Tuesday, November 28, 2017

Renewing spatial concepts via projective geometry

The following thoughts were set in motion by the prospect of my possible participation in upcoming workshops related to projective geometry.  I found myself uncomfortable with providing (in response to invitations to participate) a simple listing of relevant themes. When I paid more attention to this feeling I became aware of a conceptual "context" that clarified my discomfort. At the end of this process I in fact arrived at a listing of relevant themes -- but the sense of this listing is rooted in this context, which I now attempt to sketch.


More and more people are awakening to the limitations of scientific thinking. Despite its impressive achievements in the inorganic realm, it is more and more perceived as the source of global problems such as climate change and social unrest.  Its negative quality is often characterized as "materialistic", meaning that it looks to matter as the underlying source of all real phenomena.  The initial naive view of matter as "stuff filling space" has had to be revised in the last hundred years since the discovery  of quantum mechanics has effectively destroyed the initial sense-perceptible aspect of this definition.  One could say, the "stuff" has evaporated into abstract mathematical formulas, but the underlying concept of space has remained stubbornly fixed.  One reason for this is that no viable alternative has been proposed (curved space-time of relativity is not qualitatively different from euclidean space for the events of daily life).   


According to an ancient tradition in medicine, illnesses are always accompanied by the requisite healing plant.  The skill of the healer is to locate the healing plant that corresponds to the given illness.  In the case of the "space" crisis referred to in the  previous paragraph, it turns out that at the same time that the scientific revolution was being born (out of the work of Kepler, Galileo, Descartes, Newton, and others), a new geometry was discovered (by Desargues, a friend of Descartes) in reaction to the puzzling phenomena presented by perspective painting.  This geometry, which received a second powerful impulse around 1820, is called projective geometry, and this report is based on the conviction that its destiny is to heal the spatial one-sidedness afflicting our modern consciousness.  In a very literal sense, with projective geometry our concept of space can once again become "whole".


As with so many other domains of human life, it was Rudolf Steiner who first identified the healing role of projective geometry in this regard. His numerous references to it in his scientific lectures clearly showed the road leading to a renewal of scientific and, more importantly, of social thinking based on it. He laid particular emphasis in this context on the importance of developing the concept of "negative space" to balance the ordinary "cartesian" concept of space. Two examples: in the light course (the first natural science course, GA 320) he claimed that the phenomena of life required a new type of force, originating in the periphery of space and acting in a planar fashion from without.  He called such forces universal to distinguish them from the "central" forces of classical physics, acting between point centers.  A second example: in the warmth course (second natural science course, GA 321), in a discussion of the different states of matter, he indicates that the solid state corresponds to our familiar picture of objects filled with stuff, while the gas state has to be thought of as a “negative” space obtained by turning the standard space inside-out, a hollow space whose "interior" extends to the periphery; the fluid state is an intermediate one balanced between the solid and gaseous states. Finally, he also repeatedly emphasized how the expansion of natural science concepts in this direction was a prerequisite for a new social thinking.

These indications from Rudolf Steiner regarding the importance of an extended space concept for the renewal of science and society were taken up by a small group of students, notably George Adams and Olive Whicher, Louis Locher-Ernst, and Ernst Lehrs. (see references below).  The work of these pioneers came to an end in the early 1960's.  Since then, with some notable exceptions (e. g., Nick Thomas), progress in extending these results has been modest, despite the increasing severity of the unfortunate consequences of the one-sided spatial thinking.  

In the center, a hexagonal pattern in ordinary space; surrounding that, the same pattern "translated" into negative space.
In this context I consider it important to ask, what can be done to move forward in this important task?  As I look back on my experience with the anthroposophical mathematical community, beginning with my first exposure in 1979 and 1981 through 3-week courses with Olive Whicher and Lawrence Edwards, resp., one aspect jumps out at me.  I call it the education challenge.  What do I mean?


Research is a lot like gardening. The most important ingredient for a successful garden is the quality of the soil.  (In fact, one can say, there are no diseases of plants, there are only diseases of the soil.) In the same way the most important ingredient for successful collaboration is a thorough grounding in the fundamentals of the object of study, in this case, projective geometry. This shared heritage of theory and practice is the soil out of which scientific collaboration grows.  In brief: we know too little about projective geometry!  Although more and more people know something about projective geometry, very few people know enough to engage in research.   For example, we all know the cookbook recipe for the polarity on a conic section: the polar line of a point outside the conic section can be obtained by joining the tangent points of the two tangents from the point to the conic section.  But how many of us know the actual definition of this polar relationship, and can prove that it is unique?  Without this deeper grounding in the fundamentals, one remains a spectator and cannot participate successfully.  This is not in itself a problem, but it can become one if there are too few active participants, or if the presenters themselves lack a solid grounding in projective geometry.  The latter has perhaps interest in an interdisciplinary undertaking; my remarks here are addressed however at strengthening collaboration within the mathematical community.


The situation is exacerbated by the fact that projective geometry is not typically taught in university so that most participants are self-taught.  The result is inevitably a very uneven level of preparation.  And, let's be frank, it's hard to learn thoroughly when there is no one there to answer your questions and to question your answers.  To wrap up: the long-term health of our collaborative efforts depends on upgrading our qualifications in the fundamentals of projective geometry.  


Before discussing possible solutions, I want to mention another symptom that make progress difficult.  In the groups and seminars I have participated in, it is difficult to maintain a continuous thematic impulse or group participation.  Themes come and go, as well as the participants.  One response to this situation is to accept is as normal.  Taken to the extreme, this results in a "self-organizing" format.  Participants themselves suggest topics of interest they are willing to talk about; the job of the organizers is to fit the offerings with the available time slots.  This can sometimes lead to successful serendipity; more often than not, however, my experience has been disappointing. Such meetings may be enjoyable/inspiring/interesting while they are happening but -- in the absence of shared goals and questions -- durable, productive collaborations rarely develop.    


If my hypothesis with respect to the education challenge is correct, then one source of this second symptom can be seen in the poor quality of the soil: too much material is presented that lies beyond the skill level of most listeners, so synergetic interaction fails to materialize, or the material presented is so elementary that it is too far from research quality. So one might hope that improving the quality of the soil by an education initiative might also at the same time mitigate this second symptom.


Another ingredient of successful gardening is choosing a subset of the endless variety of plant life to plant and cultivate.  Also in scientific research a certain pruning of themes is a necessary condition for abundant growth.  These themes would have to be worked out together and not delegated to one or two "experts".  Only then can obtain the long-term commitment for participants necessary for lasting results. These shared themes and questions can then serve as “guiding lights” when organizing meetings and workshops.


Now that I have sketched out the context in which I am working I will turn to a discussion of possible features of such a research program.


  1. Education. Here I think it would be good to strive to offer some regular course (at regular intervals during the year and directly made available as video on-line?) leading to the mastery at the level of, for example, the content of Locher-Ernst's "Projective Geometry".  Integrating it into an on-line course platform would allow for remote learning (discussion forums, homework assignments, etc.)
  2. Counterspace
    • What is counterspace? It would be good to establish an overview of the various versions of counterspace and where they are applicable.  (I can think of 4 off the top of my head and there are certainly others).  Such an overview is long overdue, as people become confused when they notice that the same word is being used to mean different things.
    • Nick Thomas's work on counterspace, as far as I can tell, remains a closed book for most of the community. An overview of the contents of his book by someone who does understand it would be I think greatly appreciated.
  3. Path curves and path curve surfaces
    • Given the primary role of counterspace in George Adams' plant work, it's natural to look for it also in the path curve systems. Nick Thomas wrote a short article to show that the collineation underlying a path curve system can be factored as the product of two null systems (correlations) (in an infinite number of ways) thereby bringing in a counter spatial element.  And the pivot transformation that appears in the investigation of seed pods, etc., is also a correlation. It would be interesting to investigate further to see whether one can find other, deeper connection to counterspace.
    • Fibonacci numbers and golden ratio in plant forms: is there a way to use the approach of "The Plant between Sun and Earth" to understand why the discrete patterns of leaf and flower formation very often reveal the Fibonacci sequence or the related golden ratio?  
    • At the end of this life George Adams investigated the use of path curve surfaces for purifying or enlivening moving water (as part of his involvement in the institute for flow research at Herreschried, Germany).  This work (carried out with George Unger) was published briefly in the MPK but the work came to a stop with his premature death in 1963 and to my knowledge has not been revived (flow forms go in another direction).  It would be good to pick it up again and see if it can be developed further.
    • A purely mathematical question: The analytic/algebraic approach to path curves leads to the theory of Lie groups and Lie algebras (after all, Sophus Lie and Felix Klein discovered path curves). The "infinitesimal generator" of the path curve system is a traceless matrix A; the path curve orbit at time t is then given by the matrix g(t) = exp(At).  (A is an element of the Lie algebra, and g(t) is in the Lie group.) To what extent can this relationship be expressed synthetically? Is there a way to represent the infinitesimal generator geometrically?
  4. Physics
    • The polarity of kinematics and dynamics.  This theme was emphasized repeatedly by RS as essential to a renewed, reality-based physics; he also related it (in the warmth course) to the polarity of mental picture and will in the human being. A related question is "What is a force?". As far as I can tell George Adams "Universal forces in mechanics" is the only subsequent publication to address this in a serious way.  He shows how 19th century projective line geometry provides an elegant formulation of kinematics and rigid body motion where this polarity can be clearly delineated.  Since my Ph. D. thesis builds on this approach, I could present a short account of these results. Also Adams' essay "Forces in space and counterspace" deserves to be included here.
    • If there is interest I am also glad to present an introduction to using projective geometric algebra to represent rigid body mechanics (i. e., a reformulation of "Universal Forces in Mechanics" in modern terminology.)
    • Long-term, I think it is an interesting hypothesis that the paradoxes of quantum mechanics may not be saying anything about physical reality, but are primarily the expression of attempting to model physical reality using a “single space”.  I expect that when one integrates counterspace into the mathematical foundations of the theory, the paradoxes will either disappear or look very different.  For example, the currently trending phenomenon of non-locality (as evidenced in the entangled particles used for quantum computing) might look quite different in a geometric framework where planes are primitive entities along with points.
  5. Astronomy/Cosmology
    • Last year I gave a talk on the fact that the planetary orbits are in fact circles when considered in counterspace (a fact first noted by George Adams in “On etheric spaces”, 1933).  I think that there are other astronomical themes that also might reveal new aspects by incorporating counterspace. For example, the search for dark matter whose gravitational pull is hypothesized to be behind the unexpected expansion of the universe might be more simply explained using the force of "levity" based in counterspace -- a force that naturally pulls towards the periphery of space.
  6. Metamathemical themes
    • Goethe and mathematics.  Despite George Adams' prolific work in applying counterspace to botany, physics, and other fields, his work has not been taken up by our colleagues in the natural science section (to say nothing of mainstream scientists).  I have identified one possible ground for this: the belief that Goethe's scientific method excludes using mathematical terminology. I would be glad to give a talk on why this belief is false -- on the contrary, in projective geometry Goethe's method finds its ideal spatial vocabulary, promising a bridge between mainstream and phenomenological approaches.   
    • "Periphery".  It is inevitable as counterspace becomes more well-known that the polarity of center and periphery will also attract more attention.  For example, current literature discusses the "environment" of the plant as an entity that is more than the sum of the material surroundings. Or, one contrasts the centric nature of DNA with the peripheral nature of life processes themselves. Or, Rudolf Steiner (Bologna, 1913) indicated that the higher ego of the human being lives not in the bodily organization but in the "periphery".  While not a strictly mathematical question, the nature of “peripheral” in these contexts is a fundamental one for applying the mathematics of counterspace in the real world.  Hence it seems to me it is important that we as a group develop a concrete and differentiated sense of what "periphery" means in these sorts of examples if we are to engage credibly with scientists and thinkers not familiar with this usage.  


Concluding thought: Perhaps the best way to develop an appreciation of “peripheral” in the sense of the previous paragraph is immediately available in the quality of own group process.  That is, we can  begin to understand it, to the extent that our work together begins to develop a true “peripheral” nature distinct from a simple sum of individual selves.

Further reading:
George Adams and Olive Whicher, “The Plant Between Sun and Earth”, Rudolf Steiner Press, London, 1980.
Louis Locher-Ernst, “Space and Counterspace: An Introduction to Modern Geometry”, AWSNA, 2003.
Ernst Lehrs, “Man or Matter”, Project Gutenberg (gutenberg.org), 2004.
Nick Thomas, “Space and Counterspace: A New Science of Gravity, Time, and Light”, Floris Books, 2008.



Monday, November 27, 2017

Note on using the interactive Java applications in this blog

Java issues

In the years since I started this blog, security issues regarding Java applets and Java Webstart have led to severe restrictions on their use.  That means it might be difficult to get these applications running in your browser.  If you encounter difficulties here are some ideas for work-arounds:

  1. I recommend using the Mozilla Firefox browsers.  Chrome doesn't like applets at all.
  2. Instead of trying to activate the links in the blog directly, copy the linked file onto your computer and attempt to run it there directly.
  3. On a Mac, you cannot double-click on the downloaded jnlp (webstart) files to run them; instead you have to use the right-click context menu and select "Open with... -> Java webstart" menu item.  Otherwise it complains that they are from an "unknown developer" and refuses to run them.
  4. You may need to add the site "http://page.math.tu-berlin.de/" into the Exception Site List on the Java Control Panel.  This is accessible on a Mac on the System Preferences Panel.
Good luck.  I'll gratefully post any more specific directions you may have discovered for your configuration.


Webstart Directions

Once you do manage to get the webstarts working, you still may have to initialize them properly.  Each has a property file which controls how the initialization takes place, but this file cannot properly be opened by the webstart process, hence here are the manual instructions for getting the app properly configured:
  1. Select the menu item "Window->Left Slot" (only if it's not already selected).  This brings up the application-specific control panel, without which the application is quite crippled.
  2. Deselect the menu item "Window->Right Slot" (only if it's already selected).  This navigation panel is generally not needed for the casual user and takes up valuable real estate.
  3. Select the menu item "Camera->Zoom tool".  Now you can zoom in and out of the 3D window using by scrolling with the mouse (or the equivalent motion on a touch pad, etc.).

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 (cgunn3@gmail.com).  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.