Wednesday, December 15, 2004

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?

1 comment:

Plato said...

Without repeating ole cliche's and sounding like some repetitive impressionist, might we be masking the intent of what one might be saying?:)

I know it is very difficult to find the center, and topologically, impress the idea of "AGN jet" as a continued geometrical expression as well?

As part of some continued function of the mathematical expressions that you fellows seemed engaged in?

I think Greene was specific on this point?

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)