On Gauss's Mountain

You must understand that any corrections necessary are appreciated. The geometrical process spoken too here must be understood in it's historical development to undertand, how one can see differently.

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

These tidbits, would have been evidence as projects predceding as "towers across valleys" amd "between mountain measures," to become what they are today. Allows us to se in ways that we are not used too, had we not learnt of this progression and design that lead from one to another.


8.6 On Gauss's Mountains

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface.

One had to recognize the process that historically proceeded in our overviews "to non-euclidean perspectives," "geometrically enhanced" through to our present day headings, expeirmentallly.

Michelson interferometer(27 Mar 2006 wikipedia)

Michelson interferometer is the classic setup for optical interferometry and was invented by Albert Abraham Michelson. Michelson, along with Edward Morley, used this interferometer for the famous Michelson-Morley experiment in which this interferometer was used to prove the non-existence of the luminiferous aether. See there for a detailed discussion of its principle.

But Michelson had already used it for other purposes of interferometry, and it still has many other applications, e.g. for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. But for most purposes, the geometry of the Mach-Zehnder interferometer is more useful.

A quick summation below leads one onto the idea of what experimental validation has done for us. Very simply, the graduation of interferometer design had been taken to astronomical proportions?



Today the Count expands on this for us by showing other information on expeirmental proposals. How fitting that this historical drama has been shown here, in a quick snapshot. As well the need for understanding the "principal inherent" in the project below.

VLBI is a geometric technique: it measures the time difference between the arrival at two Earth-based antennas of a radio wavefront emitted by a distant quasar. Using large numbers of time difference measurements from many quasars observed with a global network of antennas, VLBI determines the inertial reference frame defined by the quasars and simultaneously the precise positions of the antennas. Because the time difference measurements are precise to a few picoseconds, VLBI determines the relative positions of the antennas to a few millimeters and the quasar positions to fractions of a milliarcsecond. Since the antennas are fixed to the Earth, their locations track the instantaneous orientation of the Earth in the inertial reference frame. Relative changes in the antenna locations from a series of measurements indicate tectonic plate motion, regional deformation, and local uplift or subsidence.

See:

Apollo Moon Measure

The Z Machine

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.— Nikolai Lobachevsky

Sandia’s Z machine exceeds two billion degrees Kelvin

Z’s energies in these experiments raised several questions.

First, the radiated x-ray output was as much as four times the expected kinetic energy input.

Ordinarily, in non-nuclear reactions, output energies are less — not greater — than the total input energies. More energy had to be getting in to balance the books, but from where could it come?

Lubos Motl

Janice Granhardt has pointed out a press release that is two days old and arguably much more serious and potentially far-reaching than the news about "sonofusion" we described yesterday.

http://motls.blogspot.com/2006/03/two-billion-kelvins-at-z-machine.html

I reference current article information that I had been working through here and here for obvious reasons. I would like to expand on this.

I am writng this article because of the references Lubos Motl offered on his blog about the need for, "energy production." The whole context of any model has to have understood that the current situation in gravitational perspective will have it's two extremes (weak and strong) held in thought, and ending within this context? A cyclical process maybe like thinking about Steinhardt maybe? :)

I know the idea of free energy machines is a quacks realm, if, the imput energy and output energy is not held in consideration. That a greater output must be sustained. How?

Klein's Ordering of Geometries

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.


So on what conditions, could you map the process consistently and geometrics, to have been all inclusive?

While one may discuss these alternatives, it might require that we see this process at work on a cosmological scale, and having reduced it to the quantum realm, the questions about the geometries, becomes held under the auspice of "new physics,". That we might ask, "what new geometries?"

The natural process then would have to acknowledge the need for many microstate blackholes to have further the context of the standard model and it's extension?

Is this not a fair statement? Even though we may talk about one event, the recogition is that, this happens many times in regards to high energy articles in a collidial region. This had been answered in Risk assessment, as to why the process developed naturally, in the production of microstate blackholes, we might have created in LHC.

This did not discount, the understanding of what "extra dimensions meant" when we were understanding the "new physics." Reference here, neutrino or strangelets. It was just part and parcel of a greater understanding that John Ellis had pointed us too, is our recognition of the poor man's accelerator.

See:

The Unity of Mathematics

What First principle was-- was it the geometry

I thought I would contrast this quote of Dirac's with the one of Feynman's.

You see the very idea of a constancy that spread through all Maxwell's equations was a necessary one which allowed Einstein to move into positive and negative valuations within the geometries? So did Dirac know how this was to be approached?

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.

Feynman:

‘Maxwell discussed … in terms of a model in which the vacuum was like an elastic … what counts are the equations themselves and not the model used to get them. We may only question whether the equations are true or false … If we take away the model he used to build it, Maxwell’s beautiful edifice stands…’ – Richard P. Feynman, Feynman Lectures on Physics, v3, c18, p2.

Paul Dirac Talk: Projective Geometry, Origin of Quantum Equations Audio recording made by John B. Hart, Boston University, October 30, 1972

The quote below is in response to Dirac's comments

[ROGER PENROSE]

"One particular thing that struck me... [LAUGHTER]...is the fact that he found it necessary to translate all the results that he had achieved with such methods into algebraic notation. It struck me particularly, because remember I am told of Newton, when he wrote up his work, it was always exactly the opposite, in that he obtained so much of his results, so many of his results using analytical techniques and because of the general way in which things at that time had to be explained to people, he found it necessary to translate his results into the language of geometry, so his contemporaries could understand him. Well, I guess geometry… [INAUDIBLE] not quite the same topic as to whether one thinks theoretically or analytically, algebraically perhaps. This rule is perhaps touched upon at the beginning of Professor Dirac's talk, and I think it is a very interesting topic."

So the question might have been, how this was viewed and what the result was through such a axiomization? What was the first principe here? Was there one that became the guiding principal?

I mentioned the compass for Einstein, as a modelled perception that grew into the later years, but here, we might have seen the beginnings Feynmans toys model for such geometries?

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.

Point--> Line-->Plane <---> Point<-- String<-- Brane

Under the heading of Klein`s Ordering of the Geometries :

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

Now the usual thinking here has been placed under intense thinking by the introduction of a new way in which to look at "geometry" that has gone through a "revision" in thinking.

New trigonometry is a sign of the times

Lubos Motl introduces this topic and link in his blog entry and from this this has caused great consternation in how I am seeing. I see Lisa Randall might counter this in terms of what the brain is capable of, in line with this revisionary seeing, and comparative examples of this geometry Lubos links.

Dangling Particles,By LISA RANDALL
Published: September 18, 2005 New York Yimes

Lisa Randall:

Most people think of "seeing" and "observing" directly with their senses. But for physicists, these words refer to much more indirect measurements involving a train of theoretical logic by which we can interpret what is "seen." I do theoretical research on string theory and particle physics and try to focus on aspects of those theories we might experimentally test. My most recent research is about extra dimensions of space. Remarkably, we can potentially "see" or "observe" evidence of extra dimensions. But we won't reach out and touch those dimensions with our fingertips or see them with our eyes. The evidence will consist of heavy particles known as Kaluza-Klein modes that travel in extra-dimensional space. If our theories correctly describe the world, there will be a precise enough link between such particles (which will be experimentally observed) and extra dimensions to establish the existence of extra dimensions.

But first before I get to the essence of the title of my blog entry, I like to prep the mind for what is seemingly a consistent move towards geometry that has it's basis in applicabilty to physics, and move through GR to a vast new comprehsnsion in non-euclidean geometries. Must we now move backwards that we had gained in insight, or was it recognition of the "length scales" that we now say, how could such a dynamcial view ever be assigned to the eucildean discription under the guise of brane world recognitions?

Moving Backwards?

What exactly do I mean here?

Well the idea is that if you move to fifth dimensional views, and there are ways to wrap this within our "Brains":) We then see the dynamcial nature of our neurons have found acceptable ways in which to see this brane feature. As well as, approaches in use of new processes in geometerical considerations as those linked by Lubos.

Dealing with 5D world



Thomas Banchoff is instrumental here is showing us that fifth dimensional views can be utilized in our computer screens, and such comparisons, reduce to a two dimensional frame, makes it very easy to accept this new way in which to attack the dynamcial nature of reality.

How indeed now could our computer screen act a liason with the reality of our world, when see from screen imagery effects, that all the rules of order have been safely applied for inspection and consistancy in physics approaches.

The Unity of Mathematics

Alain Connes

Where a dictionary proceeds in a circular manner, defning a word by reference to another, the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their defnition.

I think what intrigues me most, is that a world can be fabricated mathematically that is carefully constructed using models of math, to get to a desired visionary culmination? One had to have some culminative effect, from such model thinking, that a vision beocmes clear. In this sense I related Lenny Susskind here, for his developement and contributions to string theory.

Now having spent time delving into parts of this world, the "tidbits" help me to see that such alignmenets of the world of physics have correlations in mathematical design. This has to have it basis set, "in the Rossetta stone you might say," about how we percieve the deveopement of those physics. The math must contrast the physics?

So to set things straight here, in case I gave the wrong link, I thought I should attribute proper link to words in case this mistake was made.



So too, information in blogs can be readily adapted too, where previous articles might have made some feel that the article not worth maintaining in their blog? That it might have been removed? I was thinking of the B-field topic that Lubos had written briefly on, that when I went to look for relevant information pertaining to this current entry, it was no where to be seen.

A VIEW OF MATHEMATICS by Alain CONNES
Most mathematicians adopt a pragmatic attitude and see themselves as the explorers of this mathematical world" whose existence they don't have any wish to question, and whose structure they uncover by a mixture of intuition, not so foreign from poetical desire", and of a great deal of rationality requiring intense periods of concentration.

Each generation builds a mental picture" of their own understanding of this world and constructs more and more penetrating mental tools to explore previously hidden aspects of that reality.

Now many would have to forgive my adventurous heart. I was somehow transported in my thoughts and converted? I don't know when, that such models of the mathematical structure had easily become discernable for me(it's result)? Not it's elemental structure(although I have seen areas of string theory design developed) from basic principals. It had it's culminative effect.

Is my vision always right? Of course not. But I see where such discriptions are necessary. Solid, and in stone, so that such progression can be made. I respect this, and I respect the physics, and it's culminative approach in theoretical developement.

Nature's Greastest Puzzle



Alain Connes refers to "poetic design," much like I see beats to music:?), and artistic adventure, as the play ground of imagination. We hope such songs shared, lyrics or otherwise, will reveal what the most secluded and private individuals might have found in their own world. To seek out, good artistic drawers like Escher? Penrose, needed his help, and the ideas brought forth, interesting results.

Now there is a reason for this post besides setting the record straight. It came up a long time ago with the question of whether mathematics was natural or created.

This may seem simplistic thought to some, but to me, it forced me to consider whether mathematics and physics were directed connected to each other.:) Now as I have said it is not easy for me to follow the matheics of such abtract individuals, but once I catch sight of the world that they allude too, it is somehow easy for me to see the structure of the bubble, or a representative drawing correlated in nodes, and features of a world that is constained in the physics.

This is why I refer back to Lubos and his B-field missing post, or I cannot simply find it. I refer to it, because I made links to mathematical design, that correlated dynkin diagram as shown above, and connects to other blog. Now it was important for me to see this correlation in the archetecture of the picture I linked to its prospective author, in relation to the dynkin diagram. Not the E11 asscoiation, but with that I had linked in image in comments to the B-field post.

My whole blog is based on visionary developement, theoretically, as well as nurturing physics association as best as I can, to show that the envelope is being pushed theoretically.



Interpretations of the magnetic field, in all its desgin is easily comprehensible once we align our thinking to hard fact and design reprsentation. Magnetic field lines on paper, is a child's toy, but easily experimetally done. Much more abstract then, that we see the field created, it's north and south, and a channel through which expression can flow?



Now even this is contained, and a Gausssian representation, highly abtract, relates curvature in away that we would understand this force that nature has created for us.

You must remember I do not have the luxury or life's abilty to move through the higher avenues that scholastic carreers have venture forth in. To preview this branch or that branch in physics, so I am bombarded with information from all angles:?)

I like to wrap the gravitational field, much like we wrap the magnetic field. It's just the way I see, and in it's greater design, that vast gravitational field that is generate through our cosmos? Bubbles become very interesting whenyou wrap somehing and the inside is moving with the outside, and in the vast vacuum of space this is stretching the very fabric itself?

I won't make the mistake of calling it the aether, yet continuity of expression seen in this abstract mode, does not see "tears" and such, so it is allocated to topological relevances. Holes, that look like swiss cheese in the cosmos? Yet I know well the events, that materialize in comsological expression, I wanted to push beyond these material things, to see the greater vision that has been moved by mathematcians.

You can say the rogue man here who speaks, is a wolf cub. Has been raised in a foreign world, without the benefits of scholastic teachers to guide me. So I had to look for them who held sacred some of the vision that I see when this math leads to a comprehensive view.

Reimann lead Einstein, and it was fortunate that Grossman was able to spot Einsteins deficiences. Help him move geoemtical principal beyond the euclidean coordinated world, to one manifested in spacetime, and a new dynamcial feature called gravity. It was beyond billiards and the sound related, and not the clasical discription that now beocmes the analogy of, that strange world we now see in gravitational thought.

Was it enough to speak about theses things and theorectically develope thoughts, to describe ways, in which such sound could ring bars, or influence the flexible arms of LIGO We measure this abstract world mathematcially created, to realize, we are now engaged in something very unique about our visions developement? Kip Thorne progeny will be the new genration that sees in way that were new to bauss and Riemann and now as we see of Einstein. This has a geometrical expression and basis to it, and it leads into projective elements topologically described.

Klein's Ordering of Geometries

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.


Klien's ordering of geometries were specifc here?

The Elastic Nature?

As I have explained in a earlier link I am fascinated by the images of bubbles that were demonstrated through a way of thinking of the early universe to arise as Bubble Nucleation.

These images all show the 2-, 3-, 4-, and 5-fold eversions in the upper left, lower left, upper right, and lower right cornerse, respectively. First we see an early stage, with p fingers growing in the p-fold everion. Next we see an intermediate stage when the fingers have mostly overlapped. Finally we see the four halfway models. For p odd, these are doubly covered projective planes.

If we had understood the early universe to have this continous nature and not have any tearing in it, how would such rotations have moved according to some method, that we might have considered the klein bottle or some other concept, that would lend itself to explain some of the ways and means, such dynamics could have unfold, enfolded and everted in the actions of that same cosmo?

It would be very difficult to speak to probability statistics, if you did not envelope the possibilites in some kind of configuration, or compared it to a Dalton Board. The Bell curve, or the pascal's triangle to consider how something could arise in certain situations? Might we have called the basis for a "new math" to emerge? If we had come to accept the departue point for Euclid's fifth postulate then what has we encouter inthe dynamcial world of Gauss? Einstein includethese calculation in the evolving feature of GR, so how could we not see this developing of a geometry that would lead to smooth and topological considerations?

The statistical sense of Maxwell distribution can be demonstrated with the aid of Galton board which consists of the wood board with many nails as shown in animation. Above the board the funnel is situated in which the particles of the sand or corns can be poured. If we drop one particle into this funnel, then it will fall colliding many nails and will deviate from the center of the board by chaotic way. If we pour the particles continuously, then the most of them will agglomerate in the center of the board and some amount will appear apart the center.

UNderstanding then that such cosmological event could be unfolding in the universe, visually to me, these configurations had to follow some pattern of consideration, or it just didn't make sense that such abstract math in topologies could ever work. So in looking at a previous comparison here the dynamical nature of the orbital seemed a valid comparison not only on a cosmological scale, but on a very small one as well?

A Holographical way of thinking?

Mapping Quark Confinement and The Energy

As I moved through the thinking of those extra dimensions it became apparent to me that the conceptualization of that distance scale was a strange world indeed. How, if we had accept the move to non-euclidean views could we not of accepted the consequences of this move?




Dazzled with the amazing properties of this new mathematical realm, everything seemed a bit magical, as if, experiencing for the first time a taste that is strange indeed? How would I recognize this strange dynamical world, if I had not understood this move to include the geometry that Kaluza and Klien adopted, to gather together another reality of photon engagement with that of gravity?



Fig. 1. In quantum chromodynamics, a confining flux tube forms between distant static charges. This leads to quark confinement - the potential energy between (in this case) a quark and an antiquark increases linearly with the distance between them.

So at the same time you had this distant measure, how could we resolve what was happening between those two points?

Without some supersymmetrical reality(supergravity) how could any point emerge from the brane if it did not recognize the evolution of those dimensions?



So how does this point expand? This is a simple enough question?

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

In the above picture Michael Duff draws our attention too, I was drawn to the same principals that Klein demonstrated in his ideas of projective geometry, as the dimensions are revealed?

IN this effort and recognition of appropriate geometry, I had wondered, that if the same consistancy with which these two had demonstrated the principals, euclidean
postulates fell in line, as a basis of this method of applicabilty? Does one now see this thread that runs through the geometry?

Having accepted the road travelled to GR we have come to recognize the royal road has lead us to a strange world indeed. First it was Reimann with Gauss looking over his shoulder, and Maxwell joining Faraday in this celebration, with Einstein bringing all the happy go lucky, into a fine example of what has been implied by the harmonious nature, structure of strings in concert?



But I am not happy yet. If one could not see what was happening between those two points, what's the use of talking any math, without the co-existance of the physics?


The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This, in fact, also gives rise to quantization of charge, as waves directed along a finite axis can only occupy discrete frequencies. (This occurs because electromagnetism is a U(1) symmetry theory and U(1) is simply the group of rotations around a circle).

No Royal Road to Geometry?

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

It was interesting to me that I find some thread that has survived through the many centuries , that moves through the hands of individuals, to bring us to a interesting abstract world that few would recognize.



While Euclid is not known to have made any original discoveries, and the Elements is based on the work of his predecessors, it is assumed that some of the proofs are his own and that he is responsible for the excellent arrangement. Over a thousand editions of the work have been published since the first printed version of 1482. Euclid's other works include Data, On Divisions of Figures, Phaenomena, Optics, Surface Loci, Porisms, Conics, Book of Fallacies, and Elements of Music. Only the first four of these survive.

Of interest, is that some line of departure from the classical defintions, would have followed some road of developement, that I needed to understand how this progression became apparent. For now such links helped to stabilize this process and the essence of the departure form this classical defintion needed a culmination reached in Einstein's General Relativity. But long before this road was capture in it's essence, the predecessors in this projective road, develope conceptual realizations and moved from some point. To me, this is the fifth postulate. But before I draw attention there I wanted to show the index of this same projective geometry.

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

The move from the fifth postulate had Girolamo Saccheri, S.J. (1667 - 1733) ask the question?

What if the sum of the angles of a triangle were not equal to 180 degrees (or p radians)?" Suppose the sum of these angles was greater than or less than p. What would happen to the geometry we have come to depend on for so many things? What would happen to our buildings? to our technology? to our countries' boundaries?

The progression through these geometries leads to global perspectives that are not limited to the thread that moves through these cultures and civilizations. The evolution dictates that having reached Einstein GR that we understand that the world we meet is a dynamical one and with Reason, we come t recognize the Self Evident Truths.

At this point, having moved through the geometrical phases and recognitions, the physics of understanding have intertwined mathematical realms associated with Strings and loop and other means, in which to interpret that dynamical world called the Planck Length(Quantum Gravity).

Reichenbach on Helmholtz