## Friday, October 03, 2008

### Centroids

Conclusion:The state of mind of the observer plays a crucial role in the perception of time.Einstein

It is often I consider the Center of Gravity(Mass) as an illustration of the perspective that exists "around and within us" as a product of the natural world. We are limited by our own knowledge, yet when given, and lead to observation of what is natural in our world, we see where we had not seen before.

\bar z = \frac{\int z dm}{m}The center of gravity (or mass), abbreviated as COM, of any object is that point within the object upon which gravity (or any body force) acts, regardless of the orientation of the object. The COM of an object may be calculated by using the principle of equilibrium.

First off you must know that I work according to a set of principles that were discovered after a time through "subjective reflection" on the nature of mind and how it works. Of course I am no expert here, but the principles themself are realizable as an active function in the recognition of how one might interpret the world.

\bar y = \frac{\int y \rho dV}{\int \rho dV}In the event that the density ρ of an object is not uniform throughout, the calculation of COM may be done by a similar set of equations involving the addition of density to the analysis.

It is by taking one to recognisance of the title of this blog entry that I wanted to discuss the "subjective part" and the understanding of the "mathematical construct" that one, in my opinion exists. Just as we progress to understand the natural world around us, I believe these are synonymous with each other.

\bar Y = \frac{\sum m \bar y}{\sum m}If a body is made up of multiple sections, each of which has a unique mass, the method for evaluating the centroid of that body is to evaluate the composite body by finite element analysis of each of the sections through the use of moment balancing, as above.

Liminocentric structures

You can find much on this site in regards to this issue, and I now relate centroid for consideration in the case of this topic. I presented it at the Backreaction site today to show an advancement in the thinking as we move to incorporate the knowledge. As we come to understand the natural world around us.

Giving earth a new understanding in terms of its densities and it's relation to the gravitational contribution, it has in our new views of the globe, helped me learn and understand the feature of gravity as it extends to the cosmos.

I have taken it a step further as many of you know to include the emotive states as valued physiological relations to what manifests in the immediate environs of the body home. While this is a far cry from the understanding of gravitational research it was inevitable for me to see an extension of how one can topological relate to the world, in this inductive/deductive mode to reason.

While living on this earth, we always come home for reflection, and if any have not taken stock of what has accumulated during the day, then it is inevitable that is will come to reside very close to home as one sleeps through the nocturnality night. This research of subjectivity and mind had to entail "all of our activities." It had to include an understanding of the "mathematical constructs" that we observe, in order to relate to nature.

List of Centroids

ShapeFigure$\bar x$$\bar y$Area
Triangular area$\frac{b}{3}$$\frac{h}{3}$$\frac{bh}{2}$
Quarter-circular area$\frac{4r}{3\pi}$$\frac{4r}{3\pi}$$\frac{\pi r^2}{4}$
Semicircular area$\,\!0$$\frac{4r}{3\pi}$$\frac{\pi r^2}{2}$
Quarter-elliptical area$\frac{4a}{3\pi}$$\frac{4b}{3\pi}$$\frac{\pi a b}{4}$
Semielliptical areaThe area inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and above the $\,\!x$ axis$\,\!0$$\frac{4b}{3\pi}$$\frac{\pi a b}{2}$
Semiparabolic areaThe area between the curve $y = \frac{h}{b^2} x^2$ and the $\,\!y$ axis, from $\,\!x = 0$ to $\,\!x = b$$\frac{3b}{8}$$\frac{3h}{5}$$\frac{2bh}{3}$
Parabolic areaThe area between the curve $\,\!y = \frac{h}{b^2} x^2$ and the line $\,\!y = h$$\,\!0$$\frac{3h}{5}$$\frac{4bh}{3}$
Parabolic spandrelThe area between the curve $\,\!y = \frac{h}{b^2} x^2$ and the $\,\!x$ axis, from $\,\!x = 0$ to $\,\!x = b$$\frac{3b}{4}$$\frac{3h}{10}$$\frac{bh}{3}$
General spandrelThe area between the curve $y = \frac{h}{b^n} x^n$ and the $\,\!x$ axis, from $\,\!x = 0$ to $\,\!x = b$$\frac{n + 1}{n + 2} b$$\frac{n + 1}{4n + 2} h$$\frac{bh}{n + 1}$
Circular sectorThe area between the curve (in polar coordinates) $\,\!r = \rho$ and the pole, from $\,\!\theta = -\alpha$ to $\,\!\theta = \alpha$$\frac{2\rho\sin(\alpha)}{3\alpha}$$\,\!0$$\,\!\alpha \rho^2$
Quarter-circular arcThe points on the circle $\,\!x^2 + y^2 = r^2$ and in the first quadrant$\frac{2r}{\pi}$$\frac{2r}{\pi}$$\frac{\pi r}{2}$
Semicircular arcThe points on the circle $\,\!x^2 + y^2 = r^2$ and above the $\,\!x$ axis$\,\!0$$\frac{2r}{\pi}$$\,\!\pi r$
Arc of circleThe points on the curve (in polar coordinates) $\,\!r = \rho$, from $\,\!\theta = -\alpha$ to $\,\!\theta = \alpha$$\frac{\rho\sin(\alpha)}{\alpha}$$\,\!0$$\,\!2\alpha \rho$

Why are Planets Round?

It is always interesting to see water in space.

Image: NASA/JPL-
Planets are round because their gravitational field acts as though it originates from the center of the body and pulls everything toward it. With its large body and internal heating from radioactive elements, a planet behaves like a fluid, and over long periods of time succumbs to the gravitational pull from its center of gravity. The only way to get all the mass as close to planet's center of gravity as possible is to form a sphere. The technical name for this process is "isostatic adjustment."

With much smaller bodies, such as the 20-kilometer asteroids we have seen in recent spacecraft images, the gravitational pull is too weak to overcome the asteroid's mechanical strength. As a result, these bodies do not form spheres. Rather they maintain irregular, fragmentary shapes.

See:Isostatic Adjustment is Why Planets are Round?

## Wednesday, October 01, 2008

### Second Life

Creating secure places for meetings and creating the individuals for this interaction while retaining anonymity is part of the idea behind discussing strategy while not revealing who these individuals are in this exercise.

Some say, why not do this with legit names, and to some, see this as a cowardice for not standing up to what is being said. Well that's not the issue here and creating the purpose for advancement of "free thinking" is to allow one to speak freely and assign a character to the individual for the expression of that medium.

It is allowing a freedom to speak regardless of the restraints a public name can bring to one's public life. See, how I don't care how important your name is, and I see you as an individual first, who has an opinion, so I would want you to share that regardless of the public stature with which you operate in public life.

We want you to share even if it under the anonymity of this character you choose. So how do we go about creating the venue of this events to take place whether you might call it a Quebec summit or a summit in Banff it will be under the auspice of this technology that we meet.

Second Life (abbreviated as SL) is an Internet-based 3D virtual world launched June 23, 2003 and developed by Linden Research, Inc, which came to international attention via mainstream news media in late 2006 and early 2007. A free downloadable client program called the Second Life Viewer enables its users, called "Residents", to interact with each other through motional avatars, providing an advanced level of a social network service combined with general aspects of a metaverse. Residents can explore, meet other residents, socialize, participate in individual and group activities, and create and trade items (virtual property) and services with one another

## Sunday, September 28, 2008

### Three-body problem and WMAP

"We all are of the citizens of the Sky" Camille Flammarion

In 1858, by the set of its relations, it will allow Camille Flammarion, the 16 years age, to enter as raises astronomer at the Observatory of Paris under the orders of Urbain the Glassmaker, at the office of calculations.

See:The Gravity Landscape and Lagrange Points

Now there is a reason that I am showing "this connection" so that the jokes that go around at the PI institute in regards to Tegmark( not that I am speaking for him and have absolutely no affiliation of any kind) and the "mathematical constructs are recognized" beyond just the jeering section, that while not being a party too, will and can be shown some light.

Three-body problem

For n ≥ 3 very little is known about the n-body problem. The case n = 3 was most studied, for many results can be generalised to larger n. The first attempts to understand the 3-body problem were quantitative, aiming at finding explicit solutions.

* In 1767 Euler found the collinear periodic orbits, in which three bodies of any masses move such that they oscillate along a rotation line.
* In 1772 Lagrange discovered some periodic solutions which lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically. Those solutions led to the study of central configurations , for which \ddot q=kq for some constant k>0 .

The three-body problem is much more complicated; its solution can be chaotic. A major study of the Earth-Moon-Sun system was undertaken by Charles-Eugène Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.
The chaotic movement of 3 interacting particles
The chaotic movement of 3 interacting particles

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Poincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may be easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points maintaining the 60 degree-spacing ahead of and behind the less massive body in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.

So the thing is, that while one may not of found an anomalousness version of what is written into the pattern of WMAP( some Alien signal perhaps in a dimension of space that results in star manipulation), and what comes out, or how string theory plays this idea that some formulation exists in it's over calculated version of mathematical decor.

String Theory

In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza and Klein's early work demonstrated that general relativity with five large dimensions and one small dimension actually predicts the existence of electromagnetism. However, because of the nature of Calabi-Yau manifolds, no new forces appear from the small dimensions, but their shape has a profound effect on how the forces between the strings appear in our four dimensional universe. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.

This is/was to be part of the hopes of people in research for a long time. I have seen it before, in terms of orbitals(the analogical version of the event in the cosmos) and how such events could gave been portrayed in those same locations in space. Contribute, to the larger and global distinction of what the universe is actually doing. If it's speeding up, what exactly does this mean, and what should we be looking for from what is being contributed to the "global perspective" of WMAP from these locations??

But lets move on here okay.

If you understand the "three body problem" and being on my own, and seeing things other then what people reveal in the reports that they write, how it is possible for a lone researcher like me to come up with the same ideas about the universe having some kind of geometrical inclination?

You would have to know that "such accidents while in privy to data before us all", and what is written into the calculations by hand would reveal? Well, I never did have that information. What I did know is what Sean Carroll presented of the Lopsided Universe for consideration. This coincided nicely with my work to comprehend Poincaré in a historical sense. The relationship with Klein.

As mentioned before, at the time, I was doing my own reading on Poincaré and of course I had followed the work of Tegmark and John Baez's expose' on what the shape of the universe shall look like. This is recorded throughout my bloggery here for the checking.

What I want to say.

Given the mathematics with which one sees the universe and however this mathematical constructs reveals of nature, nature always existed. What was shown is that the discovery of the mathematics made it possible to understand something beautiful about nature. So in a sense the mathematics was always there, we just did not recognize it.:)