Space and Counterspace II

 The previous post introduced the dual euclidean plane and compared it to the euclidean plane on the basis of a hexagonal grid pattern.

In this post we'll continue with this example, but bring it into motion.

As you know there are two kinds of euclidean motions in EP: translations and rotations. We're going to explore subjecting the pattern to a euclidean translation and see how the corresponding dual translation moves the dual pattern.

A translation is an isometry that fixes an ideal point and moves all other euclidean points in the direction perpendicular to this fixed ideal point. The dual translation will fix the corresponding dual ideal line, perpendicular to the ideal point of the translation. Here's what we see when the translation is in the x-direction:

The hexagonal figure has been translated to the right so that the ideal point of the DEP now lies halfway towards the cyan point on the left boundary. The dual pattern of lines has undergone a dramatic shift and appears very unsymmetric. It is crowded into the left hand side and stretched out towards the right.  How can we understand this?

Remember that the star point (0,0,1) maps under duality to the ideal line [0,0,1]. And that the dual of a point appears on the opposite side of the origin as the point.  Since the hexagon grid is off-center and most of it is now to the right of the origin, most of the dual pattern will be seen on the left side; on the right there is only one vertical line between the cyan line on the right and the ideal line. 

If we continue to move the pattern to the right, eventually the star point will lie outside the grid pattern:

The whole pattern of dots is now to the right of the origin (and star point) so that the whole of the line pattern will lie to the left. 

In fact we won't show it here but the six boundary lines of the dual pattern viewed with euclidean eyes always lie on a conic section; at the above stage that is, not surprisingly, a hyperbola. But perhaps stranger is that viewed with dual euclidean eye, they remain a circle throughout!  Just as the grid of points moves rigidly in the euclidean plane when it's translated, the grid of lines viewed in the dual euclidean plane remain rigidly connected to each other.  The proof is trivial: the calculations involved in confirming this fact for the euclidean pattern are identical to those in the dual euclidean case; the only difference is that the results are interpreted once in EP and the other occasion in DEP.

Space and Counterspace I

The principle of duality in projective geometry creates a partner for any configuration or statement.  It is based on a dictionary of duality that allows any description to be translated into its dual partner.

For 2D geometry, this dictionary starts with the entries:

• point             line
• join               meet
• lies on           passes through
• move along.  rotate around

So for example, "Two points have a unique joining line" is dual to "Two lines have a unique meeting point." (Only true in PG since parallel lines meet in ideal points). The dictionary is symmetric, so that you can look up a word in either column and replace it with its partner.  Words that don't appear in the dictionary are left alone. The above dictionary applies to projective geometry in general. These blog posts [xxx] deal with this basic version of duality.

In this post we want to explore a special extension of this dictionary that allows us to dualize euclidean geometry to produce dual euclidean geometry. Restricted to 2D this leads to the euclidean plane (EP) and the dual euclidean plane (DEP).   We then begin to explore and compare these two partner geometries.

This blog isn't the place to give the details of this construction.  We can give some hints however.  The euclidean plane is derived from the projective plane by identifying a special line called the ideal line (sometimes called the line at infinity by careless speakers).  The points of this line are ideal points and can be identified with directions or vectors, that form the foundation of euclidean measurement.  In DEP, there is correspondingly a single ideal point, along with all the lines passing through it, the ideal lines of the point, which play an analogous role in dual euclidean measurement.

While the ideal line of the euclidean plane appears to be built into the nature of the world, the ideal point of each DEP can be chosen independently.  

We want to compare EP and DEP on the basis of a simple geometric pattern. The following image shows a regular hexagon containing within it a hexagonal pattern of colored dots centered on the origin of the coordinate system.  We choose the ideal line of the DEP to be this origin.  It's marked by a star icon.  What will the dual partner of this pattern look like in DEP?  The simple answer is (ignoring the six colored segments bounding the hexagon for the moment): each colored point in the central grid corresponds to a line of the same color in the outer figure. For example, the red point on the right side of the hexagon corresponds to the vertical red line on the left hand side close to the cyan vertex. Why: the point coordinate (.9, 0, 1) produces the line coordinate [.9, 0, 1] which corresponds to the line x = -1.11... The dual line appears on the opposite side of the origin from the original point, with a slope perpendicular to the line joining the origin to the given point.  The center point (0,0,1) corresponds to the ideal line z = 0. This equation reflects the fact that ideal points have z-coordinate 0. 

One characteristic feature of the line pattern is that there are many points where several different-colored lines meet.  To find the pendant in the original pattern, we have to look for lines upon which several different-colored points lie.  These are easy to find; the hexagon grid is criss-crossed with such lines -- although the lines themselves haven't been drawn.  For example the point on the left where red, yellow, magenta, and 2 olive green lines meet corresponds to the vertical row of dots running from the yellow dot on the upper right of the hexagon boundary to the red dot on the lower right. 

Space and Counter-space III

In a previous post, I introduced the euclidean plane (EP) and its dual partner, the dual euclidean plane (DEP).  I used the example of a hexagonal pattern of dots in EP that gives rise to a pattern of lines in DEP.  In a further post I showed how applying a translation to the pattern in EP leads to a "dual" translation in DEP that looks unfamiliar to our euclidean eyes.

In this post I want to continue with this line of thought, but replace the hexagonal pattern with another one.  Instead of having the euclidean pattern within and the dual pattern on the outside, the pattern consists of a regular 2D "crystal" of rhombi that extends out from the origin in all directions.  The following figure is scaled to show this pattern; the dual pattern is too small to see in this image.

What does the corresponding dual pattern look like?  Notice that a crystallographic pattern is characterized by a finite number of families of parallel lines. That is, all the lines in the pattern has a certain direction that it has in common with many other lines of the pattern. In this case, there are three different directions, or ideal points, that characterize the pattern of lines. The dual of this is: all the points of the dual pattern lies on one of three ideal lines.  Furthermore,  the spacing between neighboring lines in the same family is (in this case) always the same size; one moves from one line to the next by taking identical euclidean steps. 

So in the dual pattern will consist of a pattern of points, arranged on a set of three ideal lines (lines passing through the star pattern), such that you take equal-sized dual euclidean steps when moving from one point to the next.  The above image shows the dual pattern zoomed up; hopefully it's large enough to see that the points are in fact arranged on 3 lines through the origin. But what about the claim they are separated by equal steps?  By the fact that the colors get darker as they approach the origin, it's clear that the points closest to the ideal point correspond to the lines farthest away from the origin: which makes sense, since in both cases they are "approaching" the ideal. But what about the step size?

To understand that, consider the following line equations: x == i for integer i.  These describe a set of parallel vertical lines with a spacing of 1 unit. The line coordinates are [1,0,-i].  These become the point coordinates (1,0,-i) in the dual plane. To render them, we first note  that for i=0, this is the ideal point in the x-direction. Otherwise, we can dehomogenize them to obtain the equivalent points (-1/i, 0, 1).  These represent the sequence of points with x-coordinates (1,1/2,1/3, ...1/n,...) etc.  These points are equally spaced in the dual euclidean way of measuring! They approach but never reach the ideal point (0,0,1).

I close with a question. The following image shows an x-ray diffraction pattern for common table salt NaCl.  The latter lies on a cubic lattice in 3D.  The x-ray image has similarities to the image we encountered above of the counter-space partner of a regular 2D crystal in EP. Is this more than a coincidence? 

George Adams considered this connection in a short essay "The reciprocal lattice and the X-ray analysis of crystals" in the collection "Universal Forces in Mechanics" (Volume 35 of "Mathematische-Astronomische Blätter", Dornach, 1996, p. 213-224).  In the end he concluded that the X-ray crystallography produces another lattice in ordinary space, not in counter-space.  That should be easy to check, since X-ray crystallography implements a Fourier transform on the original crystal lattice.

UFiM Online Part 1

Due to a very unsatisfactory editing capacities in this blog I have prepared the post using GoogleDocs and it is available here. You can leave comments within that file or if these fail, send around an email:-)

Universal Forces in Mechanics Online

I have a great admiration for the work of George Adams, a student of Rudolf Steiner who made great progress in making the thought-forms of projective geometry accessible to a larger audience and showing how they can be applied to new perspectives in understanding both the outer world of Nature and the inner world of the human being. He is best known for his work on plant morphology (with Olive Whicher) "The Plant Between Sun and Earth" (London, 1952) based on the polarity of space and counter-space.  Less well-known is is later work on the foundations of mechanics, where he applies the same spatial polarity to the polarity of kinematics and dynamics, so often mentioned by Rudolf Steiner in his scientific lectures.

I'm very grateful that I have the chance to present some of this work at an upcoming seminar entitled "Universal Forces in Mechanics" that will take place Feb. 22-24 in Gloucester, England.  The title is taken from a set of essays that George Adams wrote in the late 1950's devoted to this research in re-interpreting rigid body mechanics in the light of projective geometry. An English translation of these essays is available here.

As part of this learning initiative, I am also offering an online course focusing on the mathematical content of the cited essays.  This blog entry provides participants with a "base station" where the various resources can be made available and where feedback can be registered using the comment facility below.

Paul Courtney has provided an English translation of a section of George Adams' "Strahlende Weltgestaltung" (1934) referenced in "Universal Forces in Mechanics" on p. 5. You can access it here. It's devoted to the theme of "projective polarity with respect to a triangle." I recommend reading that first. Thanks, Paul!

Then you might find it interesting to play with an interactive geometry demo  here, devoted to the same theme. You can interact with the figure and inspect the code used to create the figure. You can even edit the code that creates the figure (click on the menu symbol to the left of the drawing canvas and choose "Edit". The code appears below the canvas.) More information about the software environment used can be found at beta.observablehq.com.

Please leave comments and questions below.

• "There are no stupid questions", and,
• "If you're wondering about something there is a good chance the others are either wondering too -- or should be."

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.

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 is some advice to solve them:

1. If you don't have a recent version of Java installed on your computer, you can get the latest version here.
2. For running applets, I recommend using the Mozilla Firefox browsers.  Chrome doesn't like applets at all. 
3. Instead of trying to activate the Webstart links directly, copy the linked file (ends with ".jnlp")  onto your computer and attempt to run it there directly.
1. On a Mac, you cannot double-click on the downloaded jnlp (webstart) files to run them.
2.  Instead activate the right-click context menu.
3.  Then select "Open with... -> Java Web Start.app" menu item.  Otherwise it complains that they are from an "unknown developer" and refuses to run them.
4. In the Java Control Panel (accessible in System Preferences on a Mac and I-don't-know-where on Windows and Linux) you might need to make the following changes:
1. Select the "Security" tab.
2. Add the site "http://page.math.tu-berlin.de/" into the Exception Site List. 
3. Add the site "https://page.math.tu-berlin.de/" into the Exception Site List. 
4. Set the security level to "High" instead of "Very High" (the two choices that remain).
5. When starting up the Webstart, it may complain that the Java version asked for in the .jnlp file (1.5+ or 1.6+) doesn't match the installed one (which is now 1.8.x).  Just click on the option to use the installed version.
6. It may ask whether you are sure you want to run the application. These warnings may have various reasons. You will need to trust me that none of them represent any danger. Possible reasons include:
1. Because the application developer is unknown.
2. Because the certificate will expire soon.
3. Because the application asks for all permissions when it runs on your computer.
7. The loading process may take awhile, especially the first time.  Be patient.
8. As a last resort try clearing out the Java cache. 
1. Go to the Java Control Panel (see above).
2. Select the "General" tab.
3. Under "Temporary Internet Files" click on the "View..." button.
4. Select "Resources" from the ComboBox menu.
5. Select all resources (click in the list and then type ctrl-a), then select click on the big red X button at the upper left of the window.

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.).