About Spheres and their Generalizations of Higher Dimensional Spaces

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

— Nikolai Lobachevsky

If a math man were to be left alone, and devoid of the physics, would he understand what the physics world could impart if he were not tied to it in some way?:)

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, 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.

Now it is of course with some understanding, that few would recognize what this conjecture to mean, and having just read Andre' Weil's last word I do not think it to shabby to say that in this case, the Poincare conjecture has been recognized as a valid conjecture, that has taken some time in it being resolved.




For me being presented with the cosmic string scenario might seem just as complex, when in considertaion of the brane. How our universe could be contained in it. Some might of laughed it off quite easily, being part of some revolution of strings to M-theory, that it could include 11 dimensions.How would you embed these dimensions with these shapes?

What fascinates me, is how we could have found such visualizations of these topological forms within the the brane world and how this may have been described?

So I am looking for traces of literature that would help me in this direction. For example, how a torus would be looked at in a 2 dimensional sheet. Would this be relevant to brane world happenings if we considered, the example of "sound" (make sure speakers are on) in higher dimensions, as viable means of expression of these curvatures of those same shapes?



Of course I know I have to explain myself here, and the intuitive jump I am making. Could be wrong?

Part of this struggle to comprehend what has happened with bringing GR and QM together in one consistent framework, was to undertand that you have altered the perceptions of what dimension will mean? To me, this says that in order now for us to percieve what the poincare conjecture might have implied, that we also understand the framework with which this will show itself, as we look at these forms.

The second panel I showed of the graph, and then the topological form beside it, made sense, in that this higher dimension shown in terms of brane world happenings would have revealled the torus as well in this mapping. This is of course speculation on my part and might fall in response to appropriate knowledge of our mathematical minds. This gives one a flavour of the idea of those extra dimensions.



Well part of this developement of thinking goes to what these three gentlemen have developed for us in our new conceptual frameworks of higher dimenions. If you ask what higher dimension might mean, then I am certain one would have to understand how this concept applies to our thinking world. Without it, these shapes of topological forms would not make sense. It seems this way for me anyway.:)



I hazard to think that John Baez and others, might think they have found the answer to this as well, in what they percieve of the mighty soccerball, and the fifth solid I espouse:)

Hodge Conjecture

Interplay between geometry and topology.

One of the things that I am having difficulty with is if I understood the idea of a cosmic string. The understanding would have to imply that the higher dimensions would reveal themselves within the spacetime curvatures of gravity. So I have been looking to understand how the quantum mechanical nature could have ever reduced itself from those higher dimensions in string theory, and revealed themselves within the context of the cosmos that we know works very well with Gravity.



Part of my attempts at comprehending the abtractness with which this geometry evolved, was raised diffrent times within this blog as to whether or not there was a royal road to geometry?

Throughout, I have shown the processes with which a smooth topological feature would have endowed movements like the donut into the coffee cup and wondered, about this idea of Genus figures and how they to become part of the fixtures of the terrain with which mathematicians like to enjoy themselves over coffee?:)



How would this information in regards to the strings, become a viable subject with regards to non-euclidean realms, to have understood where GR had taken us and where QM had difficult combining with GR(gravity).

Large extra dimensions are an exciting new development … They would imply that we live in a brane world, a four-dimensional surface or brane in a higher dimensional spacetime. Matter and nongravitational forces like the electric force would be confined to the brane … On the other hand, gravity in the form of curved space would permeate the whole bulk of the higher dimensional spacetime …. Because gravity would spread out in the extra dimensions, it would fall off more rapidly with distance than one would expect … If this more rapid falloff of the gravitational force extended to astronomical distances, we would have noticed its effect on the orbits of the planets … they would be unstable… However, this would not happen if the extra dimensions ended on another brane not that far away from the brane on which we live. Then for distances greater than the separation of the branes, gravity would not be able to spread out freely but would be confined to the brane, like the electrical forces, and fall off at the right rate for planetary orbits.

Stephen Hawking, Chapter seven

Mirror Symmetry and Chirality?

What is the mathematical reasoning, for reducing GR down to the quantum world? How would these application be considered in a supersymmetrical world?

Is the figure-eight knot the same knot as its mirror-image? The property of "being the same as your mirror-image" is called chirality by knot theorists. The image sequence below shows the figure-eight knot being transformed into its mirror image -- such knots are called achiral

You have to remember this is the year 2000, article was produced below.

Physicists Finally Find a Way to Test Superstring Theory

As the unification quest has forged ahead, physicists have found it necessary to expand superstring theory to include vibrating membranes -- called branes for short. These are not just two-dimensional surfaces, like the skin of a drum or the world of the Flatlanders. Hard as it may be to picture, there can be branes with three, four, five or more dimensions. These "surfaces" can be tiny like the strings but they can also span across light-years.

by George Johnson

I know this may seem a little slow but if the cosmic string was considered in context of the Fresnel lens and I apologize for my ignorance, how would reverse imaging account for gravitational lensing?

The gravitational lensing has to reveal the warp field as such a possibility? Yet, in that distant time, such alteration of the shapes, amongst it event, would have show two points, but would also have indication that one was the reverse of the other. What signatures would we see of this?



So in keeping with the direction we were given by Lubos in his article, I a am trying to comprehend.

This page is the beginning of a demonstration of strong gravitational lensing about a pseudo-isothermal elliptical mass distribution (PID). You can look at a simple animation or read about the mathematics behind the PID lensing.

Dilation and the Cosmic String

One such field, called the dilaton, is the master key to string theory; it determines the overall strength of all interactions. The dilaton fascinates string theorists because its value can be reinterpreted as the size of an extra dimension of space, giving a grand total of 11 spacetime dimensions

According to T-duality, universes with small scale factors are equivalent to ones with large scale factors. No such symmetry is present in Einstein's equations; it emerges from the unification that string theory embodies, with the dilaton playing a central role.

Gabriele Veneziano


According to Einstein's theory of general relativity, the sun's gravity causes starlight to bend, shifting the apparent position of stars in the sky.

Time will also pass more slowly in a strong gravitational field than in a weak one? So what effect would this have if we consider the gravitational lensing, that had been talked about in previous post?:)

On the Effects of External Sensory Input on Time DilationA. Einstein, Institute for Advanced Study, Princeton, N.J.

Abstract: When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it's longer than any hour. That's relativity.

As the observer's reference frame is crucial to the observer's perception of the flow of time, the state of mind of the observer may be an additional factor in that perception. I therefore endeavored to study the apparent flow of time under two distinct sets of mental states.



Where spacetime is flat, there is no gravity, hence light will travel unabated. If we move this consideration in contrast to the non-euclidean realms, what have we learnt of gravity? What have we learnt of dimensions?



Mass, Photons, Gravity Dr. Lev Okun, ITEP, Russia

Warped Field Creates Lensing

The statement of this post, is distilled from the collaboration of some of the images to follow.



In cosmic string developement there are these three points to consider.

1. Cosmological expansion

2. Intercommuting and Loop Production

3. Radiation

I am always looking for this imagery that helps define further what gravitational lensing might have signified in our perception of these distances in space. How the cosmic string might have exemplified itself in some determination, as we find Lubos has done in the calculation of the mass and size of this early event. This image to follow explains all three developemental points.



Bashing Branes by Gabriele Veneziano
String theory suggests that the big bang was not the origin of the universe but simply the outcome of a preexisting state

The pre–big bang and ekpyrotic scenarios share some common features. Both begin with a large, cold, nearly empty universe, and both share the difficult (and unresolved) problem of making the transition between the pre- and the post-bang phase. Mathematically, the main difference between the scenarios is the behavior of the dilaton field. In the pre–big bang, the dilaton begins with a low value--so that the forces of nature are weak--and steadily gains strength. The opposite is true for the ekpyrotic scenario, in which the collision occurs when forces are at their weakest.

The developers of the ekpyrotic theory initially hoped that the weakness of the forces would allow the bounce to be analyzed more easily, but they were still confronted with a difficult high-curvature situation, so the jury is out on whether the scenario truly avoids a singularity. Also, the ekpyrotic scenario must entail very special conditions to solve the usual cosmological puzzles. For instance, the about-to-collide branes must have been almost exactly parallel to one another, or else the collision could not have given rise to a sufficiently homogeneous bang. The cyclic version may be able to take care of this problem, because successive collisions would allow the branes to straighten themselves.

The most strongest image that brought this together for me was in understanding what Neil Turok and Paul Steinhardt developed for us. It was watching the animation of the colliding branes that I saw the issue clarify itself. But before this image deeply helped, I saw the issue clearly in another way as well.

The processes of intercommuting and loop production.

It was very important from a matter distinction, to understand the clumping mechanism that reveals itself, after this resulting images of the galaxy formation recedes in the colliding brane scenrio viewing. If such clumping is to take place, we needed a way in which to interpret this.




Branes Reform Big Bang By Atalie Young

Catch a Wave From Space

Imagine the journey it took for us to have come to the developement of the methods to discern the nature of the Universe, and what Einstein has done for us in terms of General Relativity. A statement, about Gravity.




Imagine these great distances in space, and no way in which to speak about them other then in what LIGO will translate? That any extention of this prevailing thought could not have found relevance in the connection, from that event to now, and we have found ourselves limited in this view, with bold statements in regards to Redshifting perspectives?



Without a conceptual framework in which to look at the gravitational differences within the cosmo, how the heck would any of this variation make sense, if you did not have some model in which to regulate distances traversed, in the space that must be travelled?

Albert Einstein discovered long ago that we are adrift in a universe filled with waves from space. Colliding black holes, collapsing stars, and spinning pulsars create ripples in the fabric of space and time that subtly distort the world around us. These gravitational waves have eluded scientists for nearly a century. Exciting new experiments will let them catch the waves in action and open a whole new window on the universe - but they need your help to do it!

Cosmic strings are associated with models in which the set of minima are not simply-connected, that is, the vacuum manifold has `holes' in it. The minimum energy states on the left form a circle and the string corresponds to a non-trivial winding around this.

3 Sphere

What would mathemaics be without artistic expression, trying out it's hand at how such geometrical visions continue to form? Did Escher Gauss and Reimann, see above 3 sphere?

An expression of Salvador Dali perhaps in some religious context, who then redeems himself, as a man and author of artistic expression?

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.

So in looking for this mathematical expression what does Gabriele Veneziano allude too in our understanding of what could have come before now and after, in the expression of this universe, that it is no longer a puzzle of what mathematics likes express of itself, now a conceptual value that has encapsulated this math.



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


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.

So as if beginning from some other euclidean systemic pathway of expression, how in spherical considerations could topolgical formation consider Genus figures, if it did not identify the smooth continue reference to cosmoogical events? Where would you test this mathematics if it cannot be used and applicable to larger forms of expression, that might also help to identfy microstates?

The initial process of particle acceleration is presumed to occur in the vicinity of a super-massive black hole at the center of the blazar; however, we know very little about the origin of the jet. Yet it is precisely the region where the most important physics occurs: the formation of a collimated jet of charged particles, the flow of these particle in a narrow cone, and the acceleration of the flow to relativistic velocities.


So in looking at these spheres and their devlopement, one might have missed the inference to it's origination, it's continued expression, and the nice and neat gravitational collpase that signals the new birth of a process? Can it be so simple?

Would it be so simple in the colliders looking for those same blackholes?

Curvature Parameters

How does get this intuitive feeling embedded within thinking if it has not grokked the significance on a cosmological scale?



When one first begins to comprehend the intuitive possibilities the universe can move through, I was was struck, by the coordination of how we could see in terms of non-euclidean prospectives, and find correlation, with what was happening on a cosmological scale.



AS I looked at the the Friedmann Equation the connection to this dynamical movement in the cosmos, revealed itself, when you accepted the move to Reimannian understanding?



It was important that the connections and links to teachers be recognized. For what Gauss himself imparted, was also demonstrated in the work of Einstein, to bring Gaussian curvature along into the dynamical world gravity would reveal of itself when Einstein was completed.

But the Euclidean model stops working when gravity becomes strong

Once it came to understanding the metric and the distance function, between two points a new world was revealed. It became very interesting to see how non-euclidean was lead too, and how the work of GR blended together in a new perspective about the reality we live in. It no longer made sense to think of that space between those two points other then in the mathematical ideas of NCG.

Virtual interactions make the electron charge depend on the distance scale at which is it measured.



How would not derive some sense of this fluctuation if we did not understand the dynamical nature that has been revealed to us? On large scales it seems so easy, while in these micro states, it's all spread out and fuzzy. So at planck length, how would we describe the motions we understand of the spacetime fabric if we change the quantum mechanical description of it?

dS²=c² dT²-dX²

The amount of dark matter and energy in the universe plays a crucial role in determining the geometry of space. If the density of matter and energy in the universe is less than the critical density, then space is open and negatively curved like the surface of a saddle

Lagrange Points

How would clumping been drived from anything if such a supersymmetrical reality did not exist at some point?




Without someview that would be consistent through out the cosmo, how would such points be of value, if we could not see this variation ? In order for this view to be scalable it had to have begun in some other way, that we could sufficely say that it was strong once and all pervasive, but now?



Sun-Earth Lagrange Point forces affect spacecraft orbits.


Some would have trashed the cosmological view and the understanding of the quantum perspective that is developing with those short distances, that if they did not include this realization, what value would they hold for cosmology?

This is the golden age of cosmology. Once a data-starved science, cosmology has burgeoned as ground and space-based astronomical observations supply a wealth of unprecedently precise cosmological measurements. Questions that were recently the stuff of speculation can now be analyzed in the context of rigorous, predictive theoretical frameworks whose viability is determined by observational data. Finally, cosmological theory is being confronted by cosmological fact. The most surprising and exciting feature of cosmology's entrance into the realm of data-driven science is its deep reliance on theoretical developments in elementary particle physics. At the energy scales characteristic of the universe's earliest moments, one can no longer approximate matter and energy using an ideal gas formulation; instead, one must use quantum field theory, and at the highest of energies, one must invoke a theory of quantum gravity, such as string theory. Cosmology is thus the pre-eminent arena in which our theories of the ultra-small will flex their muscles as we trace their role in the evolution of the universe.

What is the Ultimate Theory of Physics?

It is always very interesting for me to try and understand how such short distances could have begun to have some visual world possibilities? When mathematics begins to develope a method to describing that same small world.

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

It is not always clear to me what would have arisen out of the possibilties of what could possibly lies beneath, but any emergent principal wuld have to be able to describe its origination(the geometry)?



If we thought for a moment that there was a guiding principal in the higg's field, how would we have perceptably explained this in conglomerations of students who would gather around their professors? In a strange sense, one could intuitively feel this gathering consequence, as a consolidation of dominante principles, around which the matter states could easily have resigned themselves?


Braided independence is another of the conceptual ideas going into the modern approach to quantum geometry. When electrons pass each other either in physical space or lexicographically during a calculation, their exchange involves an additional -1 factor. Building on this idea, one is led to a kind of braided mathematics in which the outputs of calculations are `wired' into the inputs of further calculations much like the way information flows inside a computer. Only now, when wires cross each other there is a nontrivial operator, in fact a different one for when one wire goes under the other and when it goes over. Here is a typical calculation in braided-mathematics

Noncommutative Geometry, Monday July 24th 2006 - Friday December 22nd 2006

Noncommutative geometry aims to carry over geometrical concepts to a a new class of spaces whose algebras of functions are no longer commutative. The central idea goes back to quantum mechanics, where classical observables such as position and momenta no longer commute. In recent years it has become appreciated that such noncommutative spaces retain a rich topology and geometry expressed first of all in K-theory and K-homology, as well as in finer aspects of the theory. The subject has also been approached from a more algebraic side with the advent of quantum groups and their quantum homogeneous spaces.

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.