Mersenne Prime: One < the Power of two

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.)

This is part of the education of my learning to understand the implications of the work of Riemann in context of the Riemann Hypothesis. Part of understanding what this application can do in terms helping us to see what has developed "from abstractions of mathematics," to have us now engaged in the "real world" of computation.

In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two multiplied by itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 10000...0, just like a power of ten in the decimal system.

Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of n is the number of ways the bits in a binary integer of length n can be arranged, and thus numbers that are one less than a power of two denote the upper bounds of integers in binary computers (one less because 0, not 1, is used as the lower bound). As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system, might limit the score or the number of items the player can hold to 255 — the result of a byte, which is 8 bits long, being used to store the number, giving a maximum value of 28−1 = 255.

I look forward to the help in terms of learning to understand this "ability of the mind" to envision the dynamical nature of the abstract. To help us develop, "the models of physics" in our thinking. To learn, about what is natural in our world, and the "mathematical patterns" that lie underneath them.

What use the mind's attempt to see mathematics in such models?

"Brane world thinking" that has a basis in Ramanujan modular forms, as a depiction of those brane surface workings? That such a diversion would "force the mind" into other "abstract realms" to ask, "what curvatures could do" in terms of a "negative expressive" state in that abstract world.

Are our minds forced to cope with the "quantum dynamical world of cosmology" while we think about what was plain in Einstein's world of GR, while we witness the large scale "curvature parameters" being demonstrated for us, on such gravitational look to the cosmological scale.

Mersenne Prime


Marin Mersenne, 1588 - 1648

In mathematics, a Mersenne number is a number that is one less than a power of two.

Mn = 2n − 1.
A Mersenne prime is a Mersenne number that is a prime number. It is necessary for n to be prime for 2n − 1 to be prime, but the converse is not true. Many mathematicians prefer the definition that n has to be a prime number.

For example, 31 = 25 − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 211 − 1 is not a prime because it is divisible by 89 and 23. And 24 -1 = 15 can be shown to be composite because 4 is not prime.

Throughout modern times, the largest known prime number has very often been a Mersenne prime. Most sources restrict the term Mersenne number to where n is prime, as all Mersenne primes must be of this form as seen below.

Mersenne primes have a close connection to perfect numbers, which are numbers equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. In the 18th century, Leonhard Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist (any that do have to belong to a significant number of special forms).

It is currently unknown whether there is an infinite number of Mersenne primes.

The binary representation of 2n − 1 is n repetitions of the digit 1, making it a base-2 repunit. For example, 25 − 1 = 11111 in binary

So while we have learnt from Ulam's Spiral, that the discussion could lead too a greater comprehension. It is by dialogue, that one can move forward, and that lack of direction seems to hold one's world to limits, not seen and known beyond what's it like apart from the safe and security of home.

Images or Numbers By Themself

“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).

I have an idea in mind here that will be slow to show because I am not sure how it is supposed to be laid out. So maybe by showing these numbers by them self? What use, if one did not, or was not able to see in another way?


Figure 22.10: Double slit diffraction


I looked at the "straight lines" of Thomas Young's trajectories of photon emission and while quite understandably shown to be of consequence in this post "Interference." I was more interested in how something could start off in one place and do this rotation of sorts, and then come back for examination again in the real world. The Spectrum

Plato:

What a novel idea to have the methods used by the predecessors like Maxwell, to have been united from Faraday's principals? To have Maxwell's equation Gaussian in interpretation of Riemann geometry, somehow, united by the geometries of Einstein and defined as gravity?

But it is also in mind "that the image" has to be put here also before the numbers can show them self. What use these numbers if I do not transcend them to what they can imply in images, to know that the thinking here has to be orientated in such a way that what was simple and straight forward, could have non-euclidean orientations about it?


Michael Faraday (September 22, 1791 – August 25, 1867) was a British scientist (a physicist and chemist) who contributed significantly to the fields of electromagnetism and electrochemistry.

So one reads history in a lot of ways to learn of what has manifested into todays thinking. What lead from "Gaussian coordinates in an "non-euclidean way" to know that it had it's relation in today's physics. To have it included in how we see the consequences of GR in the world. It had been brought together for our eyes in what the photon can do in the gravitational field.

Our Evolution to Images


The Albrecht Durer's Magic Square


Ulam's Spiral


Pascal's Triangle

Evolve to What?

Who was to know what Leonard Susskind was thinking when his mathematical mind was engaged in seeing this "rubber band" had some other comparative abstraction, as something of consequence in our world. Yet, people focus on what they like to focus on, other then what "lead the mind" to think the way they do?


Poincaré 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......

I have to rest now.

Visual Abstraction to Equations

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

Some of you might have noticed the reference to the Ashmolean Museum?


Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.

It seems only the good scientist John Baez had epitomes the construction of the Platonic solids? A revision then, of the "time line of history" and the correction he himself had to make? Let's not be to arrogant to know that once we understand more and look at "the anomalies" it forces us to revise our assessments.

The Art form

I relayed this image and quote below on Clifford's site to encourage the thinking of young people into an art form that is truly amazing to me. Yes I get excited about it after having learnt of Gauss and Reimann's exceptional abilities to move into the non euclidean world.

Some think me a crackpot here? If you did not follow the history then how would you know to also include the "physics of approach," as well? Also, some might ask what use "this ability to see the visual abstraction" and I think this art form is in a way destined, to what was kept in glass cabinets and such, even while the glass cabinet in analogy is held in the brain/space of them) who have developed such artistic abilities.

It's as if you move past the layers of the evolution of the human being(brain casings) and it evolution and the field that surrounds them. Having accomplished the intellect( your equations and such), has now moved into the world of imagery. Closet to this is the emotive field which circumvents our perspective on the greater potential of the world in the amazing thought forms of imagery. This move outward, varies for each of us from time to time. Some who are focused in which ever area can move beyond them. This paragraph just written is what would be considered crackpot(I dislike that word)because of the long years of research I had gone through to arrive at this point.

Of course, those views above are different.

Mapping



Is it illusionary or delusional, and having looked at the Clebsch's Diagonal Surface below, how is it that "abstraction" written?

The enthusiasm that characterized such collections was captured by Francis Bacon [1, p. 247], who ironically advised "learned gentlemen" of the era to assemble within "a small compass a model of the universal made private", building

... a goodly, huge cabinet, wherein whatsoever the hand of man by exquisite art or engine has made rare in stuff, form or motion; whatsoever singularity, chance, and the shuffle of things hath produced; whatsoever Nature has wrought in things that want life and may be kept; shall be sorted and included.

There is no doubt that the long road to understanding science is the prerequisite to mapping the images from an equation's signs and symbols. While not sitting in the classroom of the teachers it was necessary to try and move into the fifth dimensional referencing of our computer screen to see what is being extolled here not just in image development, but of what the physics is doing in relation.

In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27

So of course to be the historical journey was established like most things, Mandelstam current and what is happening there as an interlude, as well as helping to establish some understanding of the abstractions that had been developed.



But yes, before moving to current day imagery and abstraction, I had to understand how these developments were being tackled in today's theoretical sciences.

Gravity and Electromagnetism?

"Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimensionality, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality." from Flatland, by E. A. Abbott

Oskar Klein and Theodor Franz Eduard Kaluza

What a novel idea to have the methods used by the predecessors like Maxwell, to have been united from Faraday's principals? To have Maxwell's equation Gaussian in interpretation of Riemann geometry, somehow, united by the geometries of Einstein and defined as gravity?

Then, to have Gravity and Light United?

A black hole is an object so massive that even light cannot escape from it. This requires the idea of a gravitational mass for a photon, which then allows the calculation of an escape energy for an object of that mass. When the escape energy is equal to the photon energy, the implication is that the object is a "black hole."

It seems then that the very statement of "Unification," the "Theory of everything," does not seem so far fetched as we look at the implications of what comes after. What comes from the knowledge, extended.



I was starting to loose hope here in the efforts of blogging as well, and was thinking that the time had come to a end. But "these questions" help to fuel the understanding that I had gained by giving time to "what work" has been put out there by scientists?



To think scientists would close up shop to their elite view, would seem disastrous to me, because of the leading perspective of what the physics means along side of that math.

We need to know what is "experimentally going" on so that we can also judge what theoretical models are doing for us as we extend this knowledge gained.

I gave a few views in environmental sciences in terms of the cosmic relation as well as what Gr was being introduced using time clocks and such, for views of the topographical understanding of earth from a fluidness point of view.

Now join the "cloud cover" along side of particle collisions sources, and have we learn anything that we didn't know before, or has this push new light onto what we now see of earth, as it's placed in the cosmological frontier?

Quantum Hall Effect

This article below was set in motion by Stefan's article,"Pencils, Black Holes, and the Klein Paradox", at Backreaction. B will have to offer her perspective on the blackhole analogy. I offer mine.:)

The fractional quantum Hall effect continues to be influential in theories about topological order.

It is interesting to see the interconnecting links of recent between the different blogs on the internet in terms of what information is being relayed back and forth without some understading of what is going on?

Number theory is the type of math that describes the swirl in the head of a sunflower and the curve of a chambered nautilus. Bhargava says it's also hidden in the rhythms of classical Indian music, which is both mathematical and improvisational. He sees close links between his two loves -- both create beauty and elegance by weaving together seemingly unconnected ideas.

As part of a Morning Edition series exploring the intersection of art and science, NPR's Richard Harris reports on the beauty of mathematics, its ties to art -- and the man who straddles both worlds.

So you learn to see relations where one might not of have before. "Computerization techniques" that would help us understand new ways in which transmit information?

An Ultimate Theory in Physics?

Shahn Majid's research explores the world of quantum geometry, on the frontier between pure mathematics and the foundations of theoretical physics. He uses mathematical structures from algebra and category theory to develop ideas concerning the structure of space and time. His research philosophy drives a search for the right mathematical language for a unified expression for the ideas of quantum physics, founded on the notion of non-commutative geometry

While above I may have introduced the particular interest of Majid's in terms of beats in nature and number counting, it is with some understanding that "poetical desire" can have come "other issues" which rise up from schemas of nature?

The subject in its modern form has also been connected with developments in several different fields of both pure mathematics and mathematical physics. In mathematics these include fruitful interactions with analysis, number theory, category theory and representation theory. In mathematical physics, developments include the quantum Hall effect, applications to the standard model in particle physics and to renormalization in quantum field theory, models of spacetimes with noncommuting coordinates. Noncommutative geometry also appears naturally in string/M-theory. The programme will be devoted to bringing together these different streams and instances of noncommutative geometry, as well as identifying new emerging directions

So I mean if you are into the Riemann Hypothesis, you might wonder how such patterns sought by Ulam would have been of interest to people like Robert Laughlin and his ideas on "emergence." What "number systems" would arise from the first principle?

In a Pascalian sense" you might understand this now, as isssuing from some inherent "ordered" chaos?

Ulam's interest was on a high energy event( we know what that was, don't we?)? So what order can come out of such chaos?




This is the essence of the problems with transmitting information while paying witness to the origins of the math brought forward to the mind's eye from an understanding of the "birthing of new universes?"

Update:

I never saw his "site topic Monday, September 04, 2006 until yesterday "after" constructing my post.

Links to "previous posts linked in quantum hall effect" should give some idea about previous knowledge regardless of PP Cook's posting. Just wanted to set that straight.

See:

P. P. Cooks, "To Commute or not to Commute..."

Supersymmetry<->Simplistically<-> Entropically Designed?

So of course I am troubled by my inexperience, as well as, the interests of what could have been produced in the "new computers" of the future? So in some weird sense how would you wrap the dynamics of what lead to "Moore's law" and find that this consideration is now in trouble? While having wrapped the "potential chaoticness" in a systemic feature here as deterministic? Is this apporpriate?

In the presence of gravitational field (or, in general, of any potential field) the molecules of gas are acted upon by the gravitational forces. As a result the concentration of gas molecules is not the same at various points of the space and described by Boltzman distribution law:

What happens exponetially in recognizing the avenues first debated between what was a consequence of "two paths," One that would be more then likely "a bizzare" while some would have consider the other, the cathedral? Leftists should not be punished Lubos:)

So what is Chaos then?

The roots of chaos theory date back to about 1900, in the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. Poincaré found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, who was perhaps the first pure mathematician to study nonlinear dynamics, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

13:30 Lecture
Edward Norton Lorenz
Laureate in Basic Sciences
“How Good Can Weather Forecasting Become ? – The Star of a Theory”

So this talk then is taken to "another level" and the distinctions of WeB 2.0 raised it's head, and of course, if you read the exponential growth highlghted in communities desemmination of all information, how could it be only Web 1.0 if held to Netscape design?



I mean definitely, if we were to consider "the Pascalian triangle" and the emergence of the numbered systems, what said the Riemann Hypothesis would not have emerged also? The "marble drop" as some inclusive designation of the development of curves in society, that were once raised from "an idea" drawn, from some place?

Types of Blogging Software

Ask yourself this? What is the new kernel to be, if we had for one moment presented the opportunities for the using Riemann hypothesis, and contained the very idea as a philosophy presented within this blog?

A VIEW OF MATHEMATICS by Alain CONNES

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.

Would such a "paradigmal change" allow for insightual software development to take a turn for the better if the understanding existed, that one had already left the cave, and saw the aspects of probable outcomes, as more then the primes and it's integrations with physics mentality, along with theoretical development?

Micro-quantum structures that are exemplfiled, in Monte Carlo methods?

Are we "FREE" to Express?

While I have enjoyed the blogging experience of Blogger.com, and the integration of development that had been going on, the questions remain, as to where this information is deposited and how the moderation of "such a tool" is enforced?

Like many important concepts, Web 2.0 doesn't have a hard boundary, but rather, a gravitational core. You can visualize Web 2.0 as a set of principles and practices that tie together a veritable solar system of sites that demonstrate some or all of those principles, at a varying distance from that core.

I have a certain ideology about trying to bring together as much information as possible, by asking, if image linking, and phrase connections, do not involve copyright infringements, and allow the versatility of blogging experience, while respecting the owners of images and wording, while connected directly to their source.

Linux is subversive. Who would have thought even five years ago (1991) that a world-class operating system could coalesce as if by magic out of part-time hacking by several thousand developers scattered all over the planet, connected only by the tenuous strands of the Internet?

Certainly not I. By the time Linux swam onto my radar screen in early 1993, I had already been involved in Unix and open-source development for ten years. I was one of the first GNU contributors in the mid-1980s. I had released a good deal of open-source software onto the net, developing or co-developing several programs (nethack, Emacs's VC and GUD modes, xlife, and others) that are still in wide use today. I thought I knew how it was done.

Linux overturned much of what I thought I knew. I had been preaching the Unix gospel of small tools, rapid prototyping and evolutionary programming for years. But I also believed there was a certain critical complexity above which a more centralized, a priori approach was required. I believed that the most important software (operating systems and really large tools like the Emacs programming editor) needed to be built like cathedrals, carefully crafted by individual wizards or small bands of mages working in splendid isolation, with no beta to be released before its time.

This has been on my mind as I brought together many aspects of the information that is out there. From the respectable information posted by scientists and their personal experiences, to those shared by all, through such blogging experiences. So what was the battle brewing about from those early days and the struggle to develope communities, sharing information, and who are these people today?

AOL=Netscape? Microsoft? Google? Yahoo?

How would such blogging experiences allow the movement forward of society, and the thinking brain, this internet has become?

Are there concerns, that the human being once exposed to the vastness of this information, could bring it together in such a way, as to insight the "new idea" that would forward research and developement? Encourage our minds to percieve in other ways that we are not accustom? I gave an example at the very beginning of this post in regards to the Riemann Hypothesis.


Witten:

One thing I can tell you, though, is that most string theorist's suspect that spacetime is a emergent Phenomena in the language of condensed matter physics.

This is important to ask, because if such an ability is focused through the individuals efforts using such a medium, how could/would it be exploited, that it could be brought to the forefront of the "thinking brain/internet" and find indeed, that such information is useful?

Meddle then in the internal structure and enforce the rights of deposition as to the respository, and deal with it as you like?

The information depository costs money, I know? Image transference costs money. Then how shall "the dream of the thinking mind" ask, that if the repositories are the resources held in abeyance, until used as seen fit, then why not/should disrupt the information gathering and make it disjointed, while we/you look at it? Before it reveals it's state secret? An open society, right? People who are free?

Robert Laughlin:

Likewise, if the very fabric of the Universe is in a quantum-critical state, then the "stuff" that underlies reality is totally irrelevant-it could be anything, says Laughlin. Even if the string theorists show that strings can give rise to the matter and natural laws we know, they won't have proved that strings are the answer-merely one of the infinite number of possible answers. It could as well be pool balls or Lego bricks or drunk sergeant majors.

This would mean that the very ideas of the internet explosion and control of it becomes in question, as well as, the provders we use to express ourselves on the internet?

How will these repositiories change then in technologies that you and I are very quickly connected in ways that the human mind/internet becomes quite capable of seeing, in ways it is not accustom?

What revolution/paradigmalchange will then happen, that the very experiences we now enjoy, will be defuncedt with all the software solutons to metigate the ability for the individual to do what any of us can do freely, without any ofthe blogging software now demonstrated below?

Shall we choose carefully, read the requirements of, and what conclusion have you reached?

b2evolution

bBlog

Blogger

Bloxsom

Blojsom

Drupal

ExpressionEngine

Geeklog

Greymatter

iUpload

LifeType

LiveJournal

Movable Type

MvBlog

Nucleus CMS

PostNuke

Roller Weblogger

Serendipity

Slash

TypePad

Typo

TYPO3

WordPress

Xanga

If we are looking for the new "idea" where shall it arise from then? It is apparent that the early thinking in cosmology has been changed(to include strings ina time sequence of events evn thoguh they be micro seconds) and so too, the values of measure in "time," recognized as problematic, in terms of it's discrete value, when it is very well understood that continuity of expression can be very smooth(yet is it?)?

The count of Primes begins in Chaos. If we were to think of the Riemann Hypothesis assigned to a scale as an approximation to the prime distribution function, then how woud any pattern suffice to be an "emergent property" of that chaos?

Past Life Bleed Throughs?

In studying over the many years, certain concepts and ideas brought from other cultures stayed with me of course, while I explored the dream world enviroment on my own.

The idea here is that similar patterns used in construction of mandalic interpetations, serve to illustrate model applications, that when bleeding, overlap experiences, whether they be drean information gathered, or, historical correlations, these struck me as significant.

This is not to say that all people who dream will be good mathematicians, just that the road to cogitive realization held in context of Ramanujan should be seen as the developing subsconscious as a very important tool in moving the cogitive revelations that the creative mind can utilize, in developing math models for the future?

THatpattern had to be understood a dLimincentric structures as well. If you didn't you would not have seen the importance of the schematics that Liminocentric structures reveal of themself as mandalic models of intepretation.



See:

A Pure Love of Math

A Tribute to Hinduism

The inexhaustible Ramanujan was an observant Hindu, adept at dream interpretation and astrology. His work was marked by bold leaps and gut feelings. Growing up he had learned to worship Namagiri, the consort of the lion god Narasimha. Ramanujan believed that he existed to serve as Namagiri´s champion - Hindu Goddess of creativity. In real life Ramanujan told people that Namagiri visited him in his dreams and wrote equations on his tongue.

Ramanujan could never explain to G H Hardy how he arrived at his deep insights in mathematical terms; but he did say many of his discoveries came to him in dreams, from the goddess Namakkal, and that he had a morning ritual of awakening and writing them down.

Having heard of Ramanujan method of information gathering, it was interesting from the standpoint of how the consciousness could organzed itself and bring forth abstract notions(jumbling experiences in that dream world), to have the method of mathematical explanation bear tangible proof in our reality now.

This is of course are some of the ideas I have around cognitive recognitions of getting to the source of creation and bringing back tangible equative solutions.



Perhaps it is of some use for a example to better understand this?

While journalling for many years and recording my dreams, I came up with what I called "dream poetry," because I wanted to try and catch the rhythmn and story behind it as best I could without subjecting it to the scrutinization of my mind and analysis.

Of course having a pen and paper beside the bed was always good advice. There was always a definite difference in the lucidness of those dreams, that seemed to indicate this past life referencing, then the normal dream periods.

Also by doing this "awareness of the dream time" it became very pronouced in terms of the recall before getting busy with the day.



Mandalas:

Since ancient times, Mandalas have been used for meditation and healing. In old Tibet, Mandalas were used in the process of training the mind to be quiet, so that That Still Place Within was more easily accessible.

It was said that the Divine lives at the center of the Mandala. With that in mind, the Mandala was used as a focus point for meditation.

As the mind quiets and becomes still, the meditator finds the Divine at the center of his own being.

I read about this use of Mandalas years ago. When this series of designs came through me, I realized that that was the intent within them: that the Divine would reside at each design's center.

Research I've done indicates that Mandalas are the visual forms of the resonance traces of specific frequencies of sound/light. Some pretty amazing research done by Hans Jenny shows that if sound is transmitted through a thin layer of sand on a metal plate, the sand vibrates in such a manner as to create designs in the sand, the patterns shifting with the sounds transmitted.

If one wanted to pursue it further then it makes sense that the Chaldni plate becomes a interesting piece of equipment in analyzing the nature of such vibrations. A balloon dipped in a dye solution and subjected to different sounds, has an interestig perspective to it.

Nodes and anti-nodes



On beautiful mandalic designs see:

Flower Mandalas Project

On Gauss's Day of Reckoning

A famous story about the boy wonder of mathematics has taken on a life of its own -Brian Hayes

Illustration by Theoni Pappas

In a fanciful drawing done in the manner of a woodcut, the young Carl Friedrich Gauss receives instruction in arithmetic from the schoolmaster J. G. Büttner. As the story goes, Gauss was about to give Büttner a lesson in mathematical creativity.

To me the historical significance of this research is important to me. People could chastise me for saying that the research I do has no quality, then what should be assumed with scientific credentials? Still the romance I have for such abstractions and development of thinking is important just the same. It is about creativity to me, and looking back to the ingenuity of thought, is something I can see in everyone. One doesn't have to consider them self less then, just by being the student that would solve the problem, while insight and acute perception, might have been revealed in one who could throw down the slate the quickest.

The story is fascinating tome on a lot of different levels and to tracking down the essence of what we see passed from one hand to another, and how this ambiguity might creep in and additions make there way for added material.

I understand this in our response to writing science, with what language is supposed to be. Sure talk about chinese , Italian, Latin of ole, and we want to know what the truest expression of the language should be?

Of course this is the responsibility of math, that a common basis be found, between all languages, that the source would have described it so abstract/yet closest to the center of the circle) that all would understand and could work the abstract nature of this math.

I feel guilty, that I cannot contribute so well to this math language, that I strive to listen very good to the concepts espoused, as close as possible to the development of this Algebraic way of seeing.

Yes, it is as important, as the geometrical seeing that it be inherent in the way things abstractly can be seen. That both would have supported the continued work fo science.

Mine then is the student's plight in a vast world that I exist away from, yet, try and stay as close as I can to learn.

Do I sanction everyones abilities away from this in character, is no less then the character I assume, and has been treated. That respect be given, might have found the truer calling of sharing the insights, be as truthful as possible. We should all strive to this of course.

What is Swirling in my Mind

As I lay there many things float through my mind about how we are seeing things now.

So the article above sparked some thoughts here about Sylvestor surfaces and B field understandings, that also included Lagrangian perspective along with WMAP polarization mapping. All these things seem so disconnected?

I keep finding myself trying to wrap all of this in a gravitational perspective as it should , no less important then gauss contributions, hidden for a time, while the student of his brings the perspective for us all to see. So how familiar is protege as Riemann that his Hypothesis is so much the like of the numbers apparent, as in the youthful gaze of the student challenged.

Hyperbolic Geometry and it's Rise

Omar Khayyám the mathematician(6 april 2006 Wikipedia)

He was famous during his lifetime as a mathematician, well known for inventing the method of solving cubic equations by intersecting a parabola with a circle. Although his approach at achieving this had earlier been attempted by Menaechmus and others, Khayyám provided a generalization extending it to all cubics. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry.

Giovanni Girolamo Saccheri(6 April 2006 Wikipedia)

Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin from 1694 to 1697, and philosophy, theology, and mathematics at Pavia from 1697 until his death. He was a protege of the mathematician Tommaso Ceva and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).

Of course the question as to "Victorian" was on mind. Is non-euclidean held to a time frame, or not?

Victorian Era(wikipedia 6 April 2006)

It is often defined as the years from 1837 to 1901

Time valuations are being thought about here. In regards too, non euclidean geometry and it's rise. Shows, many correlations within that time frame. So that was suprizing, if held to a context of the victorian socialogical time frame. But we know this statement is far from the truth?


Seminar on the History of Hyperbolic Geometry, by Greg Schreiber

We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.


History (wikipedia 6 April 2006)

Hyperbolic geometry was initially explored by Giovanni Gerolamo Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by János Bolyai, Karl Friedrich Gauss, and Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.

Quantum Mechanics: Determinism at Planck Scale

Perhaps Quantum Gravity can be Handled by thoroughly reconsidering Quantum Mechanics itself?- Gerard t' Hooft

Albert Einstein used harmonic oscillators to understand specific heats of solids and found that energy levels are quantized. This formed one of the key bridges between classical and quantum mechanics.

Can harmonic oscillators serve as a bridge between quantum mechanics and special relativity?

It is nice Paul that you continue to bring perspective forward here for consideration.

I'll hope you will supply the paragraph one day that made the lights go on for you about what you are percieving, and from what you have understood having read Einstein's words in later life. Many tend to think Einstein was unproductive in his later life?



The basis of the paper you brought forward for inspection, is really quite significant, in my views. I'll tell you what I see and from this discussion, the ideas of what the Riemann's Hypothesis might mean in the expansion of cyclical processes we might have seen in the Ulam spiral perhaps?



You have been developing that perspective for a quite a while, as your numbers attest to this expression. So what are Poincare cycles? This I'll hold off for a bit, becuase I am returning to the earlier discussion wehad about what Zero actually means. Do you remember? Perhaps you could sum it up again from our consversationin the comment section.

You describe returning to the Laughlin and the foundational perspectives, for a better look. Type in "emergence" or "first principle" into the blog search feature, would be quite productive I think.

This is a good indicator to me that the route to describing the process although very difficult in ascertaing value in the "dissapation effect" of the virtual blackhole of Hooft, what value is this insight if it did not have a basis for which it could work?

THE MATHEMATICAL BASIS FOR DETERMINISTIC QUANTUM MECHANICS by Gerard ’t Hooft

One now may turn this observation around. A closed system that can only be in a finite number of different states, making transitions at discrete time intervals, would necessarily evolve back into itself after a certain amount of time, thus exhibiting what is called a Poincar´e cycle. If there were no information loss, these Poincar´e cycles would tend to become very long, with a periodicity that would increase exponentially with the size of the system. If there is information loss, for instance in the form of some dissipation effect, a system may eventually end up in Poincar´e cycles with much shorter periodicities. Indeed, time does not have to be discrete in that case, and the physical variables may form a continuum; there could be a finite set of stable orbits such that, regardless the initial configuration, any orbit is attracted towards one of these stable orbits; they are the limit cycles.

So Hooft is explaining this for us here? Only in a "positive" expression?

Before movng onthen soemthings would have had to been made clear as far as I can tell in regards to the basis of what zero actually means.

An Energy of Empty Space?

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

Once you get to th ebulk space it is extremely hard to explain how I gothere in my visual thinking but it is true that I see dynamcial spaces and all inlcusive views of the science of this original encapsulated in a geometrical process. Whether it's right or not is another question. I know this:)

While D brane analyisis had been given to another for perspective in relation to how we see Belenstein bound and the horizon of value, being describe by CFT, we know well then that the abstraction of D brane thinking has to answer to those microscopial visonistic qualites of a very dynamcial place?

That what has happen inside the blackhole, had something else as well to consider? Anomalies in perception then exist in how we see the quark Gluon plasma in relation to the principals of superfluids.

Why molasses and ice cream production might seem important to some, while others might dismiss the childest antics of the condense matter theorist?

So while these things are happening we should know that the condition elevated to bulk persepctive would have one see graviton production, as constituents of this bulk space. This derivation placed the bulk perspectve within grasp of what the harmonic oscillator means as we move our peceptions to the flat spacetime arrived at in the production of the quark Gluon plasma, that we are so boldly talking about here in views of the langrangian space.

I see in the WMAP perspective held to analogies of the sound in polarization modes as, nodes and anti-nodes and are really interesting when held to that perspective about what we might think of in relation to how we see particle physics having undergone a model change, as well as a perspective one as well.

This is a fifth dimensional view accomplished.

See:

Quantum Harmonic Oscillator

Harmonic Oscillation

Warm Dark Matter

Big Bang Nucleosynthesis

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

A Professors Fate, or Encouragement to Continue?

In regards to #72, as a layman, I continue to wonder.

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.

ftp://ftp.alainconnes.org/maths.pdf

No one can be faulted can they for developing pespective? :) No mathematician, for solving age ole problems in regards too, let's say, "Riemann Hypothesis."

What say this view or that view, without siging the death of one professor, or the death of another, as a hand sweeping gesture. As to the "illegitamacy of math models" and their relation to the physics world, lead by science?

Who(the community of scientists) shall decide, or one man?

Not so by evidence of failure? Experimental proposals continued, justification sought by tabletop experiments, or a less then concern, of energy missing from total calculation of imput energy? Within context of the standard model wrapped, and the comlete rotation within, as the signs of the time in one graviattional consideration of it's strength, or it's weakeness? Each "cosmological event" held to this perspective?

So resulting energy calculated in particle showers, leaving amount of energy missing, points too "new physics(increased dimension perspective)?" I mean that is a generalized statement, that might have been asked while the general "state of being" had been reached, in any collidial design? How would this method of travel have been encourgae fromthe event itself in colidial view to one that had traverded through superfluid characteristics and anomalies of such design?

Sustenance of any predictable future in energy production would have asked then, how such perpetuity could be been sustained? Where is that missing energy? How is inflation motivated, if held to the current state of the universe as some cosmological constant method? So the resulting missing energy had been seen in relation to manifested possibilties as strange matter or result cerenkov radational design, of the blue daylight of our world, from the many microbial blackhole events?

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

Wolfram's Ring Tone

What makes this a little bit, well more then a bit interesting is the evaluation you might try to assign reality of the unseen. IN Mendeelev's table, I like to think in a different way, and if one held the Riemann hypothesis up for grabs, what said that any elemental consideration would have been derived from some Probabilistic evaluated state, not to have formed into views with which all nature might have embued itself?

Lubos Motl:

If you're interested in the more precise isomorphisms between the cellular automata and the anthropic principle, there is a cute analogy invented by Nima that looks as follows: the negative cosmological constant is mapped to the automata that die out (big crunch) while the large positive cosmological constant is mapped to the trivial (solvable) automata - and the nontrivial automata that don't die out represent the anthropically allowed window for the cosmological constant. :-)

You need a background for this, about how such perceptions could have arisen from the very nature that all things, will continue to vibrate even in a empty space.The quantum harmonic oscillator would have something to say to this, and if held to the very nature of flat spacetime, where would this be stopped?

Wolfram's Theory of Everything


So such expansions to the entropically large valuation of all that is in materiality, had to come from some kind of "soup of thnking" that rests itself from all the myriad forms that could have been emitted? So our thnking is "colored" then from musical interludes analogistically based. But indeed, how did you get there and you have this strict regimentation to follow in the probabilitic valution that the Pascal triangle would have surmounted, when thinking of Wolfram's work.

Plato:

How do such things make there way into reality and these prime numbers as signatures of the atoms and ways in which they would relate themself to this elemental table for viewing, and something then shifts in my perception. I don't know why? :)

How can you not help but think in new ways, once your given perspective about the ways we have always done things. A Serpinski fractoriallization about the nature of the world in a myriad of ways, and the probabilistic valuation about events in the unseen. How could they ever be captured?

Tiny Bubbles

AS a child, Einsten when given the gift of the compass, immediately reocgnized the mystery in nature? If such a impression could have instigated the work that had unfolded over timein regards to Relativity, then what work could have ever instigated the understanding of the Pea as a constant reminder of what the universe became in the mind of a child, as we sleep on it?

Hills and Valley held in context of Wayne Hu's explanations was a feasible product of the landscape to work with?

'The Princess & The Pea' from 'The Washerwoman's Child'

If Strings abhors infinities, then the "Princess's Pea" was really a creation of "three spheres" emmanating from the "fabric of spacetime?" It had to be reduced from spacetime to a three dimensional frame work?

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.

Was it really fantasy that Susskind was involved in, or was there some motivated ideas held in mathematical structure? People like to talk about him without really understandng how such geometrical propensities might have motivated his mind to consider conjectures within the physics of our world?

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.

You had to realize that working in these abstractions, such work was not to be abandon because we might have thought such abstraction to far from the tangible thinking that topologies might see of itself?


Poincaré Conjecture Proved--This Time for Real
By Eric W. Weisstein

In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953, pp. 486 and 498), Poincaré's conjecture stated that every closed simply connected three-manifold is homeomorphic to the three-sphere. Here, the three-sphere (in a topologist's sense) is simply a generalization of the familiar two-dimensional sphere (i.e., the sphere embedded in usual three-dimensional space and having a two-dimensional surface) to one dimension higher. More colloquially, Poincaré conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes. This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3.

While it is very dificult for me "to see" how such movements are characterized in those higher spaces, it is not without some understanding that such topologies and genus figures would point to the continuity of expression, as "energy and matter" related in a most curious way? Let's consider the non-discretium way in which such continuites work, shall we?

From one perspective this circle woud have some valuation to the makings of the universe in expression, would identify itself where such potenials are raised from the singular function of the circular colliders. Those extra dimensions had to have some basis to evolve too in those higher spaces for such thinking to have excelled to more then mathematical conjectures?

We can also consider donuts with more handles attached. The number of handles in a donut is its most important topological information. It is called the genus.

It might be expressed in the tubes of KK tower modes of measure? That such "differences of energies" might have held the thinking to the brane world, yet revealled a three dimensional perspective in the higher diemnsional world of bulk. These had to depart from the physics, and held in context?



Clay Institute

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. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

While three spheres has been generalized in my point of view, I am somewhat perplexed by sklar potential when thinking about torus's and a hole with using a rubber band. If the formalization of Greene's statement so far were valid then such a case of the universe emblazoning itself within some structure mathematically inclined, what would have raised all these other thoughts towards quantum geometry?

In fact, in the reciprocal language, these tiny circles are getting ever smaller as time goes by, since as R grows, 1/R shrinks. Now we seem to have really gone off the deep end. How can this possibly be true? How can a six-foot tall human being 'fit' inside such an unbelievably microscopic universe? How can a speck of a universe be physically identical to the great expanse we view in the heavens above?

(Greene, The Elegant Universe, pages 248-249)


Was our thoughts based in a wonderful world, where such purity of math structure became the basis of our expressions while speaking to the nature of the reality of our world?


Bubble Nucleation

Some people do not like to consider the context of universe and the suppositions that arose from insight drawn, and held to possibile scenario's. I like to consider these things because I am interested in how a geometical cosistancy might be born into the cyclical nature. Where such expression might hold our thinking minds.


Science and it's Geometries?


Have these already been dimissed by the physics assigned, that we now say that this scenario is not so likely? Yet we are held by the awe and spector of superfluids, whose origination might have been signalled by the gravitational collapse?

Would we be so less inclined not to think about Dirac's Sea of virtual particles to think the origination might have issued from the very warms water of mother's creative womb, nestled.

Spheres that rise from the deep waters of our thinking, to have seen the basis of all maths and geometries from the heart designed. Subjective yet in the realization of the philosophy embued, the very voice speaks only from a pure mathematical realm, and is covered by the very cloaks of one's reason?

After doing so, they realized that all inflationary theories produced open universes in the manner Turok described above(below here). In the end, they created the Hawking-Turok Instanton theory.

The process is a bit like the formation of a bubble
in a boiling pan of water...the interior of this tiny
bubble manages to turn itself into an infinite open
universe. Imagine a bubble forming and expanding at the
speed of light, so that it becomes very big, very quickly.
Now look inside the bubble.

The peculiar thing is that in such a bubble, space and time
get tangled in such a way that what we would call today's
universe would actually include the entire future of the
bubble. But because the bubble gets infinitely large in
the future, the size of 'today's universe' is actually infinite.
So an infinite,open universe is formed inside a tiny, initially
microscopic bubble.

String Theory Displays Golden Ratio Tendency?

Srinivas Ramanujan (1887-1920):

Ramanujan was a mathematician so great his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last hundred years. "His leaps of intuition confound mathematicians even today, seven decades after his death. ..the brilliant, self-taught Indian mathematician whose work contains some of the most beautiful ideas in the history of science. His legacy has endured. His twenty-one major mathematical papers are still being plumbed for their secrets, and many of his ideas are used today in cosmology and computer science. His theorems are being applied in areas - polymer chemistry, computers, cancer research - scarcely imaginable during his lifetime. His mathematical insights yet leave mathematicians baffled that anyone could divine them in the first place.'

Namagiri, the consort of the lion god Narasimha. Ramanujan believed that he existed to serve as Namagiri´s champion - Hindu Goddess of creativity. In real life Ramanujan told people that Namagiri visited him in his dreams and wrote equations on his tongue.

Artist's impression of the setup.

The disks represent the bosonic condensate density and the blue balls in the vortex core represent the fermionic density. The black line is a guide to the eye to see the wiggling of the vortex line that corresponds to a so-called Kelvin mode, which provides the bosonic part of the superstring (image and text:

http://arxiv.org/abs/cond-mat/0505055

Plato:

When I was a kid, I liked to take buttons and place a thread through them. Watching Mom, while I prep the button, she got ready to sew. I would take both ends of the thread and pull it tightly. I liked the way the button could spin/thread depending on how hard I pull the thread

I was thinking about this toy model developed for strng theory comprehension and all of a sudden the attempts by Lubos of Solving the Riemann Hypothesis came into view?

Now some of you know that such consistancies built up from the very idea of "Liminocentric structures" are always pleasing to me. Because of the energy valuations I might have associated to the "circles within circles" as ideas manifest( their degrees of manifest).



A KK tower about 1r radius valuation seen in the varing shapes of tubes? At what stage were these and what could I tell about the idea as it merged from that deep source and probabilstic value of where we all draw from.




That soothing watery world( our dream world ) of ideas that could manifest for us into nature, taken as an consequence relayed, from the continued circles of action? We are better predictors then we think? We did not know where this idea could manifest from, and what energy relations could have given such suttle thoughts repercussions in the very world they could have manifested into?

The relation and perplexing problem I had with identify how such a structure intrigued by Sklar would make it difficult to identify which circle is describing which stage of whee we are at with the innner/outer, was raised when it came to the developing the understanding and differences on how rubber bands placed over a apple, might have a different connotation, when moved over a donut?

Continuity of this action as a color vaiation would have made me then think of Mendeleev in his table of constituents, as I looked at the relation in the world of such discrete things.



Imagine the complexity of music that could be most pleasing, could also be very destructive in the "fields of thought"? I had espoused this in Plato's academy? All of this contained in the light sensation in a little music disc?

What stories indeed have we converted to light, in our apprehensions? Philosophically, I could be committed for my heresy, for all the things I might have assigned to "Heavens ephemeral qualities." Verging on the crackpotism, I know.:)

See:

Fool's Gold

Big Horn Medicne Wheel

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.