Projective Geometries

Action at a Distance

Now ths statement might seem counterproductive to the ideas of projective geometry but please bear with me.

In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often with early theories of gravity and electromagnetism to describe how an object could "know" the mass (in the case of gravity) or charge (in electromagnetism) of another distant object.

According to Albert Einstein's theory of special relativity, instantaneous action-at-a-distance was seen to violate the relativistic upper limit on speed of propagation of information. If one of the interacting objects were suddenly displaced from its position, the other object would feel its influence instantaneously, meaning information had been transmitted faster than the speed of light.

Test of the Quantenteleportation over long distances in the duct system of Vienna Working group Quantity of experiment and the Foundations OF Physics Professor Anton Zeilinger

Quantum physics questions the classical physical conception of the world and also the everyday life understanding, which is based on our experiences, in principle. In addition, the experimental results lead to new future technologies, which a revolutionizing of communication and computer technologies, how we know them, promise.

In order to exhaust this technical innovation potential, the project "Quantenteleportation was brought over long distances" in a co-operation between WKA and the working group by Professor Anton Zeilinger into being. In this experiment photons in the duct system "are teleportiert" of Vienna, i.e. transferred, the characteristics of a photon to another, removed far. First results are to be expected in the late summer 2002.

One of the first indications to me came as I looked at the history in regards to Klein's Ordering of Geometries. Now I must admit as a layman I am very green at this understanding but having jumped ahead in terms of the physics involved, its seems things have been formulating in my head, all the while, this underatnding in terms of this "order" has been lacking.

In Euclidean geometry, the basic notions are distances and angles. The transformations that preserve distances and angles are precisely the rigid motions. Effectively, Klein's idea is to reverse this argument, take the group of rigid motions as the basic object, and deduce the geometry. So a legitimate geometric concept, in Euclidean geometry, is anything that remains unchanged after a rigid motion. Right-angled triangle, for example, is such a concept; but horizontal is not, because lines can be tilted by rigid motions. Euclid's obsession with congruent triangles as a method of proof now becomes transparent, for triangles are congruent precisely when one can be placed on top of the other by a rigid motion. Euclid used them to play the same role as the transformations favored by Klein.

In projective geometry, the permitted transformations are projections. Projections don't preserve distances, so distances are not a valid conception projective geometry. Elliptical is, however, because any projection of an ellipse is another ellipse.

So spelt out here is one way in which this progression becomes embedded within this hisotry of geometry, while advancing in relation to this association I was somewhat lifted to question about Spooky action at a distance. WEll if such projective phase was ever considered then how would distance be irrelevant(this sets up the idea then of probabilistic pathways and Yong's expeirment)? There had to be some mechanism already there tht had not been considered? Well indeed GHZ entanglement issues are really alive now and such communication networks already in the making. this connection raised somewhat of a issue with me until I saw the the phrase of Penrose, about a "New Quantum View"? Okay we know these things work very well why would we need such a statement, so I had better give the frame that help orientate my perspective and lead to the undertanding of spin.



Now anywhere along the line anyone can stop such erudication, so that these ideas that I am espousing do not mislead. It's basis is a geometry and why this is important is the "hidden part of dirac's mathematics" that visionization was excelled too. It is strange that he would not reveal these things, all the while building our understanding of the quantum mechanical nature of reality. Along side of and leading indications of GR, why would not similar methods be invoked as they were by Einstein? A reistance to methodology and insightfulness to hold to a way of doing things that challenegd Dirac and cuased sleepless nights?



Have a look at previous panel to this one.

While indeed this blog entry open with advancements in the Test in Vienna, one had to understadn this developing view from inception and by looking at Penrose this sparked quite a advancement in where we are headed and how we are looking at current days issues. Smolin and others hod to the understnding f valuation thta is expeirmentally driven and it is not to far off to se ehosuch measure sare asked fro in how we ascertain early universe, happening with Glast determinations.

Quantum Cryptography

Again if I fast forward here, to idealization in regards to quantum computational ideas, what value could have been assigned to photon A and B, that if such entanglement states recognize the position of one, that it would immediately adjust in B?

Spooky At any Speed

If a pair of fundamental particles is entangled, measuring an attribute of one particle, such as spin, can affect the second particle, no matter how far away. Entanglement can even exist between two separate properties of a single particle, such as spin and momentum. In principle, single particles or pairs can be entangled via any combination of their quantum properties. And the strength of the quantum link can vary from partial to complete. Researchers are just beginning to understand how entanglement meshes with the theory of relativity. They have learned that the degree of entanglement between spin and momentum in a single particle can be affected by changing its speed ("boosting" it into a new reference frame) but weren't sure what would happen with two particles.

So there is this "distance measure" here that has raised a quandry in my mind about how such a projective geometry could have superceded the idea of "spooky things" and the issues Einstein held too.

So without understanding completely I made a quantum leap into the idealization in regards to "logic gates" as issues relevant to John Venn and introduced the idea around a "relative issues" held in my mind to psychological methods initiated by such entanglement states.

As far a one sees here this issue has burnt a hole in what could have transpired within any of us that what is held in mind, ideas about geomtires floated willy Nilly about. How would such "interactive states" have been revealled in outer coverings.

The Perfect Fluid

Again I am fstforwding here to help portray question insights that had been most troubling to me. If suych supersymmetrical idealizations arose as to the source and beginning of existance how shall such views implement this beginning point?

So it was not to unlikely, that my mind engaged further problems with such a view that symmetry breaking wouldhad tohave signalled divergence from sucha state of fluid that my mind encapsulated and developed the bubble views and further idealizations, about how such things arose from Mother.

What would signal such a thing as "phase transitions" that once gauged to the early universe, and the Planck epoch, would have revealled the developing perspective alongside of photon developement(degrees of freedom) and released information about these early cosmological events.

So I have advance quite proportinately from the title of this Blog entry, and had not even engaged the topological variations that such a leading idea could have advanced in our theoretcical views of Gluonic perceptions using such photonic ideas about what the tragectories might have revealled.

So indeed, I have to be careful here that all the while my concepts are developing and advanced in such leaps, the roads leading to the understanding of the measure here, was true to form and revalled issues about things unseen to our eyes.

It held visionistic qualities to geometric phases that those who had not ventured in to such entanglement states would have never made sense of a "measure in the making." It has it's limitation, though and why such departures need to be considered were also part of my question about what had to come next.

Science and the Mind: Sir Roger Penrose

Above picture, belongs to this article and titled above, of frames that Sir Roger Penrose wrote in 1999.

Roger Penrose, a professor of mathematics at the University of Oxford in England, pursues an active interest in recreational math which he shared with his father. While most of his work pertains to relativity theory and quantum physics, he is fascinated with a field of geometry known as tessellation, the covering of a surface with tiles of prescribed shapes.

Being reminded of Roger Penrose I am actually going to contribute this blog entry to him, and sources that I had collected.

Twistor Theory


The motivation and one of the initial aims of twistor theory is to provide an adequate formalism for the union of quantum theory and general relativity. Twistors are essentially complex objects, like wavefunctions in quantum mechanics, as well as endowed with holomorphic and algebraic structure sufficient to encode space-time points. In this sense twistor space can be considered more primitive than the space-time itself and indeed provides a background against which space-time could be meaningfully quantised.

Twistor Program

http://twistor-theory.rdegraaf.nl/index.asp?sND_ID=436182

R. Penrose and M. A. H. MacCallum, Phys. Reports. 6C (1972) p. 241

pages:

242-243,244-245,246-247,248-249,250-251,252-253,254-255,256-257,258-259,260-261,262-263,264-265,266-265,268-269,270-271,272-273,274-275,276-277,278-279,280-281,282-283,284-285,286-287,288-289,290-291,292-293,294-295,296-297,298-299,300-301,302-303,304-305,306-307,308-309,310-311,312-313,314-315,316-317,318-320,320-322,322-324,324-326,326-328,328-330,330-332

or download the unix tar-ball to get all the pages at once. (WinZip is able to unzip this archive)

Sir Roger Penrose

Science and the Mind

Einstein's Equation and Twistor Theory

Gravitationally Induced Quantum State Reduction

Quantum State reduction as a real phenomenon

Schrödinger's Cat in Space

Fedja Hadrovich
In the past 30 years a lot of work has been done on developing twistor theory. Its creator, Roger Penrose, was first led to the concept of twistors in his investigation of the structure of spacetime and it was he who first saw the wide range of applications for this new mathematical construct. Yet 30 years later, twistors remain relatively unknown even in the mathematical physics community. The reason for this may be the air of mystery that seems to surround the subject even though it provides a very elegant formalism for both general relativity and quantum theory. These notes are based on a graduate lecture course given by R. Penrose in Mathematical Institute, Oxford, in 1997 and should give a brief introduction to the basic definitions. Let us begin with the building blocks: spinors.

R. Penrose, F. Hadrovich
Twistor Theory

The motivation and one of the initial aims of twistor theory is to provide an adequate formalism for the union of quantum theory and general relativity. Twistors are essentially complex objects, like wavefunctions in quantum mechanics, as well as endowed with holomorphic and algebraic structure sufficient to encode space-time points. In this sense twistor space can be considered more primitive than the space-time itself and indeed provides a background against which space-time could be meaningfully quantised.

Lecture I
Lecture II
Lecture III

Rapping Our Way to Einstein

Now you know us platoists and our academy had issues about music. The influence it has in soothing it's way into the minds heart of hearts. So I thought, why not, make an exception here, because I like the example associated of "the pretty girl and the hot stove", taken to a new level of conceptualization in society.

A Chladni plate consist of a flat sheet of metal, usually circular or square, mounted on a central stalk to a sturdy base. When the plate is oscillating in a particular mode of vibration, the nodes and antinodes set up form a complex but symmetrical pattern over its surface. The positions of these nodes and antinodes can be seen by sprinkling sand upon the plates;

Although, like the segregation of a racial divide and people, I do not like the innuendo's of association, by gender either. I thought the feelings here, were more important and explained things in a way that help to define the "emotive suffering" and "mental fleetingness of happiness," in ways that we are not accustomed too.

For me this is a philosophical difference, much as one would see the value of music as analogy of the field and the allowable concept of penetration(holes), deep into consciousness. It would not be unfair to see mental abstractions of mathematics held in this light, devoid of any "emotional connotation?"

DJ Vader told BBC Radio 4's Today programme he had not thought much of physics at school and had spent most of his time playing truant.

But he had been inspired by Einstein's explanation of his theory, which reads: "When a man sits with a pretty girl for an hour it seems like a minute.

"But let him sit on a hot stove for a minute and its longer than any hour. That's relativity."

DJ Vader said: "At the time I was thinking of me and my girlfriend being together and definitely it does go really quickly."

Rap the key to explain relativity

Trademarks of the Geometer II

John g,



Lubos had some claim about Martian ancestry, but we know that he jests?:)

So I do not want to use up to much more of Lubos's blog for this conversation even though he pushes the envelope. Perhaps, you will start your own blog?

Genuis at Work
(Picture credit: AIP Emilio Sergè Visual Archives)

Lastly, I have know certain "trademarks of people" like Dirac as "the geometer" is inherent at the foundations of such psychologies(even I like to dabble in model developement ex: John Venn), with current information Peter Woit brought forward, are key indicators to me of visualization capabilties that are every advanced for this abstract world. Clifford demonstrates like a Rorshach Ink blot as an experiment, with the picture on that "blackboard"?

Wassily Kadinsky

His art and in composition? As a reference made in the comment section of another artist in realtion to Clifford's article, Wassily came to mind.

The term "Composition" can imply a metaphor with music. Kandinsky was fascinated by music's emotional power. Because music expresses itself through sound and time, it allows the listener a freedom of imagination, interpretation, and emotional response that is not based on the literal or the descriptive, but rather on the abstract quality that painting, still dependent on representing the visible world, could not provide.

How would it be possible to extend let's say the idealization to a history of geometries without establishing this basis in thought? There had to be expanded frontiers that would let people develope towards objective goals in science, based on science and herein lies the difficulites with the INKBLOt. As by subjective interpretaion based on current knowledge bases, these views would be very much different then what someone "well trained might see"? Let alone, classify it to any geometric formulation.



Surely inkblot below is a mask? I have one in relation, drawn from the antiquities of evolution. If you ever visit the Drumheller museum, in Alberta Canada, you'll identify it for sure?:) So what is this "projection" based on?



Keep it simple

I like to keep it simple, and fragmentary indications of my blog entries can be accumulative of something deeper and very revealing about such a nature of these geometers I like to talk about. I had to learn this history in order to understand where we had been taken with Einstein's General Relativity. Another one, who understood after Grossman that such geoemters were needed to bring consistancy to the undertanding of theoretical developement.

I would not have gotten this far without bloggists, like Lubos, Peter, Sean, Clifford, Mark and the rest of the Cosmic Variance group, who are most kind in helping us lay people to recognize issues in ways and helping to develope info according to the academic world. This has been truly a grace.

Entries of my own, would have past as incoherent states of unfamiliar words, on a very simple level dealing with the societal world we live in. I now find comfort, that I am not so strange, in this geometer sense.

Have I excelled myself? On the contrary, its about learning about ourselves and who we are, is all. If it past the stage of pure mathematics( towards that center), then why would we not see that this outward development had some psychological model in which to adorn oneself in this mandalic sense.

Sean, makes brief link entree in that blog of Cliffords on Cosmic Variance.

Indeed, this is where such models helped me understand from a Jungian sense, that such a map had to exist, and models built. This can only come from experience, and from the direction of coming from that center. Why I ask Lubos, or anyone for that matter, about where ideas come from. Here you would see such a flavour and distinction in Plato's ideology, about what could manifest in any mind, and not just any one select part of this society.

No doubt, that like any fisherman's hook, you would need to have some valuation and inclination to manifest. As you develope through any model apprehension, where you could add more ideas to the pot. For a further invitation for probilities to manifest in our everyday conversations. Are some of these "inductions and deductions" always right? Of course not, and this is where our education comes in, and the saving grace of bloggists in general.

Who would of thought by using "internet world" the bloggists could have ever reached the "periphery" of this society? I'll intoduce you to another foreigner whose concept defintiely challenges the mind in this bubble sense. In a way I helped him to develope further, and him, I.

The Succession of thinking

How far indeed the the imagination can be taken to see such processes enveloped in how we percieve these changes all around us. Why is gravity so weak, here and now. I have jumped ahead but will lead into it from the other end of this article.

Never before had I encountered the reasoning of imaging behind the work of "conceptual frameworks" now in evidence. In how a mathmatician, or a scientist, like Einstein or Dirac, had some basis at which the design, of all that we endure, would have its's counterpart in this reality as substantial recognition of what must be done.

I don't think anyone now in the scientific arena needs to be reminded about what it takes to bring theory into the framework of cultural and societal developement, to see how it all actually is working. On and on now, I see this reverberating from Lisa Randall to all scientists that we encounter from one blog to the next, a recognition and developement of this visualization ability.

That Famous Equation and You , By BRIAN GREENE Op-Ed Contributor in New York Times, Published: September 30, 2005


Brian Greene:

After E = mc², scientists realized that this reasoning, however sensible it once seemed, was deeply flawed. Mass and energy are not distinct. They are the same basic stuff packaged in forms that make them appear different. Just as solid ice can melt into liquid water, Einstein showed, mass is a frozen form of energy that can be converted into the more familiar energy of motion. The amount of energy (E) produced by the conversion is given by his formula: multiply the amount of mass converted (m) by the speed of light squared (c²). Since the speed of light is a few hundred million meters per second (fast enough to travel around the earth seven times in a single second), c² , in these familiar units, is a huge number, about 100,000,000,000,000,000.

There are two links here.One by Peter Woit with reference to article and one toSean Carroll who further illucidates the article by Brian Greene.

So here I am at the other end of this referenced article, that other thoughts make their way into my mind. Previous discussison ongoing and halted. To todays references continued from all that we had encountered in what General Relativity surmizes.

That this issue about gravity is very real. So that's our journey then, is to understand how we would percieve the strength and weakness through out the spacetime and unification of a 3 dimension space and one of time, to some tangible reality within this coordinated frame Euclidean defined.

The Succession of Thinking

Mark helps us see in a way we might not of considered before.

Dark Matter and Extra-dimensional Modifications of Gravity

But the issue is much more complicated then first realized if we take this succension of thinking beyond the carefuly plotted course Einstein gave us all to consider.

Plato on Sep 27th, 2005 at 10:23 pm We were given some indications on this site about the state of affairs with Adelberger. Do you think this time span of proposed validation processes, were constructively and experimentally handled appropriately through it’s inception? As scientists would like to have seen all such processes handled in this respect?

So indeed I began to see this space as very much alive with energy that had be extended from it's original design to events that pass through all of creation, then how indeed could two views be established in our thiniking, to have Greene explain to us, that the world holds a much more percpetable view about what is not so understood in reality.

An Energy of Empty Space?

Einstein was the first person to realize that empty space is not nothingness. Space has amazing properties, many of which are just beginning to be understood. The first property of space that Einstein discovered is that more space can actually come into existence. Einstein's gravity theory makes a second prediction: "empty space" can have its own energy. This energy would not be diluted as space expands, because it is a property of space itself; as more space came into existence, more of this energy-of-space would come into existence as well. As a result, this form of energy would cause the universe to expand faster and faster as time passes. Unfortunately, no one understands why space should contain the observed amount of energy and not, say, much more or much less.

All the while the ideas that would leave gravity without explanation in a flat euclidean space, gravity would have been left to that solid response without further expalnatin in a weak field manifestation. But it was always much more then this I think.

While being caution once on what the quantum harmonic oscillator is not, Smolin did not remove my thinking of what was all pervasive from what this "empty space" might have implied, that heretofor "it's strength" was a measure then of a bulk, and what better way in which to see this measure?

Taken in context of this succession, this place where such conceptual framework had been taken too, it was very difficult not to encounter new ways in which to understand how gravity could changed our perceptions.

Thalean views were much more then just issues about water and all her dynamical explanations. It presented a new world in which to percieve dynamical issues about which, straight line thinking could no longer endure. A new image of earth in all it's wander, no less then Greene's analysis to how this famous equation becomes evident in our everyday world. It presented a case for new geometries to emerge. Viable and strengthened resolve to work in abstract spaces that before were never the vsion of men and women who left earth. Yet it all had it's place to endure in this succession that we now have adbvanced our culture in ways that one would not have thought possible from just scientific leanings.

So now I return myself to Einstein's allegorical talk on what concept had taken, when a scientist had wondered on the valuation of time.

On mathematics, imagination & the beauty of numbers

It's always nice to see this kind of infomration, because indeed if one were to start later on in life, then why not learn new things like mathematics. Especially, if it seems to be thta there is some consistancy in thought about geometry, that had been taken hold of, and leads the thinking mind capable.

Dialogue between Barry Mazur & Peter Pesic
Barry Mazur:

I can’t answer that question, but I can offer some comments. A person’s ½rst steps in his or her mathematical development are exceedingly important. Early education deserves our efforts and ingenuity. But also here is a message to any older person who has never given a thought to mathematics or science during their school days or afterwards: You may be ready to start. Starting can be intellectually thrilling, and there are quite a few old classics written in just the right style to accompany you as you begin to take your ½rst steps in mathematics. I’m thinking, for example, of the old T. C.Mits series, or Tobias Dantizg’s wonderful Number: The Language of Science, or Lancelot Hogben’s Mathematics for the Millions. Moreover, one should not be dismayed that there are many steps– there is no need to take them all. Just enjoy each one you do take

.

Make it Fun

Like Alice in Wonderland or views on the Looking Glast, it was not to hard to figure out that mathematicians like to tell stories too. Bring the latest together in a way that the layman can accept at the level societal minds do. It is interesting indeed in facing the strange and wonderful world of scientists and mathematicians in their abstract mood.

The New World of Mr. Tompkins, by George Gamow and Russell Stannard, Cambridge, ISBN 0 521 63009 6
After sending the piece to several large circulation magazines and receiving impersonal rejection slips, Gamow put it to one side until his physicist friend, Sir Charles Darwin (the grandson of the author of The Origin of Species), suggested sending it to C P Snow, then the editor of Discovery magazine, published by Cambridge University Press. The text was immediately accepted and the discerning Snow demanded more.
Mr Tompkins tries valiantly to follow dry science lectures, but easily falls asleep. However, all becomes clear in his vivid dreams. Soon the articles were collected into Mr Tompkins in Wonderland, published in 1940, followed by Mr Tompkins Explores the Atom in 1944. Each was a major success and the two volumes were reissued with additional material as a single volume in 1965. This reissue alone was reprinted some 20 times.

Plectics

Murray Gellman:

The name that I propose for our subject is "plectics," derived, like mathematics, ethics, politics, economics, and so on, from the Greek. Since plektos with no prefix comes from *plek- , but without any commitment to the notion of "once" as in "simple" or to the notion of "together" as in "complex," the derived word "plectics" can cover both simplicity and complexity.

It is appropriate that plectics refers to entanglement or the lack thereof, since entanglement is a key feature of the way complexity arises out of simplicity, making our subject worth studying.

Dirac's Hidden Geometries

I find this interesting because I like to visualizze as much as possible, and I sometimes think the basis of the leading ideas in science would had to follow a progression. Klein's Ordering of Geometries was one such road that seem to make sense. The basis of relativity lead through in geometrical principals?

Such an issue with string theory had to have such a basis with it as well, although how do you assign any views to the very begininngs of the universe below planck length? Well there are images to contend with what are these and how are they derived? Rotations held in context of te progression of this universe and all thoughts held to the very nature of particle creation and degrees fo freedom?

[PAUL DIRAC]

When one is doing mathematical work, there are essentially two different ways of
thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.

While I am very far from being the mathematician, I understand that this basis is very important. Such summations in mathmatical design, leave a flavour, for conceptiual ideas to form in images, so I understand this as well. It is a progression of sorts I think, as I read, and learn. Geometry lies at the very basis of all such progressions in science?

So Feynmans toy models arose from the ideas of Dirac?

Professor Shiing-Shen Chern

HOUSTON JOURNAL OF MATHEMATICS
Editorial, Electronic Edition Vol. 28, No. 2, 2002

Shiing-Shen Chern -- famed mathematicianCarrie Sturrock, Chronicle Staff Writer
Thursday, December 9, 2004

He was a great mathematician partly because of the quality of his research as well as his ability to convey it to other people, UC Berkeley mathematics Professor Robin Hartshorne said. Professor Chern turned the once- dormant field of differential geometry, which deals with the mathematical description of geometric figures, into a lively field of study. He had the greatest impact on global differential geometry and complex algebraic geometry, which are fundamental to many areas of mathematics and theoretical physics.

Big Ideas.....To String Theory

Plato said:

yes to Gauss and gaussian coordinates, not forgetting, Saccheri, Bolyai and Lobschevasky along this lineage of geometers

On the Hypotheses which lie at the Bases of Geometry

So A continuation from this, and reference to important papers for consideration.

I was actually looking for papers on S.S.Chern and I have been having difficulty tracking down one of his papers entitled,"Relativity and Post Reimannian Differential Geometry," published in 1980. As I look, I usually come across interesting sites for consideration. They do indeed lead from one spot to another willy-nilly.

So I thought I would show the transition to topics that I compiled for reference in relation to string theory.

So having gone through a list here as follows, I came upon the article from a site called "Big Ideas". It was nice then, that I link to the site in question and the article for consideration. Talk about getting off the beaten path.:)

My intentions was to see how Gauss's and S.S. Chern's work correlated together and developed in line with Reimann. Hence the paper in question I was looking for. If anything had change my perspective, Gauss and Reimann were instrumental here and the understading of the metric. Gaussian coordinates help united much for me into the picture General Relativity had taken me too in see the dynamcial nature of the graviational field.

Big Ideas



It is of course from 2003, but always interesting nonetheless.




I understand Clifford's hesistancy on articles that have come out and some trepidation also seen by P.P. Cook on the issue Horizon of Hawkings in his article here. My focused is well set to this horizon as well, as th e question of blackhole types etc, and how such theoretical positions arise fromthis horizon. This was important to me that I move to the understanding of conformal ideas from tha horizon.

But articles, as best they can, hopefully can bring the lay person up to speed on what these ladies and gentlemen are doing with string theory and such. They help me in the generalized direction, so I hope all things are not to lost for Clifford and Paul in their entrancement of observation. "Disgust" to something fine in the media of consideration.

I noticed Paul's link to Jan Troost's site and have seen that site develope from inception, so it was interesting to continue to see the summation of string theory on his site as well. There are really good sites out there that I have kept track of, to help orientate my thinking in regards to to theoretical thinking at it's finest.