the center point is inside.) Then dualize this to a definition to decide when a plane is inside the cube.  Extend or generalize this result to dualize the actual faces and edges of the cube (as finite pieces of infinite planes and lines).  What is dual to moving a point along an edge of the cube between the two endpoints of the edge?
2.  We could have begun by defining the center point of the cube as the intersection of the 4 space diagonals of the cube.  What is the dual of this point in the octahedron?
P. S. If you're interested in meeting more dual polyhedra, try out this interactive application for exploring the platonic and archimedean solids and their duals.  A screenshot is shown below.

Thursday, April 19, 2012

Where parallels meet ...

In a previous post, I introduced projective geometry, directly,  by means of two interactive applications demonstrating the quality of projective phenomena.  In this post I want to introduce the basic concepts of projective geometry.  The story begins in renaissance Italy.

 In 1420 Massacio painted a fresco in a chapel in Florence that is considered the first example of perspective painting.   When one attempts to define what "perspective painting" means one is led naturally to the thought constructs that lie at the basis of projective geometry.  The French architect and mathematician  Rene Desargues (1591-1661) is considered the father of projective geometry based on his book Brouillon Projet, which represents the culmination of a gradual coming-into-consciousness of the new thought forms revealed by perspective painting.  This is a fascinating story which lies outside the scope of this post, which restricts itself to  the fundamental connection between the art of perspective painting and the science of projective geometry.

A perspective image of a scene is defined by a process, called central projection, involving the following ingredients:
  • the lines passing through the eye of the painter (called the center of the projection), 
  • the points at which those lines first intersect an object in the world, and finally,
  • a flat screen that is inserted between the eye and the objects of the world.  
The perspective image is created by transferring, for each line through the center, the color of the intersection point with the world to the intersection point with the screen. In this way a colored image is created that reproduces the visual impression of an observer whose eye is positioned at the center.

This interactive application illustrates central projection with some simple scenes.  The center of the projection (the "eye") is on the left, the lines ray out to the checkerboard (the "world") and the image is created on the vertical screen in the middle by transferring the colors from the checkerboard to the screen.

Perspective images contain interesting features.  For example, consider a scene consisting of a strip of constant width leading away from the screen, like parallel train tracks disappearing in the distance.  (Use the '4' key in the application to obtain such a scene.) What is the perspective image of such a scene?  It's easy to see that the image of a line is again a line.  The parallel lines will be mapped to lines which are not parallel.   Indeed, consider the line through the eye which is parallel to the train tracks.  It's not hard to see that where this line intersects the screen will be where the train tracks appear to meet. It's called a vanishing point.   There is one vanishing point for every set of parallel lines.

The step from central projection to projective geometry is a small but significant one.  It occurs when one assumes that the train tracks themselves -- and not just their perspective images -- have a point in common. After all, I do see such a point -- the vanishing point!  Such a point is called an ideal point and they form the foundation of projective geometry.  One could characterize an ideal point as a point which one sees, but which one can never reach.  Here one sees how the tension between sight and touch -- mentioned at the end of the previous post --  is built into the foundation of projective geometry.

The set of ideal points is organized in a nice way.  In every plane, in every direction, there is an ideal point, where all lines having this direction meet.  Taken together, all these ideal points behave just like a line -- it's called the ideal line of the plane.  The horizon line is an example of such an ideal line.  This interactive application allows you to experiment with this concept.  It shows 4 sets of parallel lines in a plane; seen from above (the left image) one experiences the euclidean parallelism; when one rotates the scene away from the viewer (right image) one sees the four ideal points on the horizon line where the sets of parallel lines meet and experiences the horizon line as a real entity.

Finally, all the ideal points of all the planes in space form a plane, the ideal plane of space.  Projective geometry arises when one takes all the ordinary points of space and appends this ideal plane, with all its ideal lines and ideal points.  Further posts on this blog will explore the consequences of this inconspicious extension.

Wednesday, March 21, 2012

Jump in!

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 (  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.