Pasquale Del Pezzo and E8 Origination?

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Donald (H. S. M.) Coxeter

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.

Clifford of Asymptotia drew our attention to this for examination and gives further information and links with which to follow.

He goes on to write,"Let’s not get carried away though. Having more data does not mean that you worked harder to get it. Mapping the human genome project involves a much harder task, but the analogy is still a good one, if not taken too far."

Of course since the particular comment of mine was deleted there, and of course I am okay with that. It did not mean I could not carry on here. It did not mean that I was not speaking directly to the way these values in dimensional perspective were not being considered.

Projective 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.

There had to be a route to follow that would lead one to think in such abstract spaces. Of course, one does not want to be divorced from reality. So one should not think that because the geometry of GR is understood, that you think nothing can come from the microseconds after the universe came into expression.

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

The Holy Grail sure comes up lots doesn't it:) Without invoking the pseudoscience that Peter Woit spoke of. I thought, if they could use Babar, and Alice then I could use the Holy Grail?

See more info on Coxeter here.

Like Peter I will have to address the "gut feelings" and the way Clifford expressed it. I do not want to practise pseudoscience as Peter is about the landscape.:)



When ones sees the constituent properties of that Gossett polytope 4₂₁ in all it's colours, the complexity of that situation is quite revealing. Might we not think in the time of supergravity, gravity will become weak, in the matter constitutions that form.

As in Neutrino mixing I am asking you to think of the particles as sound as well as think them in relation to the Colour of Gravity. If you were just to see grvaity in it's colourful design and what value that gravity in face of the photon moving within this gravitational field?

We detect the resulting "wah-wah-wah" in properties of the neutrino that appear and disappear. For example, when neutrinos interact with matter they produce specific kinds of other particles.

For example, when neutrinos interact with matter they produce specific kinds of other particles. Catch the neutrino at one moment, and it will interact to produce an electron. A moment later, it might interact to produce a different particle. "Neutrino mixing" describes the original mixture of waves that produces this oscillation effect.

The "geometry of curvature" had to be implied in the outcome, from that quantum world? Yet at it's centre, what is realized? You had to be lead there in terms of particle research to know that you are arriving at the "crossover point." The superfluid does this for examination.

5. Regular polytope: If you keep pulling the hypercube into higher and higher dimensions you get a polytope. Coxeter is famous for his work on regular polytopes. When they involve coordinates made of complex numbers they are called complex polytopes.

Pasquale Del Pezzo, Duke of Cajanello, (1859–1936), was "the most Neapolitan of Neapolitan Mathematicians".

He was born in Berlin (where his father was a representative of the Neapolitan king) on 2 May 1859. He died in Naples on 20 June 1936. His first wife was the Swedish writer Anne Charlotte Leffler, sister of the great mathematician Gösta Mittag-Leffler (1846-1927).

At the University of Naples, he received first a law degree in 1880 and then in 1882 a math degree. He became a pre-eminent professor at that university, teaching Projective Geometry, and remained at that University, as rector, faculty president, etc.

He was mayor of Naples starting in 1919, and he became a senator in the Kingdom of Naples.

His scientific achievements were few, but they reveal a keen ingenuity. He is remembered particularly for first describing what became known as a Del Pezzo surface. He might have become one of the strongest mathematicians of that time, but he was distracted by politics and other interests.

So what chance do we have, if we did not think this geometry was attached to processes that would unfold into the bucky ball or the fullerene of science. To say that the outcome had a point of view that is not popular. I do not count myself as attached to any intelligent design agenda, so I hope people will think I do not care about that.

NATHAN MYHRVOLD

I found the email debate between Smolin and Susskind to be quite interesting. Unfortunately, it mixes several issues. The Anthropic Principle (AP) gets mixed up with their other agendas. Smolin advocates his CNS, and less explicitly loop quantum gravity. Susskind is an advocate of eternal inflation and string theory. These biases are completely natural, but in the process the purported question of the value of the AP gets somewhat lost in the shuffle. I would have liked more discussion of the AP directly

See here for more information

So all the while you see the complexity of that circle and how long it took a computer to map it, it has gravity in it's design, whether we like to think about it or not?

But of course we are talking about the symmetry and any thing less then this would have been assign a matter state, as if symmetrical breaking would have said, this is the direction you are going is what we have of earth?

Isostatic Adjustment is Why Planets are Round?

While one thinks of "rotational values" then indeed one would have to say not any planets is formed in the way the sun does. Yet, in the "time variable understanding" of the earth, we understand why it's shape is not exactly round.



Do you think the earth and moon look round if your were considering Grace?

On the moon what gives us perspective when a crater is formed to see it's geological structure? It's just not a concern of the mining industry, as to what is mined on other orbs, but what the time variable reveals of the orbs structure as well.



Clementine color ratio composite image of Aristarchus Crater on the Moon. This 42 km diameter crater is located on the corner of the Aristarchus plateau, at 24 N, 47 W. Ejecta from the plateau is visible as the blue material at the upper left (northwest), while material excavated from the Oceanus Procellarum area is the reddish color to the lower right (southeast). The colors in this image can be used to ascertain compositional properties of the materials making up the deep strata of these two regions. (Clementine, USGS slide 11)

See more here

Finiteness of String Theory and Mandelstam

It might be that the laws change absolutely with time; that gravity for instance varies with time and that this inverse square law has a strength which depends on how long it is since the beginning of time. In other words, it's possible that in the future we'll have more understanding of everything and physics may be completed by some kind of statement of how things started which are external to the laws of physics. Richard Feynman

I was lead into this subject of Quantum Gravity, by Lee Smolin's book called, "Three Roads to Quantum Gravity." As a lay person reading what our scientist's have to say, I have a vested interest in what can start one off and find, that changes are being made to the synopsis first written. Did I understand his position correctly from the very beginning? I'll have to go back over my notes.

But with this format now I have the opportunity to...ahem... get it..directly from the horses mouth(no disrespect intended and written based on knowing how to read horses). As I said, I tried early on to see how the situation of string theory could be refuted. I "instigated" as a comparative front for Lubos Motl and Peter Woit to speak from each of their positions. I had to disregard "the tones" set by either, as to the nature of whose what and how ignorant one might be, and comparatively, one might be to intelligent design? To get "some evidence" of why string theory might not be such a good idea?

Now I believe this is a more "civil situation" that such a format has been proposed and that Lee Smolin can speak directly. As well as, "further information" supplied to counter arguments to Lee's position.


A sphere with three handles (and three holes), i.e., a genus-3 torus.

Jacques Distler :

This is false. The proof of finiteness, to all orders, is in quite solid shape. Explicit formulæ are currently known only up to 3-loop order, and the methods used to write down those formulæ clearly don’t generalize beyond 3 loops.

What’s certainly not clear (since you asked a very technical question, you will forgive me if my response is rather technical) is that, beyond 3 loops, the superstring measure over supermoduli space can be “pushed forward” to a measure over the moduli space of ordinary Riemann surfaces. It was a nontrivial (and, to many of us, somewhat surprising) result of d’Hoker and Phong that this does hold true at genus-2 and -3.

Just a reminder about my skills. While I do things like carpetry, plumbing, electrical, I do not call myself a Carpenter, a Plumber or a Electrician. Nor shall I ah-spire to be more then I'm not, as I am to old this time around.

Greg Kuperberg:

The string theorists are physicists and this is their intuition. Do you want physical intuition or not?

Okay, Smolin is also a physicist and his intuition is radically different from that of the strings theorists. So who is right?

Yet, least I not read these things, can I not decipher "the jest" while it not being to technical? Shall I call it a Physicists intuition or I will only call my intuition what it is?

Jacques Distler:

When most people (at least, most quantum field theorists) use the term “finiteness,” they are referring to UV finiteness.

While the things above talked about from Jacques are served by hindsight, "the jest" follows what comes after this point.

The Jest of the Problem?

My present research concerns the problem of topology changing in string theory. It is currently believed that one has to sum over all string backgrounds and all topologies in doing the functional integral. I suspect that certain singular string backgrounds may be equivalent to topology changes, and that it is consequently only necessary to sum over string backgrounds. As a start I am investigating topology changes in two-dimensional target spaces. I am also interested in Seiberg-Witten invariants. Although much has been learned, some basic questions remain, and I hope to be able at least to understand the simpler of these questionsStanley Mandelstam-Professor Emeritus Particle Theory

Gina has asked questions in context of "academic excellence" in relation to what is being seen in relation to string theory. Of course we thank Clifford for providing the format for that discussion.

“The Trouble With Physics,” by Lee Smolin, Index page 382, Mandelstam, Stanley, and string theory finiteness, pages 117,187, 278-79, 280, 281, 367n14,15

For reference above.

Gina:

I raised 16 points that I felt Lee’s arguments were not correct or problematic. This is an academic discussion and not a public criticism, and I truly think that such critique can be useful, even if I am wrong on all the 16 points.

Three of my 16 points were on more technical issues, but I feel that I can understand Lee’s logical argument even without understanding the precise technical nature of “finiteness of string theory” (I do have a vague impression of what it is.) I think that my interpretation of this issue is reasonable and my critique stands.

I find this interesting based on what information has been selected to counter the arguments that Lee Smolin used to support his contentions about what is being defined in string theory.


Stanley Mandelstam Professor Emeritus Research: Particle Physics

My research concerns string theory. At present I am interested in finding an explicit expression for the n-loop superstring amplitude and proving that it is finite. My field of research is particle theory, more specifically string theory. I am also interested in the recent results of Seiberg and Witten in supersymmetric field theories.

So of course, here, I am drawn to the content of his book and what is the basis of his argument from those four pages. I hope my explanation so far summarizes adequately. For the lay person, this information is leading perspective as to the basis of the argument.

Lee Smolin:

Perturbative finiteness is a major element of the claim of string theory as a potential theory of nature. If it is not true then the case for string theory being a theory of nature would not be very strong.

-Perturbative finiteness has not been proven. There is evidence for it, but that evidence is partial. There is a complete proof only to genus two, which is the second non-trivial term in an infinite power series, each term of which has to be finite. The obstacles to a complete proof are technical and formidable; otherwise we would certainly have either a proof or a counterexample by now. There is some progress in an alternative formulation, which has not yet been shown to be equivalent to the standard definition of string theory.

-This is not an issue of theoretical physicists rigor vrs mathematical rigor. There is no proof at either level. There is an intuitive argument, but that is far from persuasive as the issue is what happens at the boundaries of super-moduli space where the assumption of that argument breaks down. In the formulation in which there is a genus two result it is not clear if there is an unambiguous definition of the higher order terms.

Is string theory in fact perturbatively finite? Many experts think so. I worry that if there were a clear way to a proof it would have been found and published, so I find it difficult to have a strong expectation, either way, on this issue.

It should be known here and here that all along I have been reacting to Lee Smolin's new book. The title itself should have given this away?

The explanation of scientific development in terms of paradigms was not only novel but radical too, insofar as it gives a naturalistic explanation of belief-change. Thomas Kuhn

So of course knowing the basis of my thought development is a "good idea" as the links show what spending our dollars can do, having bought what our good scientist Lee Smolin has written.

There is a little "tit for tat" going on right now, but I think the point has been made sufficiently clear as to where Gina's thoughts in regards to the points on Finiteness is being made beyond 2?

In these lectures, recent progress on multiloop superstring perturbation theory is reviewed. A construction from first principles is given for an unambiguous and slice-independent two-loop superstring measure on moduli space for even spin structure. A consistent choice of moduli, invariant under local worldsheet supersymmetry is made in terms of the super-period matrix. A variety of subtle new contributions arising from a careful gauge fixing procedure are taken into account.

Yes I think I have to wait now to see if the discussion can now move beyond the first three points raised? Hopefully Lee will respond soon?

How do you fight sociology

Because this by any of the leaders of string theory. it was left to someone like me, as a quasi "insider" who had the technical knowledge but not the sociological commitment, to take on that responsibility. And I had done so because of my own interest in string theory, which I was working on almost exclusively at the time. Nevertheless, some string theorists regarded the review as a hostile act.

The trouble with Physics, by Lee Smolin, Page 281

I have discovered one of Lee Smolin's objection to a string theorist. They are only craftsman, and not seers.

Part of Facing the Trouble With Physics

It might be that the laws change absolutely with time; that gravity for instance varies with time and that this inverse square law has a strength which depends on how long it is since the beginning of time. In other words, it's possible that in the future we'll have more understanding of everything and physics may be completed by some kind of statement of how things started which are external to the laws of physics. Richard Feynman

Faced with the task of showing the connection between string theory and reductionistic consideration is quite a task, as I am sure in most eyes? To me it just seems that everytime we adjust our view and include new views, what shall we say of "gamma ray detection" when we look at high energy photons describing the early universe for us?



Hey, it makes my heart jump too.

Here is a case, with which I like to make my point. Having someone corrected makes it that much better now to make comparisons like I do. The simple point of "order" enlightened greatly the situation for us, in what I am exemplifying here. We wil not forget the paper offered up after, in that comment thread either. Thanks

A realization 1; 2; 3 that QGP at RHIC is not a weakly coupled gas but rather a strongly coupled liquid has lead to a paradigm shift in the field. It was extensively debated at the “discovery” BNL workshop in 2004 4 (at which the abbreviation sQGP was established) and multiple other meetings since.

In the intervening three years we had to learn a lot, some new some from other ranches of physics which happened to have some experience with strongly coupled systems. Those range from quantum gases to classical plasmas to string theory. In short, there seem to be not one but actually two difficult issues we are facing. One is to understand why QGP at T ∼ 2Tc is strongly coupled, and what exactly it means.

In Extracting Beauty From Chaos I am recognizing this depth of perception enhancement that is supplied by JoAnne of Cosmic Variance. Would you rather look at "Seans moon" in gamma?

CERN planned a global-warming experiment in 1998?

Experimentalists at CERN will use a cloud chamber to mimic the Earth's atmosphere in order to try and determine whether cloud formation is influenced by solar activity. According to the Danish theory, charged particles from the Sun deflect galactic cosmic rays (streams of high-energy particles from outer space) that would otherwise have ionized the Earth's lower atmosphere and formed clouds.

What shall I say to you as SNO investigated the "cerenkov effect" from the cosmos ray particle collisions? Shall I speak about the "weather predictions" that arise. This is a interference and a "weak measure" of what is fast becoming the thought in my mind of the diversity of global painting, to include, that blue light as each of the detectors "pick" the overall pattern of high energy exchanges in the detectors as inherent image understanding. It has been transcribed from the "sun's energy value" and applied to high energy considerations?

"Atmospheric" neutrinos, produced by interactions of cosmic ray particles with the earth's atmosphere, might be useful for studying the properties of neutrinos. But if you're hunting sources of neutrinos in the universe, atmospheric neutrinos are nothing but noise.

Now, I may reference Glast indications here in the experimental validation of those high energy photons, gamma ray indication is a wonderful jesture to extending the depth of perception, as I have tried to do here by helping Q see the relevance of the quantum dynamical perception. From ,the beginning of this universe.

So we see where the " Window of the universe" has helped me to see in ways that we were not accustomed. It is "the physics" that has taken us there.

So, while the picture of JoAnnes is highlighted, the lesser of the views is the "gamma ray detection" while I have pointed to the neutrino here in experimentation.

Should we loose sight of what the KK tower exemplifies?



I am sorry about the "dead link picture to topology" but blogger does not go back to 2004 so that I can adjust it.

Now why would I then reference "quantum gravity" behind the picture of the KK tower, and the information about topology? Possibly, that we have for the first time thought here that the Navier-Stokes equations could have been applied at a fundamental level while thinking of what the QGP has given us, as we witness "cerenkov radiation" from a long line of reductionistic reasoning? Is this worth a million from the Clay Instituted by generalization alone?:)

If not, at least, if held in line with lagrangian views of gravitonic perceptions in the bulk as we phyically see the relation between the sun and earth?

It is thus my mind has been held to the idea of the "conical flows[Volcanos, to jet engines in analogy of the laval nozzle]" as the energy is released for the dissemmination from the collider of nature enhanced, to all that follows from the cosmic particle interactions. Right to the neutrinos resulting from the fluidity of the QGP pertaining to viscosity?

What was not present before? Muon detectors hmmmm..... and the road from muon neutrinos too?? What am I missing here?

The muons are stopped by the rock. Impervious to all such obstacles, the muon neutrinos will leave the CERN tunnels and streak through the rock on their 732 kilometre journey to Italy.

Hold that picture of JoAnnes, while you think of Glast. In the determinates of the gamma ray detection, we have therefor faced the "Trouble with Physics?":)

What is the False Vacuum?

Quantum Field Theory

Quantum Vacuum:

In classical physics, empty space is called the vacuum. The classical vacuum is utterly featureless. However, in quantum theory, the vacuum is a much more complex entity. The uncertainty principle allows virtual particles (each corresponding to a quantum field) continually materialize out of the vacuum, propagate for a short time and then vanish. These zero-point vibrations mean that there is a zero-point energy associated with any quantum field. Since there are an infinite number of harmonic oscillators per unit volume, the total zero-point energy density is, in fact, infinite. The process of renormalization is usually implemented to yield a zero energy density for the standard quantum vacuum, which is defined as no excitation of field quanta, i.e., no real particles are present. In other word, the quantum vacuum is at a state of minimum energy - the ground state.

You have to be able to envision this movement in what our universe is doing. What is WMAP saying? Other events say what, in the node/anti-nodal?

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;

The "quantum harmonic oscillator" and "zero point as a ground state, are the basis of my thinking. :)Energy densities. I needed a way in which to see these events unfolding in the universe. Why I look at WMAPing very seriously. Why I looked at the chaldni plate very early on.

Physically, the effect can be interpreted as an object moving from the "false vacuum" (where = 0) to the more stable "true vacuum" (where = v). Gravitationally, it is similar to the more familiar case of moving from the hilltop to the valley. In the case of Higgs field, the transformation is accompanied with a "phase change", which endows mass to some of the particles

If you look at things in this way I have covered a lot of ground work in terms of what the basis of this universe is? "Nothing," is a extremely hard thing for me to accept when I accept the quantum harmonic oscillator, as the basis of my thinking. I had to be able to describe what I was seeing. So "sound" in analogy became a very important aspect of my research. Became discriptive of what Higgin's the graviton is doing?

If you sprinkle fine sand uniformly over a drumhead and then make it vibrate, the grains of sand will collect in characteristic spots and figures, called Chladni patterns. These patterns reveal much information about the size and the shape of the drum and the elasticity of its membrane. In particular, the distribution of spots depends not only on the way the drum vibrated initially but also on the global shape of the drum, because the waves will be reflected differently according to whether the edge of the drumhead is a circle, an ellipse, a square, or some other shape.

In cosmology, the early Universe was crossed by real acoustic waves generated soon after Big Bang. Such vibrations left their imprints 300 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chaldni patterns resulting from a complicated three-dimensional drumhead that

String theory is only "topologically equivalent" to the shape and values of those events microscopically/macroscopically at a certain plac einthe unfolding universe? I learnt that the energy densities flunctuations, would give meaning to the place dynamically and geometrically speaking, to the place in time, that is unfolding. What evidence do you have for that if "Higgin's" is strong in some event places and not in others? :)

The star Eta Carina is ejecting a pair of huge lobes that form a "propeller" shape. Jet-like structures are emanating from the center (or "waist"), where the star (quite small on this scale) is located.

Yes, there are many event shapes and they are diverse. But they happen within context of a "larger false vacuum" scenario as I am explaining it, while they make their way to what is "True?"

I had to "go further" then the microseconds, strings inhibit?

The plates can be made visible by mounting a mirror behind the row of plates, angled so that the top of the plates are visible to the audience (same idea as in Polarization by Scattering). Create the optimum angle for the front rows, as the back rows will be looking down on the plates anyway. Make sure the cello bow is nice and tactile by treating it with rosin before the performance. Sprinkle the sand on the plates so that it forms an even cover. Don't overdo the amount.

Why Higher Energies?

I guess I don't have to tell anyone how confusing all this stuff is and the need for a consistent picture to arise out of it.

New physics beyond the standard model of particle physics and parallel universes by Rainer Plaga

top-quark masses - for which the standard model predicts such a decay - cannot be interpreted as evidence for new physics at low energy scales.

The history of Risk Assessment, was a exercise into understanding the developing role as to what new physics should be? Strangelets and strange quarks arose from this?

The search for the very small requires very high energies. The discoveries necessary for the electroweak unification were near the upper end of available energies in the current generation of particle accelerators. Establishing Grand Unification is beyond the practical limits of earthbound laboratories. This forces particle physicists to look outward to astrophysical phenomena which may have enough energy to shed some light on further attempts at unifying the four fundamental forces.

This map defines the whole standard model and the phase transitions. We are talking about a "certain time" in the planck epoch. So what is happening "in" the Planck epoch?

If such energies had recognized the current state of the superfluid created, then anomalies in "this scenario" would have allowed such "geometrical presence to be channelled" as part of the cyclical features contained in the expression of the universe?

So you take this universe and apply the backhole on a cosmlogical scale eqaul to it's inflation, as a distance in the blackhole's radius? Such a crunch would have recognized the boundary conditions as a the furthest point this universe could have grown, from the original blackhole that created this universe?

So what evidence is left? That the universe and it's "dark matter" as the false vacuum is creating the scenarios for the universe to have found it's temeperature today, started from some "other condition" seen in the planck epoch? Okay how did you get there?

The bubble conditions would then have to existed in the superfluids? How would have geoemtrically arrived at such a "topology expressed" in this one universe?

Professor Satyendra Nath Bose, the founder of Bose-Einstein statistics and the discoverer of the “Boson,” is well known as a giant in the world of physics and science as the man who, along with Albert Einstein, revolutionized the world of theoretical physics and showed the world a new way to imagine how the world works.

The topological genus figure of the sphere, to a torus and it's rotation seen in characteristic, housed the equallibrium state arrived at, as to the channelling of that extra energy and the resulting "new physics" in the strange quarks created?

So what is "that cylinder" created as the jet is expressed, in the gravitational collapse

See: John Bahcall and the Neutrinos

Thus, this cycle is completed in the bulk perspective? Would have created the situation again in strong concentrations? Why cosmologically the conditons are "many" and such evidence pointing to ICECUBE, as to the conditons beyond the standard model, leads to questions about "cerenkov radiation?"

Is there no backreaction created, if we were to lets say look at the Laval nozzles, and understand that what is expressed in the standard model energy once ejected in the jet, would have had counter proposals manifest in the geomerical presence held to a whole universe. The Anti-matter? Non Qui

If it's Not a Soccer Ball, What is it?

Timaeus concludes

And so now we may say that our account of the universe has reached its conclusion. This world of ours has received and teems with living things, mortal and immortal. A visible living thing containing visible things, and a perciptible God, the image of the intelligible Living Thing. Its grandness, goodness, beauty and perfection are unexcelled. Our one universe, indeed, the only one of its kind, has come to be.

WMap currently expressed, has some explaining to do. While they talk about ekroptic universe and such, such cyclical natures need some reference with which to speak, in order for such idealizations and perspective form around the information WMap has currently released.

The fifth element, i.e., the quintessence, according to Plato was identified with the dodecahedron. He says simply "God used this solid for the whole universe, embroidering figures on it". So,I suppose it's a good thing that the right triangles comprising this quintessence are incommensurate with those of the other four elements, since we certainly wouldn't want the quintessence of the universe to start transmuting into the baser subtances contained within itself!

Higher abstractness, in the case of leaving euclidean perspective, are part of the realization when you look at the WMAP, that is being presented. While sound implications are being implied, they are necessary if it is understood the role they will play in such analogies of a larger global pewrspectve then the one seen in how gaussian coordinates and curvatures are implied, in Omega features, and critical density.

A Finite Dodecahedral Universe

According to the team, who published their study in the 9 October 2003 issue of Nature, an intriguing discrepancy in the temperature fluctuations in the afterglow of the big bang can be explained by a very specific global shape of space (a "topology"). The universe could be wrapped around, a little bit like a "soccer ball", the volume of which would represent only 80% of the observable universe! (figure 1) According to the leading cosmologist George Ellis, from Cape Town University (South Africa), who comments on this work in the "News & Views" section of the same issue: "If confirmed, it is a major discovery about the nature of the universe".

So having a greater perspective on this kind of mapping is necessary. While I reference tunneling and such, implicate bee's travelling or satelittes using pathways of least resistance, how does such states have been reached, if you did not have the perspective that is necessary in seeing how the universe is laid out geometrically?

Scientists Get Glimpse of First Moments After Beginning of Time by DENNIS OVERBYE

"If this holds up to the test of time, it's a real landmark," said Max Tegmark, a cosmologist and cosmic microwave expert at M.I.T. "I really feel like the universe has given up one more clue," he said.

The quantum gravity issues implored, and speculated, have a diverse model selection with which to talk about? John Baez's view came up when I seen the model in which he choose for such understanding. Having seen the "membrane" idealizations used in qauntum gravity models, helped with the perspective that he used, but like Plato his definition fell short to me of the substructure(a discrete meaasure in bubble facets) of the cosmo that would relegated some model, to perfect our view microscopically, as well as macroscopically.




Tegmark and others disproved this, so what is the nature of the cosmo in question, and it's shape? So having understod the idealization of flat gemetries of the universe, how could such negative features be understood as the hyperbolic understanding became part and parcel of GR in our understanding of the gravity issues?

With the discovery of sound waves in the CMB, we have entered a new era of precision cosmology in which we can begin to talk with certainty about the origin of structure and the content of matter and energy in the universe.-Wayne Hu

Such analogies were very important in raising our understanding of what might have been seen in the membrane ideals of how gravity could act in bubble boundaries conditions. While such faces might have been understood in such analogies of John Baez's, a spherical harmonics would have had to been much smoother in our understanding?

Imploring the analogies and models in this respect made it much easier to see on another level, in non euclidean realities. Further then, the understanding of gaussian coordinates. You knew this was part and parcel of a larger picture understood in GR?



The Sound of Gravitational Waves

We can't actually hear gravational waves, even with the most sophisticated equipment, because the sounds they make are the wrong frequency for our ears to hear. This is similar in principle to the frequency of dog whistles that canines can hear, but that are too high for humans. The sounds of gravitional waves are probably too low for us to actually hear. However, the signals that scientists hope to measure with LISA and other gravitational wave detectors are best described as "sounds." If we could hear them, here are some of the possible sounds of a gravitational wave generated by the movement of a small body inspiralling into a black hole.

So there is a culmination in my views as I now look at the new WMap presented. A greater understanding implored in geometric realization, that had to be taken down to the microscopic where quantum gravity existed, yet, the geometry used, what new math would this be? It was encapsulted in the overall understanding of cyclical natures.



Is there such a thing, as isometrical relations of orbitals, in cosmological designs? A Classical definition of the Quantum World perhaps?

Some patterns are telling to me of the way in which the universe, and the galaxies in which had formed, would have followed some geometrical all inclusive pattern, that we see unfolding from place to place, and assigning, specific polarization points within the view of the WMAP.



Now, look at the map below.

The WMAP (Wilkinson Microwave Anisotropy Probe) mission is designed to determine the geometry, content, and evolution of the universe via a 13 arcminute FWHM resolution full sky map of the temperature anisotropy of the cosmic microwave background radiation. The choice of orbit, sky-scanning strategy and instrument/spacecraft design were driven by the goals of uncorrelated pixel noise, minimal systematic errors, multifrequency observations, and accurate calibration. The skymap data products derived from the WMAP observations have 45 times the sensitivity and 33 times the angular resolution of the COBE DMR mission. The WMAP mission characteristics are summarized in the table below.

So by using information in the Chladni plate exhibit within this site, it becomes a interesting picture when such "spectrum analysis" make themselves representable in "color" as a variation of the landscape.

The temperature fluctuations of the Cosmic Background Radiation may be decomposed into a sum of spherical harmonics , much like the sound produced by a music instrument may be decomposed into ordinary harmonics. The "fundamental" fixes the height of the note (as for instance a 440 hertz acouctic frequency fixes the "A" of the pitch), whereas the relative amplitudes of each harmonics determine the tone quality (such as the A played by a piano differs from the A played by a harpsichord). Concerning the relic radiation, the relative amplitudes of each spherical harmonics determine the power spectrum, which is a signature of the geometry of space and of the physical conditions which prevailed at the time of CMB emission.

But if you are really interested in the way we see the universe how would such patterns illustrate what is being shown in the WMAP. A Chladni plate perhaps as spoken in reference. I needed a model with which to work the whole geometrical picture.

If you sprinkle fine sand uniformly over a drumhead and then make it vibrate, the grains of sand will collect in characteristic spots and figures, called Chladni patterns. These patterns reveal much information about the size and the shape of the drum and the elasticity of its membrane. In particular, the distribution of spots depends not only on the way the drum vibrated initially but also on the global shape of the drum, because the waves will be reflected differently according to whether the edge of the drumhead is a circle, an ellipse, a square, or some other shape.

In cosmology, the early Universe was crossed by real acoustic waves generated soon after Big Bang. Such vibrations left their imprints 300 000 years later as tiny density fluctuations in the primordial plasma. Hot and cold spots in the present-day 2.7 K CMB radiation reveal those density fluctuations. Thus the CMB temperature fluctuations look like Chaldni patterns resulting from a complicated three-dimensional drumhead that

See:

Plato's Defintion of God

Sound of the Landscape

B Field Manifestations

Resonance:Brownian Motion

All Particles of the Standard Model and Beyond

Polchinski Elected Member Of National Academy of Sciences

Polchinski's discovery of D-branes and their properties is, according to the Academy citation, "one of the most important insights in 30 years of work on string theory."

Can I tell a little story before I head into the essence of this posted thread below?

From one mechanic to another

I am not a mechanic by trade. Yet I had taken apart, and put back together motors which ran and ran well. Through a transition period, and without a place in which to do this work myself, I decided to give it to "a mechanic" to work on. Pay the price, which was well beyond my means at that time. With three children a wife, and barely making it, I asked for help financially. It was cold, and snow blowing.

After picking up my motor and installing it. Making sure everything was right, I went for a slow drive to seat my rings in newly honed out cylinders. Well, much to my dismay and lots of dollars, blue smoke clouded the world behind me.

Taking it back home, I called the mechanic, and told him what was happening. "It was something you must of done," he siad.

So, I called another mechanic. He compression tested the cylinders for me, and to my dismay and his, one of the cylinders was not up to par.

So what things did I learn?

That I could have "one mechanic go against another," for the shoddy work that was done? No, it doesn't work that way.

After tearing off the head, I had found they had broken the oil and compression rings, as they pushed the compressed rings and piston, back into the cylinder. They had cracked them while doing this. The cracked ring gouged the cylinder wall, as it went up and down on the crankshaft.

Were there things I might have done different now? Maybe pressure tested the cylinders before hand?

Anyway, on to the subject of this post.

After doing my research and investigations into how the standard model itself might have been displayed, I selected two events, that were very discriptive of what might have happened, when taken as a whole story of the science in progress.



These were censored by Peter Woit on his site and removed. These lead to questions that might have implicated "string theory" as part of the process of inquiry beyond the standard? See Icecube.

If one holds to the idea that they had assumed a counter position to currents trends, then would it not include the theoretical approach well understood, that it also attached, not just a geometrical association, but one described in the physics process as well?

As a layman, this was proving itself, as I looked at the diversity of the geometrical models choosen to represent that abstract world. See B Field and Hitchins. Genus Figures, and topology, on this site.

More and more, it had weighted heavily on my mind, that the consistancy through which selected comments were shown, were to hold validation processes as to anti-string theory. As tones of select comments, as very disconcerting to me, but through his awareness Peter did strived to referee.

The overall message, was not one with the care which Cosmic Variance had ascertained it's caution of String evangelistism, or Lubos Motl's declaration as well, that the underlying motivation, was more to provide a "general widesweping statement" that applied to the string model development as a whole.

IMpressional Minds
If as a student, having now moved toward my senior years, how could I have turned back the clock of time, that I might have stood beside any of these leaders of science?

That I had to accustom myself to the very level on which my opinion would not have mattered coming from layman status. So being on the bottom of the totem pole, I accept the resolve to which such treatment was dealt. It was a small price to pay.

So imagine then, what the overall message by Peter has done to those prospective entries into the world of, might now have said, why should we now enter, being the brunt of what good science men hate, would have us believe?

The Reductionistic Process
Is it incorrect to say that the events of the collision process are incapable of decribing all fawcetts of the standard model?



So by concentrating on the collision process itself, what factors would have said that no, the standard model does not fit the current processes in LHC? Does not fit the process in high energy collision process to earths atmospheric conditions, for evdience of? See Pierre Auger expeirments here. See John Bachall and the Ghost particle.



So by closely looking at the poor man's version, what process would lead one to believe that the standard model was inclusive in this interactive process as well?

Here's the post in full. It was in response to Jack Safartti's comments and the document in which he had wrote was in contradiction of what I had learnt of the "possible new physics?" THis is of course held within context of collider results and the micro perspective results, created the form of quark Gluon Plasma. A superfluid?

So both events involved, "microstate blackhole" recognitions.

Post removed from Peter Woits comment section

In regards to facing nightmares

In recent years the main focus of fear has been the giant machines used by particle physicists. Could the violent collisions inside such a machine create something nasty? "Every time a new machine has been built at CERN," says physicist Alvaro de Rujula, "the question has been posed and faced."

The link was added here now.

If one follows the logic development, Jack's position becomes a interesting one to question. As well, such thoughts about cosmic collisions, and the high energy particles cosmological events. Microstate blackhole processes are the poor man's experimental pallete. Just as valid the dissipative state created in the collider.

The resulting end product is what is being explore with ICECUBE. It is all consistent with the standard model. Right from, the start of the collision process, to the resulting shower created.

Jack has some explaining to do?

Update

(To help anonymous understand better I hope the student does not feel s/he has to learn string theory in order to be valid in existance. Also, the interactive shower from the collison process with high energy article is well understood and what comes from it.

He deletes yours too.! Oh look, what we have in common?:) What drivel have you drummmed up?)

Anyway. As I was saying.

This is not to slight Peter Woit in the slighest, but to move him to consider the enormity with which the process of string/M theory is involved in the standard model expression. As fundamental particles and the interactions thereof.

To reject the model on the basis of preference, is of course for any who choose to follow which road. But to say that such a process should not be followed would have been a erroneous statement, as well as influencing the general population by such ascertions of preference aghast and in reaction.

Of course I recognized it is his blog and his comment section. On the basis of his dislike for anyone, can do anything they like, within reason right?

See:

History of the Universe and the Standard model

Big Bang:One Man's Change of Heart

Thanks Paul

One definitely needs some perspective around this and how such information is given. I refer here for consideration, about perspective, and how it can be exploited for further consideration on what is emitted, and what manifests in weak gravitational field measure, as neutrino effects(quantum gravity).

Microperspective and methods of examination, raise the issue fo cerenkov radiation and what it tells us about such interactive phases?

Here in refractive consideration, ICECUBE, paints a different picture of what began somewhere else in cosmological high energy collisions. "Neutrinos and strangelets" are part of the developing scenario with which the universe has consequences, if held to the initial conditons of our universe. You had to know where to look for these.

Plato:

"Nothing" in stated form was and always is "nothing" which would have not allowed any further discussion. "Zero" in our conversation is a much different kind of thinking. I understood that as well. "Zero" would have been the equivalent to "i" in the Dirac's matrices?

Physics at this high energy scale describes the universe as it existed during the first moments of the Big Bang. These high energy scales are completely beyond the range which can be created in the particle accelerators we currently have (or will have in the foreseeable future.) Most of the physical theories that we use to understand the universe that we live in also break down at the Planck scale. However, string theory shows unique promise in being able to describe the physics of the Planck scale and the Big Bang.

I wanted to add this post, and to centralize some references that were found that helped form my perspective on "nothing." What! I guess I'm done?:)

Seriously, this had to be confronted, and who better then from our layman perspectve, then the admission of a leaders in science, who can change theirs mind after some thinking?

Cosmological Constant SeeSaw in Quantum CosmologyMichael McGuigan

Lubos shares his perspective on linked section of titled paper above.

One interpretation of the coupling of Wheeler-DeWitt functions is that it originates from topology changing effects. Topology change seems to be inevitable in quantum gravity. To treat topology change properly is a very complicated calculation using today’s mathematical tools.

I wanted to add these links here for consideration, as well what link given by Paul for consideration in regards to Penrose, the figure of the man's change of heart that ighlight's this post. In Phase transitions the comments have been quite enlightening.

Before the Big Bang BBC News, with Stephen Sackur
Sir Roger Penrose has developed a new theory on what happened before the Big Bang.

These pages were created by Jack "Turtle" Wong, Spring 1999

First of all, how do we think the universe began?

The Big Bang theory.

Resolving the inadequacies of the big bang theory.

The Hawking-Turok Instanton theory: Stephen Hawking's
ideas.

The Hawking-Turok Instanton theory: Neil Turok's ideas.

The Hawking-Turok Instanton theory: the result of merging
two interesting theories.

Is the search over?

Bibliography / Further Reading

See Also:

Cycle of Birth, Life, and Death-Origin, Indentity, and Destiny by Gabriele Veneziano

Ekpyroptic and cyclical models

A New Way of Seeing?

Sorry couldn't resist. How many before us in our speculations and way of seeing? :)

When looking at Gaussian coordinates, the very idea that our views of "length of lines" had to have another way in which to interpret how we would see such divergences in the UV considerations. Now I might use UV differently then most, but it is always in context of gaussian coordinates that I always do this.

If I had created a triangle in much the same context(empheral qualties) we might have seen in 2D idealizations then, in how would you transfer such thinkng to three dimensional view held in context of the spiral, and the ever widening primes? These views would have to be locked in Gaussian interpretations, whether they came from Riemann or not. But such trancendance to 5d worlds had to have interpetation that would make you see this in other ways?

Saccherri introdces us to the 5th postulate and we move accordingly into the views of Riemann and others, in ways that are different.

Observations on the Regularity of Prime Number Distribution Peter Marteinson

Stanislaw Ulam’s (1964: 516) most general observation on his famous spiral, that a “property of the visual brain” allows patterns relating to the characteristics of primes to be discovered, may indeed stimulate the mathematical imagination, and inspire further creative attempts at visual pattern recognition in this area, but his spiral (fig.1), like its derivatives, has yet to be successfully interpreted in terms of possible arithmetic principles that can explain the genesis of the known distribution of prime numbers. Of his spiral he says only that it “appears to exhibit a strongly nonrandom appearance” (Stein et al. 1964). A corollary of this somewhat disappointing observation is that Euler’s pessimistic prognosis has yet to be disproved: “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate” (cited by Ivars Peterson in Science News, 5/4/2002).

While it is never easy to take it all in it seems certain phrases and sentence structure stand out as important. While they may seem familiar I refrain from specify what this is, while I continue the search.

Topological Primes

The combinatorial concept of cartesian product allows decomposition of topological structure. Thus, a cylinder is the cartesian product of the circle and the line segment.

This provides for what might be called "topological primes" -- comparable to prime numbers in arithmetic wich form composite numbers. Thus, every mathematical structure may be considered as formed of the basic components:

vertex;
edge;
face;
hole;
cross-cap.

See Also:

Riemann's Hypothesis: A Pure Love of Math

Michael Atiyah and the Geometry

Michael Atiyah:

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should
never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’.

I think it is hard sometimes to keep a global view about all the things that are included in this process, yet at some level, such geometries would have to be explained in relation, between the very small, and the every large.

So how would you take this advancement of all that Michael has talked about and included it in a real world picture? I am trying to answer this but am having difficulties. The words in support of the geometries had to be included in that global perspective.

plato Says:

January 10th, 2006 at 2:20 pm
Your censorship of legitmate questions is not a very good practise.

If one had developed in that bulk perspective one would have gained in results, the question would have revealled this but you are are to quick with the button, protecting your point of view.

Has nothing to do with keeping the thread on track.

You'll notice which one he kept?:)

This was posted on Peter Woit's comment section that is censored continously serves no one but those who have drawn the line and any relation to the valuations attributed to bulk perspective. While I have been moving to this feature held in context of experimental basis developed in LHC and RHIC features, what pray tell has the extra dimesional scenario given to us, as we move beyond the idealization that Dirac gave to us for viewing in his equative understandings? Where is this beginning?

It was much more then this and the viewing with which I have become so intrigued that runs through the vein of all our discussions. This is the commonality as I am discovering, that has to have some basis, so that if you talk about "topology" how would this be comparative to our viewings of the universe and the events within it?

Do you see comparative functions as we relay our veiws of the microstate world and how such developements could have lead us all through GR to have come face to face with strangelets?

The bulk perspective is being exemplified, whether you are a Peter Woit or not, who wants to limit these views by casting doubt on any roads that lead beyond GR to the understanding of the inclusion of the microstate valutions from a geometrical consistancy. That comes from, the beginning and end.

If I keep the universe in perspective, am I holding the global perspective and including all that we have learnt. I certainly hope so.

We have to have explanation of the dark matter/energy scenarios do we not?

Special holonomy manifolds in string theory

So what instigated my topic today and Hypercharge make sits way for me to reconsider, so while doing this the idea of geoemtries and th eway in which we see this uiverse held to the nature of it's origination are moving me to consider how we see in ths geometrical sense.

The resurgence of ideas about the geometries taking place are intriguing models to me of those brought back for viewing in the Sylvester surfaces and B field relations held in context of the models found in the >Wunderkammern.

This paragraph above should orientate perception for us a bit around methods used to see in ways that we had not seen before. This is always very fascinating to me. What you see below for mind bending, helps one to orientate these same views on a surface.



Hw would you translate point on a two dimensional surface to such features on the items of interest on these models proposed?



Part of my efforts at comprehension require imaging that will help push perspective. In this way, better insight to such claims and model methods used, to create insight into how we might see those extra 10 dimensions, fold into the four we know and love.



G -> H -> ... -> SU(3) x SU(2) x U(1) -> SU(3) x U(1).

Here, each arrow represents a symmetry breaking phase transition where matter changes form and the groups - G, H, SU(3), etc. - represent the different types of matter, specifically the symmetries that the matter exhibits and they are associated with the different fundamental forces of nature

If one held such views from the expansitory revelation, that our universe implies at these subtle levels a quantum nature, then how well has our eyes focused not only on the larger issues cosmology plays, but also, on how little things become part and parcel of this wider view? That the quantum natures are very spread, out as ths expansion takes place, they collpase to comsic string models or a sinstantaneous lightning strikes across thei universe from bubbles states that arose from what?

So knowing that such features of "spherical relation" extended beyond the normal coordinates, and seeing this whole issue contained within a larger sphere of influence(our universe), gives meaning to the dynamical nature of what was once of value, as it arose from a supersymmetrical valuation from the origination of this universe? If Any symmetry breaking unfolds, how shall we see in context of spheres and rotations within this larger sphere, when we see how the dynamcial propertties of bubbles become one of the universes as it is today? Genus figures that arise in a geometrodynamcial sense? What are these dynacis within context of the sphere?



So as I demonstrate the ways in which our vision is being prep for thinking, in relation to the models held in contrast to the nature of our universe, how are we seeing, if we are moving them to compact states of existance, all the while we are speaking to the very valuation of the origination of this same universe?



Holonomy (30 Dec 2005 Wiki)

Riemannian manifolds with special holonomy play an important role in string theory compactifications. This is because special holonomy manifolds admit covariantly constant (parallel) spinors and thus preserve some fraction of the original supersymmetry. Most important are compactifications on Calabi-Yau manifolds with SU(2) or SU(3) holonomy. Also important are compactifications on G2 manifolds.

Special Lagrangian geometry

Dr. Mark Haskins

On a wider class of complex manifolds - the so-called Calabi-Yau manifolds - there is also a natural notion of special Lagrangian geometry. Since the late 1980s these Calabi-Yau manifolds have played a prominent role in developments in High Energy Physics and String Theory. In the late 1990s it was realized that calibrated geometries play a fundamental role in the physical theory, and calibrated geometries have become synonymous with "Branes" and "Supersymmetry".

Special Lagrangian geometry in particular was seen to be related to another String Theory inspired phemonenon, "Mirror Symmetry". Strominger, Yau and Zaslow conjectured that mirror symmetry could be explained by studying moduli spaces arising from special Lagrangian geometry.

This conjecture stimulated much work by mathematicians, but a lot still remains to be done. A central problem is to understand what kinds of singularities can form in families of smooth special Lagrangian submanifolds. A starting point for this is to study the simplest models for singular special Lagrangian varieties, namely cones with an isolated singularity. My research in this area ([2], [4], [6]) has focused on understanding such cones especially in dimension three, which also corresponds to the most physically relevant case.

I am execising the geometrical tendencies here in how Sylvester surfaces might have revealled the interior space of a Reimann sphere( Calabi Yau rotations exemplified and complete), while these points located on the sphere's surface, brane, reveal a deeper interactive force within this sphere. Again I am learning to see here, hopefully it's right. The bloggers out there who work in this direction are most helpful, P.P Cook, Lubos Motl and others, who help point the way.

Differences in the gravitational forces speak directly to dimensional relevances In Lagrangian, by association to the energy valuations? Euclids postulate from 1-4, had to be entertained in a new way, from a non-euclidean world of higher dimensions? It was well evident that supergravity, would find solace in the four dimensional relevances of spacetime? How did Kaluza and Klein get there? Cylinders?

Yet the dynamical world of the way in which the satelitte can move through space helps one to adjust to how these dynamcial avenues can propel this satelitte through that same space. Circular orits chaotically predictable, yet quite diverse shown in the poincare model representation, shows how bizzare the ability of the Lagrangian points become. Can one see well with this new abstractual quality?

Einstein's equations connect matter and energy (the right-hand side) with the geometry of spacetime (the left-hand side). Each superscript stands for one of the 4 coordinates of spacetime; so what looks like one equation is actually 4 x 4 = 16 equations. But since some are repeated there are really 10 equations. Contrast this with the single gravitational law of Newton! That alone gives a hint of the complexity of these equations. Indeed, they are amongst the most difficult equations in science. Happily, however, some exact solutions have been found. Below we discuss one such exact solution, the first, found in 1916 by Karl Schwarzchild.

So it was important to understand how this view was developed further. The semantics of mathematical expression was a well laid out path that worked to further our views of what could have been accompished in the world of spacetime, yet well knowing, that the dynamcial revealled a even greater potential?



So now you engaged the views inside and out, about bubble natures, and from this, a idea that is driven. That while Michio Kaku sees well from perspective, the bridge stood upon, is the same greater comprehension about abstract and dynamical processes in that same geometrical world. Beyond the sphere, within the sphere, and the relationship between both worlds, upon Lagrangian perspective not limited.

Placed within the sphere, and this view from a point is a amazing unfoldment process of views that topological inferences to torus derivtives from boson expressed gravitational idealizations removed themself from the lines of circles to greater KK tower representations?

The following is a description of some of the models for the hyperbolic plane. In order to understand the descriptions, refer to the figures. They may seem a bit strange. However, a result due to Hilbert says that it is impossible to smoothly embed the hyperbolic plane in Euclidean three-space using the usual Euclidean geometry. (Technical note: In fact it is possible to have a C^1 embedding into R^3, according to a 1955 construction of Nicolaas Kuiper, but according to William Thurston, the result would be "incredibly unwieldy, and pretty much useless in the study of the surface's intrinsic geometry."[William Thurston, "Three Dimensional Geometry and Topology," Geometry Center Preprint, 1991, p.43.]) Since there is no such smooth embedding, any model of the hyperbolic plane has to use a different geometry. In other words, we must redefine words like point, line, distance, and angle in order to have a surface in which the parallel postulate fails, but which still satisfies Euclid's postulates 1-4 (stated in the previous article). Here are brief descriptions of three models:

This process had to be thought of in another way? Point, line, plane, became something else, in terms of string world? M theory had to answer to the ideas of supergravity? How so? Great Circles and such? Topological torus forms defined, inside and out? Completed, when the circle become a boson expressed? A point on a brane now becomes something larger in perspectve? Thanks Ramond.

Three Sphere

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

— Nikolai Lobachevsky

Bernhard Riemann once claimed: "The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean." His prophesy was realized later with Einstein's general theory of relativity. It is futile to expect one "correct geometry" as is evident in the dispute as to whether elliptical, Euclidean or hyperbolic geometry is the "best" model for our universe. Henri Poincaré, in Science and Hypothesis (New York: Dover, 1952, pp. 49-50) expressed it this way.

A sphere is, roughly speaking, a ball-shaped object. In mathematics, a sphere comprises only the surface of the ball, and is therefore hollow. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball).

More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.

Spheres can be generalized to higher dimensions. For any natural number n, an n-sphere is the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number. Here, the choice of number reflects the dimension of the sphere as a manifold.

a 0-sphere is a pair of points

a 1-sphere is a circle

a 2-sphere is an ordinary sphere

a 3-sphere is a sphere in 4-dimensional Euclidean space


Spheres for n ¡Ý 3 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere. The notation Sn is also often used to denote any set with a given structure (topological space, topological manifold, smooth manifold, etc.) identical (homeomorphic, diffeomorphic, etc.) to the structure of Sn above.

An n-sphere is an example of a compact n-manifold.

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. A regular sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, R3. A 3-sphere consists of all points equidistant from a single point in R4. Whereas a 2-sphere is a smooth 2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold.

In an entirely analogous manner one can define higher-dimensional spheres called hyperspheres or n-spheres. Such objects are n-dimensional manifolds.

Some people refer to a 3-sphere as a glome from the Latin word glomus meaning ball.

Poincare Conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not...

In mathematics, the Poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. It is widely considered to be the most important unsolved problem in topology.

The Poincaré conjecture is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute is offering a $1,000,000 prize for a correct solution. As of 2004 it is becoming accepted that a proof offered by Grigori Perelman in 2002 may have disposed of this question, after nearly a century. Perelman's work is still under review.

Artists such as M. C. Escher have become fascinated with the Poincaré model of hyperbolic geometry and he composed a series of "Circle Limit" illustrations of a hyperbolic universe. In Figure 17.a he uses the backbones of the flying fish as "straight lines", being segments of circles orthogonal to his fundamental circle. In Figure 17.b he does the same with angels and devils. Besides artists and astronomers, many scholars have been shaken by non-Euclidean geometry. Euclidean geometry had been so universally accepted as an eternal and absolute truth that scholars believed they could also find absolute standards in human behavior, in law, ethics, government and economics. The discovery of non-Euclidean geometry shocked them into understanding their error in expecting to determine the "perfect state" by reasoning alone.