Algebraic Topology

A first course in Algebraic Topology, with emphasis on visualization, geometric intuition and simplified computations. Given by Assoc Prof N J Wildberger at UNSW. The really important aspect of a course in Algebraic Topology is that it introduces us to a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. What does this exactly mean? That is a key question... The course has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical. We try to use pictures and models to guide our understanding.

See Also:

• Algebraic Topology

Artifacts in the Exploration of Geometry

Ashmolean Museum, Oxford, UK

It should not be lost on individuals who have followed this blog, that there is a range of connection to Platonic Forms idealization, that such an artifact in Ashmolean Museum although modeled to represent a reality and constituent forming basis, it is by this choice,  that I exercised a" foundational attitude"  about what I can use to push my own perspective forward in science. What others were using.

"The Artist and his Museum"

The first public showing of the mastodon (also known as the "Mammoth", the American incognitum and the "animal de l'Ohio") took place next door to Independence Hall, the building in which both the Declaration of Independence and Constitution were finalized. The venue, known variously as Peale's Museum, the American Museum or simply as The Museum, was the remarkable product of a resourceful, versatile and passionate artist and showman, Charles Wilson Peale.

Peale (1741-1827) was born and raised in Maryland. A vocal opponent of the Stamp Act, he was effectively driven from his first trade, saddle making, when loyalist merchants cut off his credit. He turned to a traveling life of a self-taught, itinerant portrait painter. After a short apprenticeship with Benjamin West in London, Peale returned to Maryland in 1769 to paint wealthy patrons throughout the Chesapeake region.
In 1776 he moved to the largest city of the colonies, Philadelphia, in the hopes of further developing his career. Through his contacts made while serving as a captain of the Continental Army, Peale painted a remarkable assemblage of Revolutionary War figures, including the most comprehensive portrait series ever painted of George Washington. See:Charles Willson Peale's Museum

After doing quite a bit of reading over the years it is surprising what one can come across as they look at the historical perspective with artifacts which sat on shelves to curious onlookers as they examine these items.

Shown here are the models in the mathematical wunderkammer located in the Department of Mathematics at the University of Arizona. Like those in most modern mathematics departments, the collection is a combination of locally-made student and faculty projects together with a variety of commercially produced models. Sadly, a century since their Golden Age, many of the models are in disrepair and much of their documentation has been lost. However, some recent detective work, with the help of the Smithsonian Institution in Washington, has helped the department identify models by the American educators W. W. Ross and R. P. Baker in the collection.

Also see here for further thoughts on this




So you have in fact "forerunners of museums today" revealed in pursuits by individuals to catalog items according to the range of professions and undertakings. In this case, I was interested on geometrical forms as it was some interest to me that we could move our minds around in abstract spaces . I followed the surfaces of "dynamic movement"  issued forth by theoretical application. These would be,  modular forms or Genus figures of string theory, that raised my interest about the space we are working in.

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.

Now you must know that I do not have the education of the universities but this did not stop me from trying to understand what these artifacts in geometry actually represented. Where they were placed by theoreticians to represent the figurative evolution of what actual begins in this universe, from beyond time and space and arrived to a direction of expressions unfolding in the arrow of time. This was a recognition of the times in microseconds that had been "used in minutes" of Steven Weinberg.

A giddy craze was sweeping across Europe at the turn of the 17th century. The wealthy and the well-connected were hoarding things—strange things—into obsessive personal collections. Starfish, forked carrots, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, and unicorn tails were acquired eagerly and indiscriminately. Associations among these objects, if they were made at all, often reflected a collector's personal vision of an underlying natural "order". Critical taxonomy was rarely in evidence.

So this historical perspective of the artifacts moved my perspective to today and what is going on in mathematical abstraction. What are these shapes actually representing in reality? Is there such a thing once perception has been granted of the close correlative function of the description of that microscopic reality?

It would be that the mind has become capable of moving into the realm of the microscopic, that by measure of energy used, details the plethora of particle and constituents of that energy, that each artifact is leading toward ever finer issues of what began in the formation of the matter, to allow us to see it's constitutions as they are revealed today macroscopically.

Tipping LightCones and the Escape Velocity of a Photon

"Black Hole" by Tamsin van Essen Also see: Tamsin van Essen-Ceramic Design

Wonderful and creative thinking to what remains a mystery to a lot of people who do not ever get to see what cancer looks like or what it can do to one's family. This physical process, and the creative representation is a very interesting one to me.


Figure 2. Clebsch's Diagonal Surface: Wonderful.

We are told that "mathematics is that study which knows nothing of observation..." I think no statement could have been more opposite to the undoubted facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world, ...that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. ...Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another.

It also reminded me of the Wunderkammern and the move to "geometrical design," which was housed in Glass cases and for a time lost to the public eye. Sylvester surfaces is a case in point when looking at the nature of these geometrical models

I am interested in determining how one can detect a blackhole.

So the following post is in latex language that can be copied and those who have latex can place for examination. Clifford's spam checker(just recently checked and see that it was posted.) will not allow me to complete the rest of the comment entry so I will just put it here and go to bed. I tried putting in his latex sandbox(this now worked as well), but to no avail either.

Text book or not, it gives a clearer picture of what a "strong gravitational space" does to the photon.

Gravity and the Photon

The relativistic energy expression attributes a mass to any energetic particle, and for the photon

[tex]E=mc^2=hv[/tex]

The gravitational potential energy is then

[tex]\LARGE U=\frac{-GMm}r=\frac{-GMh}{rc^2}{vo}[/tex]

When the photon escapes the gravity field, it will have a different frequency

[tex]\large hv=hv_o[{1-}\frac{GM}{rc^2}] \hspace9 v=v_o[{1-}\frac{GM}{rc^2}] \hspace9 \frac{\bigtriangledown v} {v_o}={-}\frac{GM}{rc^2}[/tex]

Since it is reduced in frequency, this is called the gravitational red shift or the Einstein red shift.

--------------------------------------------------
Escape Energy for Photon

If the gravitational potential energy of the photon is exactly equal to the photon energy then

[tex]\normal hv_o=\frac{GM}{rc^2}{v_o} \hspace9 \text or r=\frac {GM}{c^2}\\ \text so if Mass M collapses to radius r a photon will be redshifted to zero frequency[/tex]

Note that this condition is independent of the frequency, and for a given mass M establishes a critical radius. Actually, Schwarzchilds's calculated gravitational radius differs from this result by a factor of 2 and is coincidently equal to the non-relativistic escape velocity expression

[tex]v_e_s_c_a_p_e_ = \sqrt {\frac{2GM}{r}} \hspace9 \\ \text which if V is set equal\\to c gives a radius r=\frac {2GM}{c^2}\hspace9 \text Schwarzchild Radius[\tex]

This equivalence is used as a mnenomic, but does not imply this is a valid way to derive the Schwarzchild Radius

You can delete from your tipping light thread. Have a nice day. I acknowledge fully I am the student. While we see tipping light cones there is an actual qualitative understanding for the determination of the blackhole in this context? By your definition you were right to let me know, how you are presenting this for better consumption and how I might be interfering with that process. So my apology (my bad).

Previously, I left a comment in relation to Susskind's thought experiment about the elephant and Bob on the back of the elephant B moving toward the horizon of the blackhole. My thoughts were about the "entanglement process" and how Alice on the back of elephant A would reveal aspects of the nature of the blackhole as elephant B move closer to that horizon.

This point, while understanding the representation of CFT in this regard, I thought it quite humorous that Susskind did in fact use the elephant as a representative thinking in relation to Quantum gravity? I do not know if people picked up on this?

IN Search of Mandelstam's Holy Grail

There are two posts that reflect the purpose of this post today. One is Clifford's linked through Lee Smolin's comment and the other, at Backreaction. Good Physics is Conflict

A lot of you may never understand the significance of the mystery that follows the thinking of the Holy Grail. Yet is it more the knowledge that can be gained from all soul's day, that on this occasion we may have called it Halloween.

We celebrated the past, in the living of today? You philosophize, while you become the thoughts of models created by science leaders shared? I do not think any have a "personality disorder" like I do:)

Lee Smolin:

Here is an example of the kind of question I found I needed a book to explore: what to think of the problems that arise from the need for higher dimensions in string theory, such as the problem of moduli stabilization and the vast number of static solutions. To approach this I read books on the early history of GR and unified field theories and learned that higher dimensional compactifications were explored many times between 1914 and 1984 and that close to the beginning these problems were appreciated and discussed by Einstein and others. I weave this story into my book because I find it useful when trying to judge how serious the present issues in string theory are to know how Einstein and many others struggled with the same issues over decades.

So of course when we think of the persons of science who walked before us (shoulders of giants), what are their whole stories, but what is evidenced to us as we read those words? So you compile your data accordingly, and from it, we say at certain spots, how are we to react to the challenge now facing us?



Stanley Mandelstam, Professor Emeritus, Particle Theory

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

As a lay person watching the debate it is difficult for me to discern the basis of these arguments. But I strive to go past what you think is surface in conduct in science's response, as some may show of themself in a reactionary pose. Should we all be so perfect, that the human condition is not also the example by which we shall progress in science?

Dealing in the Abstract



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

Of course the thinking may seem so detached from reality that one asks for some reason with which to believe anything. It required, that the history of this approached dust off models in glass cabinets, that were our early descendants of the museum today.

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.

How many of you know how to work in such abstract spaces, and know that what you are talking about has it's relationships in the physics of today? Or that, what satellites we use in measure of, have some correlation to how one may have seen "UV coordinates supplied by Gauss?"

Artifacts of the Geometrical WunderKammern

As one visits the mathematical puzzles and conjectures, what value these insights to the physics or our universe if we did not see things in this way? As artifacts of some other kind of geometrical thinking that we could then apply it to how we see the micro-perspective and macro-perspective working within the Quantum or cosmological realms?

So abstract and foreign to our eyes that we let it escape our attention while we talk about all our theoretical points of view and divergences from what is symmetrical?

In the past, new scientific discoveries, strange finds, and striking pieces of original artwork were greeted with awe and wonder. It became popular during the Renaissance to build a "cabinet of curiosities" to display a private collection of art and natural objects of which the owner was extremely proud. These groups of objects were at first housed in an actual cabinet or ornate piece of furniture, known as Wunderkammern or Wunderkabinetts. They are simultaneously pieces of furniture and the collections of items within them.

In the exhibits of these early Wunderkammern, owners might display strange, beautiful, mysterious, and precious marvels like starfish, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, Egyptian figurines, and “unicorn tails.” Rich art patrons would display their new art acquisitions in the intimate backdrop of a prized spot in an ornate carved cabinet. At Kensington castle, Sir Walter Cope is said to have displayed, “holy relics from a Spanish ship; earthen pitchers and porcelain from China; a Madonna made of feathers; a back-scratcher; a Javanese costume, Arabian coats; the horn and tail of a rhinoceros; the baubles and bells of Henry VIII's fool; and a Turkish emperor's golden seal.” The collections demonstrated manmade wonders and the diversity of God’s creations as well as a fascination with new scientific approaches to the study of natural phenomena. Each collection’s commitment to miscellany dependended on the idiosyncratic interests of the collector.

So it was again important to bring people back to the ways of geometry working in spaces, that although seemingly detached from our reality is the underlying basis of the physics involved.

Mine is a layman's perspective so I cannot say for certain that all I write here will be of value. It is up to you whether you think it important or not.

Figure 2. Clebsch's Diagonal Surface: Wonderful.

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.

The Museum when thought in context of "Platonic solids" was some what of a contention when I showed it's location in early historical context as a artifact.

Shown here are the models in the mathematical wunderkammer located in the Department of Mathematics at the University of Arizona. Like those in most modern mathematics departments, the collection is a combination of locally-made student and faculty projects together with a variety of commercially produced models. Sadly, a century since their Golden Age, many of the models are in disrepair and much of their documentation has been lost. However, some recent detective work, with the help of the Smithsonian Institution in Washington, has helped the department identify models by the American educators W. W. Ross and R. P. Baker in the collection.

Also see here for further thoughts on this

I thought it important to quickly post this so that people understand that "a glass case" can hold many things for inspection, but in this case ,I was referring to the geometrical forms. If any of these form were to show symmetry in action which of these would do so? Sylvestor surfaces?

An attempt in Latex to map these functions from a layman's perspective. What use if I cannot understand the mathematical language as it is written, yet, I can see "acrobatically" the way geometry works in space?

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.

Beyond the Dance of the Sun

When we first start facing truth, the process may be frightening, and many people run back to their old lives. But if you continue to seek truth, you will eventually be able to handle it better. In fact, you want more! It's true that many people around you now may think you are weird or even a danger to society, but you don't care. Once you've tasted the truth, you won't ever want to go back to being ignorant!

If we concretize thngs and leave no room, then other theories seem like a waste of time compared to our views? Is their no room, to see what perfection the sun has for us in it's rays?


SOHO is a project of international cooperation between ESA and NASA

A lot of times it is much easier to accept the cosmological review of the universe in such a grand scale why would we think we need something more then what is already here? What has the subject of helioseismology to do with the way Wayne Hu may look at his universe?

One of the lessons I learnt as I tried to understand how they tied together the cosmological and quantum world, was to understand that relativity only spoke to that cosmo at large, and that to think anythng more, we would have to bring quantum perspectve in line with it.

We talked lots about micro perspective and particle creation and understood that the beginning of the universe is tied directly to how we micro perceptively deal with it's origins?

You may look at the sun and then realize the dynamical nature such quantum perception reveals as this process continues to unfold for us, as long as the energy is there to support it?

Now that you have shifted your views to the "nature of the dance," I had some choreographies for you to consider.

A Microperspective of the Cosmological world.

A giddy craze was sweeping across Europe at the turn of the 17th century. The wealthy and the well-connected were hoarding things—strange things—into obsessive personal collections. Starfish, forked carrots, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, and unicorn tails were acquired eagerly and indiscriminately. Associations among these objects, if they were made at all, often reflected a collector's personal vision of an underlying natural "order". Critical taxonomy was rarely in evidence.

I'll just point towards Greg Egans animations in this article.

What are Holonomy figures

While you are seeing these dynamics in context of cosmological reviews would you discard images of the quantum world?), how is abstractness(not real?). Holding many others in thoughts about geometrical propensities from a historical course in projective geometries?

Withn context of a complete revolution, the noting of the solar body and polarity shifts are quite natural, yet, how would you not think of these geometrical dynamcis as how we might look at the "B field" and Cayley shapes?

Just something to add to your thoughts as you concretize your views about let's say, quasars.



I like the Latin name of Sun(Sol). Plato's use of "the sun" in the analogy of the Cave?

And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: --Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.

Holding a "ideal or image" in mind "as to perfection," can be a guiding light in terms of what possibly "enlightenment" may do for society? What any one moment might do in our realization of what "truly rings the basic core of our understanding" about what we thought long and hard about.:)

A "aha" moment perhaps? A "greater depth of seeing" beyond the "shadows on the walls."

Concepts of the Fifth Dimension

"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 September 15, 1894 - February 5, 1977

Dealing With a 5D World

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."

Herein, I also make the assumption:

The Spacetime Fabric, "is" the Fifth dimension.

Now of course, how do such assumptions make their way into my thinking and visualizations that I do? The seeing, of what mathematics and it's symbology had done for those who might see the geometry of expression, as a very vital way of thinking in the abstract world of mind, analogous, to the computer screen in front of us?


Juan Maldacena:

The strings move in a five-dimensional curved space-time with a boundary. The boundary corresponds to the usual four dimensions, and the fifth dimension describes the motion away from this boundary into the interior of the curved space-time. In this five-dimensional space-time, there is a strong gravitational field pulling objects away from the boundary, and as a result time flows more slowly far away from the boundary than close to it. This also implies that an object that has a fixed proper size in the interior can appear to have a different size when viewed from the boundary (Fig. 1). Strings existing in the five-dimensional space-time can even look point-like when they are close to the boundary. Polchinski and Strassler1 show that when an energetic four-dimensional particle (such as an electron) is scattered from these strings (describing protons), the main contribution comes from a string that is close to the boundary and it is therefore seen as a point-like object. So a string-like interpretation of a proton is not at odds with the observation that there are point-like objects inside it.

(Wikipedia 23 April 2006)

Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature of path of a moving infinitesimal (test) particle. 't Hooft has speculated that the fifth dimension is really the spacetime fabric.

Linked paragraph above was pointed out to a link you to further thoughts on this. It was a strange revelation of sorts to think that such a process could lead you to such thinking, as well, as leading one to understand how General Relativity becomes a result of of String theory.

It just made so much sense as I watched this developement take place in this geometrical extension of thought, that to a beginning, from a point to a line to a plane, was raised in mind, as a short cut to the brane world understandings. I really do not undertsand how I made this jump, but never the less, it took me to a fifth dimensional referencing.



The work of Banchoff helped in this understanding. In using image production, our 2d computer screens, as example, shows the work we are doing in the abstract space of mind.

While I am still ever the student, such thinking moved from the ideas of General Relativity, and it's geoemtrical nature, moves one into the dynamical regions of thought. Held, in regards to those curvatures. I just tend to see them in this way after understanding the "geometrical nature." So too, the undertanding of General Relativity means, and in this assumption, "gravity" becomes the terminology that I see in the dynamcis of that universe.

Can I help seeing the thought of humanity so capable in the mind, to relate choices to the heart and the feather weighting truth, that I also had come to see the gravity of that situation? IN such thoughts, Einsteins analogy of the Pretty girl always come to mind. It was a conceptual leap of sorts, as well as beautifully laid out model of GR as to our understanding in terms of what gravity means.

From strong to weak, and all the understanding of the place, where a flat plane of which no gravity exists, is a place where such transitions take place in my mind. Is this true or not? The very thinking of brane developement lead me to think in a 2 dimensional framework, yet I am well aware of the fifth dimensional views that this framework supplies. Is it wrong? I would have to rely on competent readers of the Brane world to have them say ye or nay, as to the thoughts being portrayed here.

(Wikipedia 23 April 2006)

In physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. When N=5, one of these numbers is sometimes colloquially called the fifth dimension. This usage may occur in casual discussions about the fourth dimension, which, in the context of physics, refers to time, coming after the first three spatial dimensions (up/down, left/right and forwards/backwards). Abstract five-dimensional space occurs frequently in mathematics, and is a perfectly legitimate construct. Whether or not the real universe in which we live is somehow five-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics.

Lisa Randall:

My most recent research is about extra dimensions of space. Remarkably, we can potentially "see" or "observe" evidence of extra dimensions. But we won't reach out and touch those dimensions with our fingertips or see them with our eyes. The evidence will consist of heavy particles known as Kaluza-Klein modes that travel in extra-dimensional space. If our theories correctly describe the world, there will be a precise enough link between such particles (which will be experimentally observed) and extra dimensions to establish the existence of extra dimensions. Dangling Particles,By LISA RANDALL, Published: September 18, 2005 New York Yimes

The extensions beyond what we had always taken for meaning as "seeing," is the undertanding that all 3 space coordinated directions with time, are embedded in some "design" beyond that frame of reference held to General Relativity. If it wasn't, how could anything working beyond this, be found as a coordinated result?

(Wikipedia 23 April 2006)











Figure 2. Clebsch's Diagonal Surface: Wonderful

Arthur Koestler and Creativity

True creativity often starts where language ends.
Arthur Koestler

For those who engaged the issue of intuition, can we say that this is very close to what creativity is ,and the quote suplied above, is in essence. Is the mind, having come to a point on the Aristotlean arch, as, having fully understood, that work with reason and insight would have been to see that, the medium what ever it is, is held to this regard?



If the question is held in context of the mind and foci is strong, in what probabilistic venue would we see such events as issuing from someplace? This I would say would be the "unconscious" and in having diagramed this in a schematic way, how is it that such causations, might have been tied to the fisherman's line and lure, that is sent deep into a future for an examination result.

Of Koestler’s many books, his powerfully anti-Communist novel Darkness at Noon (1941) is still the most famous, but he wrote one book that focused squarely on the paranormal – The Roots of Coincidence (1972). Here, he attempts to find a basis for paranormal events in coincidence, or more precisely synchronicity, so that there is only one phenomenon to explain rather than many. He proceeds to seek the roots of coincidence in the Alice-in-Wonderland world of quantum physics, the infinitesimally small subatomic realm where our everyday logic no longer holds sway, where particles can be waves and vice versa, where forces that only mathematical equations can glimpse swim in the dark, unfathomable ocean of probability before the manifestation of either matter or mind. Towards the end of the book, Koestler pleads that parapsychology be made “academically respectable and attractive to students”, otherwise the “limitations of our biological equipment may condemn us to the role of Peeping Toms at the keyhole of eternity”.

So it was gathered all around, everything that we were involved with, and out of it, a solution abstractually engaging the mind in symbolisms of a language not understood. But still relevant. What would this new language be, if it had run it's course previously, and we needed new insight. We were careful then in understanding the porgress can be made in our expectancy, as well as having full confident in the self, to explore these unknown regions.

Who better then to create the dialogue necessary in bringing forth the creative flow, if we had acknowledged the teacher and student, within ourselves?

Art Mirrors Physics Mirrors Art, by Stephen G. Brush

The French mathematician Henri Poincaré provided inspiration for both Einstein and Picasso. Einstein read Poincaré's Science and Hypothesis (French edition 1902, German translation 1904) and discussed it with his friends in Bern. He might also have read Poincaré's 1898 article on the measurement of time, in which the synchronization of clocks was discussed--a topic of professional interest to Einstein as a patent examiner. Picasso learned about Science and Hypothesis indirectly through Maurice Princet, an insurance actuary who explained the new geometry to Picasso and his friends in Paris. At that time there was considerable popular fascination with the idea of a fourth spatial dimension, thought by some to be the home of spirits, conceived by others as an "astral plane" where one can see all sides of an object at once. The British novelist H. G. Wells caused a sensation with his book The Time Machine (1895, French translation in a popular magazine 1898-99), where the fourth dimension was time, not space.

So, would we have recognized some of these features, in the way the words are written, or how the question mark, would transcended the inspiration sought and found from others, who would propel us forward? The conditions then and foundational attitudes had to rely on what history had already gone through, that we might have recognized also the work that Poincare might have relinquished in that dialogue. To have propelled other minds, like Picasso or Einstein forward?

Is this where "time" became something of a issue with the space coordinates, that such resolution might have paved the way for a spacetime? Answered, what the fourth dimension actually was? Such progression then would have been important, as we move forward in society that not only had Poincare provided the prospect, but that also Grossman in the geometrically refined views sought out as well, to contribute to the troubles Einstein was facing?

Where these might have thought to be random, the events are tied together? Are seen in the actualization of what trasncended these two random events? Or were they?



We talk about the historical time and around then, what was happening if we had seen information on Flatland and Abbott? Issues of mysticism held in context of what those extra dimensions might actually mean.

Out of this, a new found responsibility, as to how such mysticism once held in the spookiness of Einstein, has now an explanation that has been further refined in what a Anton Zeilinger might have been doing for us?

Working the Angles UNtil They Add Up Too?

I assume now, you are in the non-eucldiean inferences?

If people had thought "the negative" always evil, then what value any "dynamic of thinking" if we could not resolve what we had been doing by changing the shape of our attitudes? :)



Helen Joyce:

Both spherical and hyperbolic geometries are examples of curved geometries, unlike Euclidean geometry, which is flat. In spherical geometry, the curvature is positive, in hyperbolic geometry, it is negative.

Some might have never understood the dynamics going on, and if a "backreaction" was created, what value would this serve in our thinking? I ponder. Some might think of Dirac? Some of anti-matter? In opposition, some might think of Reimann and what was encapsulate in Gauss's Mountain.

So maybe in churches, or concert halls, we might think about the ways "sound reverberates," within an enclosure? The lines in the architecture? How this resonance passes through and changes the very matters in their oscillations?

Ways IN which To Percieve Landscape?

What a Cosmologist Wants from a String Theorist?




Emotion versus Reason?

3.1 As Cytowic notes, Plato and Socrates viewed emotion and reason as in a kind of struggle, one in which it was vitally important for reason to win out. Aristotle took a more moderate view, that both emotion and reason are integral parts of a complex human soul--a theory proposed by Aristotle in explicit opposition to Platonism (De Anima 414a 19ff). Cytowic appears to endorse the Platonic line, with the notable difference that he would apparently rather have emotion win out.

Emotion can be used as a catelysct into higher abstractual/dimensional thinking, if, it can be used to counter research into?:)

Figure 2. Clebsch's Diagonal Surface: Wonderful.

Mein Gott. :) If seeing on distance scales, had relevances in regards to "all the issues" of the standard model, would this not in effect change the way we see in those distances?

Peter Woit:I’ve looked very carefully in landscape papers and Susskind’s book for any sort of plausible idea about how this stuff will ever lead to a prediction of anything and I can’t find it.

Thanks Peter, that's it?


The Hills are Alive with the Sound of M theory?

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 universeWayne Hu

By exercising the imagination I thought Wayne Hu did a fine job of relating these things on a "cosmological scale." Hills and Valleys. But in a more detailed quantum look, what value, conformal field theory of point particles?

In effect, the 5-D universe is recorded like a hologram on the 4-D surface at its periphery. Superstring theory rules in the 5-D spacetime, but a so-called conformal field theory of point particles operates on the 4-D hologram. A black hole in the 5-D spacetime is equivalent to hot radiation on the hologram--for example, the hole and the radiation have the same entropy even though the physical origin of the entropy is completely different for each case. Although these two descriptions of the universe seem utterly unalike, no experiment could distinguish between them, even in principle.

Les Houches




ROBBERT DIJKGRAAF:Map of the world, as used in my Les Houches lectures

I like this picture better Clifford. Is the landscape, as barren, or is it, the hope that we see such beautiful things of which the seed bed wil allow such things to arise from it?

For some, the "creative" outlet? Maybe, a Shangri-la high" in the mountains of abstractual thinking?

IN the Wunderkammern

James Joseph Sylvester (September 3, 1814 - March 15, 1897) was an English mathematician and lawyer.

We are told that "mathematics is that study which knows nothing of observation..." I think no statement could have been more opposite to the undoubted facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world, ...that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. ...Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another.

While I always point upward in Rapheal's painting, I mention often, the "One thing."

Gold or wisdom, while leadng "the alchemist" in the search of that elucive material, mining, has to note the glimmer's as a sun shines on the landscape of ideas. So you work it, use a sluicebox, or a gold pan. Watch how river flow's and the bends in it. Where some deposits might have laid themself while others are carried off further down stream, left to some "eddie" or "pool of thinking." See flowers emerge in rocks crevices of all places.

However, don't be fooled! The charm of the golden number tends to attract kooks and the gullible - hence the term "fool's gold". You have to be careful about anything you read about this number. In particular, if you think ancient Greeks ran around in togas philosophizing about the "golden ratio" and calling it "Phi", you're wrong. This number was named Phi after Phidias only in 1914, in a book called _The Curves of Life_ by the artist Theodore Cook. And, it was Cook who first started calling 1.618...the golden ratio. Before him, 0.618... was called the golden ratio! Cook dubbed this number "phi", the lower-case baby brother of Phi.

See:

Fool's Gold

The Alchemist in You

String Theory Displays Golden Ratio Tendency

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.

Foundations of Mathematic

Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

--BERTRAND RUSSELL, Study of Mathematics

In a Question below, is it worth it, to look at the context of what groups who gather might spark to the rest of society(click on it)? Look at what it has done for myself, and the reasons why such inductive/deductive features seem to be a part of the origins of cognitive functions that mathematically display itself?

Is there a theme in this regard through my blog that I had questioned earlier and links brought forth to raise awareness of what might have been implied in that true "consciousness sense" about the very nature of our involvement in the nature of reality?

But then too, awareness, about the death of such sensations. This is most troubling to me, if such model consumptions had made this impression then what had happened to the views as they exploded into the other realms? Other Realms? Why would I introduce Thales as a culminative vision about what could emerge and the father of geometry? Models make our view culmnative and increase the vision capabilites. Is there no one here that see differently after they had crossed a page to find that in our new tomorrows we see reality a little different now?

You have been touched at a most deep level, that goes beyond the death of such sensations as Toposense, or momentums of curvatures. A microscopic eye now, to the quantum nature, right next to your reading from this screen. It's in the air all around you, this potential? :)

Plato:

Mathematics(logic?) and experiment?

I respond in that thread, and although it would seems disjointed from the rest of the commentaries, I thought I was talking directly to Sean's opening post. So I have linked the post on the very title as I have done with previous entires, as they have been setting the pace for my thinking about what views they share and what safety net is placed out there for us lay readers.

Would this impede my question as to the relation of philosphy in Sean's opening statement, to find that it had found a trail that leads to reasons why funding and perspective on it, should be thought about most carefully. Held in the esteem, with which one's adventures in physics and mathematics might have benefited society?

I understand this need for determination, and as well, the need to reaffirm what philosophy might hold in regards to truly active memebers of the science community and the projects they are engaged in. Would they have a distain for the philosophy of mathematics?

I left a question mark out there, and this question although never answered did see some slight comment in relation to the philosophy that where such logic might have gained in relation, being mentioned. I'll have to explain this some more so you understand that I am working hard to make sense of what is out there and viewed, whether in the tabloids, or what ever generalizations made by mathematicians, or the physicist who looks that little bit further.

Shall I quickly respond to the thread commetary or should I continue? I thnk it important that I respond to the comments rasied but I'll do this after by highlighting the area that spoke to me in relation to this train of thought.

I linked a quote from Plato on the idea of philosophy in my comment. I wil be moving from that position.

Philosophy of Mathematics

Foundations Study Guide: Philosophy of Mathematics by David S. Ross, Ph.D.
The philosophy of mathematics is the philosophical study of the concepts and methods of mathematics. It is concerned with the nature of numbers, geometric objects, and other mathematical concepts; it is concerned with their cognitive origins and with their application to reality. It addresses the validation of methods of mathematical inference. In particular, it deals with the logical problems associated with mathematical infinitude.

Among the sciences, mathematics has a unique relation to philosophy. Since antiquity, philosophers have envied it as the model of logical perfection, because of the clarity of its concepts and the certainty of its conclusions, and have therefore devoted much effort to explaining the nature of mathematics.

You have to understand that although I am deficient in the math skills many have, it is not without effort that I am enaging myself in what appears to be beautiful and simplistic design when completed as a model. When we look at what the Wunderkammern had to offer in a revitalizing and dusting off of, models that were concretized for us. Did they lanquish until they were refurbished to the museums of time, so that we may again look at what mathematics has accomplished for us. In ways, that are very abstract and beautiful? What then exist as you gazed into the magnetic field, the dynamcis of brane held issues and the exemplification of design in those branes? It had to follow consistent and progressive developement in the physics of.



The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner

The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

[3 M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago
Press, 1958), says: "All these difficulties are but consequences of our
refusal to see that mathematics cannot be defined without acknowledging
its most obvious feature: namely, that it is interesting" (p 188).]

Social constructivism or social realism

Now here is the part, that while I saw the devloping nature of the tread of thinking and comments how would I answer and stay in tune? I previously spoke of John Nash and the inherent nature of mathematics as it could pierce the bargaining process, that to have this moved t a dynamcial social and constructive pallette developed in the ongoing relations of nations, why would such a scoial construct not be recognized as to the direction and strength of what mathematics might mean from a cognitive and developing brain that we have.

This theory sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly compared to 'reality' and may be discarded if they don't agree with observation or prove pointless. The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints (the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated) that work to conserve the historically defined discipline.

This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current Mathematical Community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-belief in axiomatic proof and peer review as practices.

This gets very comlicated for me. Yet I recognize the inhernet pattern at the basis of these negotiatons and the games involved. More to follow, and short on time.

Angels and Demons

Now how could such good thinking minds have not seen that the publics understandings might have been warped by the very underpinnings of good science men/woman, and all the issues become some fictional story for what evils and saints can do for us.



So was it some distant function of creation that we should not recognize the negative effect of all "good things" that will emerge from the actions of what is revealled to us in our "rainbows and aurora's," that we would not seem pleased as to the emissions have to say in the wave forms that surround such things?

Can we hope to use antimatter as a source of energy? Do you feel antimatter could power vehicles in the future, or would it just be used for major power sources?

There is no possibility to use antimatter as energy "source". Unlike solar energy, coal or oil, antimatter does not occur in nature: we have to make every particle at the expense of much more energy than it can give back during annihilation.

You might imagine antimatter as a possible temporary storage medium for energy, much like you store electricity in rechargeable batteries. The process of charging the battery is reversible with relatively small loss. Still, it takes more energy to charge the battery than what you get back out of it. For antimatter the loss factors are so enormous that it will never be practical.

If we could assemble all the antimatter we've ever made at CERN and annihilate it with matter, we would have enough energy to light a single electric light bulb for a few minutes.

So while good thinking men and woman dance with the ideas of Einsteins geometrical propensities to answer thse functions, what spherical relation would have said, that for every sun that burns out, it will rejuvenate itself, by strict geometrical functions in anti-matter creation to bring forth this "new vision" of the world.

Create this wonderful unlimited resource of energies that exist around us now?

So again let's take this back to the Pierre Auger examination of what is taking place outside of the collider expeirments. While it is nice to have these controls, why were we not informed about the potentiality of what exists as you pursue your visons to the very beginnings of this universe? That this beginning would take place right next to you? Is this wrong that we not assign astronomical valuations to the very nature of our world now, as such interactions take place between the sun and earth? That in those compacted dimensions, such calculations would reveal the thinking of relative and mathematical entities, as signals of the events that can take place everyday around us as well.

Einstein was very revealling in what could be taken to a larger scale for what could split apart, so it is not so unlikely that ourvisions have been curtailled,just becuase we did not se the actions that could take plac ein a larger scenario?

So did Heisenberg see what was revealling towards these geometerical propensities, as events unfolded themselves?

Update:

if your foci is "string" enough, you might realize it is less than K=0 :)


All M.C. Escher works (c) 2001 Cordon Art BV - Baarn - the Netherlands. All rights reserved. www.mcescher.com

While some believe in positive curvatures they also understand that the inception could have a negative effect, yet it would not be "angel and demons" they espoused?

We are all better then that, right? There is a "greater whole" we are each part of? To further extend this empowerment beyond "good and evil in religion" think of sound then, and the related entry below. Maybe, it will have a certain resonance for you?

Music of the Spheres

About how the brain's neuronic vitalities of vison are enhanced, and related?

Wunderkammern