Descriptive geometry

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

The search for this "cup that overflow" is at the heart of all who venture for the lifeblood of the mystery of life. While Atiyah speaks to a unification of Quantum theory and Relativity, it is not without a understanding on Einstein's part that having gained from Marcel Grossmann, that such a descriptive geometry could be leading Einstein to discover the very basis of General relativity?

Marcel Grossmann was a mathematician, and a friend and classmate of Albert Einstein. He became a Professor of Mathematics at the Federal Polytechnic Institute in Zurich, today the ETH Zurich, specialising in descriptive geometry.

So what use "this history" in face of the unification of the very large with the very small? How far back should one go to know that the steps previous were helping to shape perspective for the future. Allow for perspective to be changed, so that new avenues of research could spring forth

Gaspard Monge, Comte de Péluse-Portrait by Naigeon in the Musée de Beaune Born: 9 May 1746 in Beaune, Bourgogne, France
Died: 28 July 1818 in Paris, France-was a French mathematician and inventor of descriptive geometry.

Monge contributed (1770–1790) to the Memoirs of the Academy of Turin, the Mémoires des savantes étrangers of the Academy of Paris, the Mémoires of the same Academy, and the Annales de chimie, various mathematical and physical papers. Among these may be noticed the memoir "Sur la théorie des déblais et des remblais" (Mém. de l’acad. de Paris, 1781), which, while giving a remarkably elegant investigation in regard to the problem of earth-work referred to in the title, establishes in connection with it his capital discovery of the curves of curvature of a surface. Leonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him. Monge's memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795. (Monge's 1781 memoir is also the earliest known anticipation of Linear Programming type of problems, in particular of the transportation problem. Related to that, the Monge soil-transport problem leads to a weak-topology definition of a distance between distributions rediscovered many times since by such as L. V. Kantorovich, P. Levy, L. N. Wasserstein, and a number of others; and bearing their names in various combinations in various contexts.) A memoir in the volume for 1783 relates to the production of water by the combustion of hydrogen; but Monge's results had been anticipated by Henry Cavendish.

Descriptive geometry

Example of four different 2D representations of the same 3D object

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. [1] The theoretical basis for descriptive geometry is provided by planar geometric projections. Gaspard Monge is usually considered the "father of descriptive geometry". He first developed his techniques to solve geometric problems in 1765 while working as a draftsman for military fortifications, and later published his findings. [2]

Monge's protocols allow an imaginary object to be drawn in such a way that it may be 3-D modeled. All geometric aspects of the imaginary object are accounted for in true size/to-scale and shape, and can be imaged as seen from any position in space. All images are represented on a two-dimensional drawing surface.

Descriptive geometry uses the image-creating technique of imaginary, parallel projectors emanating from an imaginary object and intersecting an imaginary plane of projection at right angles. The cumulative points of intersections create the desired image.

So given the tools, we learnt to see how objects within a referenced space, given to such coordinates, have been defined in that same space. Where is this point with in that reference frame?

What is born within that point, that through it is emergent product. Becomes a thing of expression from nothing? It's design and all, manifested as a entropic valuation of the cooling period? Crystalline shapes born by design, and by element from whence it's motivation come? An arrow of time?

Melancholia Magic Square

The most famous 4 x 4 magic square is the so-called Melancholia magic square. In 1514, Albrecht Dürer used the square in his engraving Melancholie which depicts the indecision of the intellectual. The two numbers, 15 and 14, in the centre of the bottom row date the engraving.

This image above I have discussed before in this website. It is also in the legend on the right hand side of this web page.

See:Number Patterns-Fun with Curves & Topology-Jill Britton

Projective Geometries

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

It is always important to see the progression geometries follow. So you have to know "the origins of geometry" before you can begin to expand into the abstract spaces that space will allow and accomadate.

Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false. This means that it is possible to assign meanings to the terms "point" and "line" in such a way that they satisfy the first four postulates but not the parallel postulate. These are called non-Euclidean geometries. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.

In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead:

The projective axiom: Any two lines intersect (in exactly one point).

Gino Fano

Gino Fano (5 January 1871 - 8 November 1952) was an Italian mathematician. He was born in Mantua, Italy and died in Verona, Italy.

Fano worked on projective and algebraic geometry; the Fano plane and Fano varieties are named for him.

Ugo Fano and Robert Fano were his sons.

There are reasons with which I wanted to share information about this gentlemen. What has been written in context of "finite geometry." You must know I am never the expert, but one who aspires to learn what is needed to learn and understand what is happening with regards to model presented by Garrett Lisi.

You must know that my mind thinks in abstract spaces and is involved in a wide range of variables expressed in terms of the dimensional attributes of actions within that space.

Diagram of the Fano plane

In finite geometry, the Fano plane (after Gino Fano) is the projective plane with the least number of points and lines: 7 each.

See: Elements of Finite Geometry

Self Evident Dimensional Perspective

Where a dictionary proceeds in a circular manner, defining a word by reference to another, the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their definition. Alain Connes

John Merryman in comment section:

Can they propose these dimensions as anything more then the copyrighted product of their own imagination and not loose control over the idea?

Okay I have a problem with the term "static."

I'll just give you an example of what I am thinking in relation to how we may perceive dimension and then of course, there is a mathematical interpretation of topological spaces that others are better qualified to speak on. How could there be such a geometrical interpretation at such quantum levels.

Is there such thing as "a breakdown of time" within the context of measure? It is my ignorance that separates me from the more educated here, yet it is not without wanting to understand, that I am pushing this point further.

Think about the following concept for a moment.

Savas Dimopoulos:

Here’s an analogy to understand this: imagine that our universe is a two-dimensional pool table, which you look down on from the third spatial dimension. When the billiard balls collide on the table, they scatter into new trajectories across the surface. But we also hear the click of sound as they impact: that’s collision energy being radiated into a third dimension above and beyond the surface. In this picture, the billiard balls are like protons and neutrons, and the sound wave behaves like the graviton.

Here we are given a new look into another dimension? A shift from what is euclidean, to what is now non-euclidean. It is really quite simple to understand "what Einstein did" when we now talk about gravity.




Juan Maldacena:

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.

While it is abstract, the move to thinking in the new way is important while we are looking at the whole picture.

Albert Einstein

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.Albert Einstein p. 83 of his Relativity: The Special and the General Theory

There are deeper philosophical questions here about being a realist and an anti-realist.?

René Thom

See:René Thom:René Thom (September 2, 1923 – October 25, 2002) was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became celebrated for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Christopher Zeeman). He received the Fields Medal in 1958.



Photograph by Paul Halmos

Much emphasis has been placed during the past fifty years on the reconstruction of the geometric continuum from the natural integers, using the theory of Dedekind cuts or the completion of the field of rational numbers. Under the influence of axiomatic and bookish traditions, man perceived in discontinuity the first mathematical Being: "God created the integers and the rest is the work of man." This maxim spoken by the algebraist Kronecker reveals more about his past as a banker who grew rich through monetary speculation than about his philosophical insight. There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity. If one has any consciousness at all, it is consciousness of time and space; geometric continuity is in some way inseparably bound to conscious thought.

The Ring of Truth

Savas Dimopoulos:Here’s an analogy to understand this: imagine that our universe is a two-dimensional pool table, which you look down on from the third spatial dimension. When the billiard balls collide on the table, they scatter into new trajectories across the surface. But we also hear the click of sound as they impact: that’s collision energy being radiated into a third dimension above and beyond the surface. In this picture, the billiard balls are like protons and neutrons, and the sound wave behaves like the graviton.

On the title it is important to understand what is being implied within the context of this post. What came to mind immediately when Bee wrote"Ring of Truth" in her post, "A Theoretically Simple Exception of Everything." Joseph Weber came to mind.

Joseph Weber 1919 - 2000

Joseph Weber, the accomplished physicist and electrical engineer, has died at the age of 81. Weber's diverse research interests included microwave spectroscopy and quantum electronics, but he is probably best known for his investigations into gravitational waves.

In the late 1950s, Weber became intrigued by the relationship between gravitational theory and laboratory experiments. His book, General Relativity and Gravitational Radiation, was published in 1961, and his paper describing how to build a gravitational wave detector first appeared in 1969. Weber's first detector consisted of a freely suspended aluminium cylinder weighing a few tonnes. In the late 1960s and early 1970s, Weber announced that he had recorded simultaneous oscillations in detectors 1000 km apart, waves he believed originated from an astrophysical event. Many physicists were sceptical about the results, but these early experiments initiated research into gravitational waves that is still ongoing. Current gravitational wave experiments, such as the Laser Interferometer Gravitational Wave Observatory (LIGO) and Laser Interferometer Space Antenna (LISA), are descendants of Weber's original work.

Weber was born in 1919 in Paterson, New Jersey, and graduated in 1940. He spent eight years as an electrical engineer in the US Navy, and was assigned as navigator on the aircraft carrier Lexington during World War II. After his resignation from the Navy in 1948, Weber went on to obtain his PhD in 1951 from the Catholic University of America. He was appointed professor of electrical engineering at the University of Maryland, and he moved into the physics department in 1961 when he began his investigations into gravitational waves.

Weber died on 30 September in Pittsburgh, Pennsylvania. He is survived by his wife, the astrophysicist Virginia Trimble.

Bee writes about "Ring of Truth" from Lee Smolin's book,

"But we are also fairly sure that we do not yet have all the pieces. Even with the recent successes, no idea yet has that absolute ring of truth." p. 255 (US hardcover).

So I pulled this above from Bee's comment blog for further reference. To help make my point about gravitational wave detection and all the kinds of wav(Y)es in which gravity can now be looked at.

So of course it is necessary to include the commentary from Bee's reference too, Garrett Lisi's comment section, to help one see the complex rotations that speaks to all manifestations(geometrical foresight on complex rotations in dimensional spaces), from the origins of all a particle creations to the elemental understanding given in context of the post by Bee.

"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 universe-Wayne Hu

Stefan,

Maybe I have a better chance to understand them when their relation to the original post is more than just the word "gravity" in both of them?

Your "toying with the way we see gravitational and gravity waves?" Dealing with the objective world with ancient ideas?

I pointed to the differences.

Plato:Wherever there are no gravitational waves the spacetime is flat. One would have to define these two variances. One from understanding the relation to "radiation" and the other, "to the perfectly spherically symmetric."

But still to see such dynamics in terms of the "mathematical abstract" I see see no reason why you would "lesson my points" on helping one to see these differences in the space around us.

This recording was produced by converting into audible sounds some of the radar echoes received by Huygens during the last few kilometres of its descent onto Titan. As the probe approaches the ground, both the pitch and intensity increase. Scientists will use intensity of the echoes to speculate about the nature of the surface.

So I may point to the ways in which one may synthesized the views of the world in relation to not only "sound" as Kris just talks about, but also about how one may transform that sound "to colour."

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.

Cosmic variance may talk about synesthesia yet you cannot stop the changes such understanding brings to the emotive forces that surround earth and us.

Such a shift to bulk perspective is not without it's lessons on progressing the views of gravity in "all situations."

I am not so smart, just that I may see differently then you Stefan. :)

We can't actually hear gravitational waves, even with the most sophisticated equipment, because the sounds they make are the wrong frequency for our ears to hear. This is similar in principle to the frequency of dog whistles that canines can hear, but are too high for humans. The sounds of gravitational waves are probably too low for us to actually hear. However, the signals that scientists hope to measure with LISA and other gravitational wave detectors are best described as "sounds." If we could hear them, here are some of the possible sounds of a gravitational wave generated by the movement of a small body in spiralling into a black hole.

Does anybody really understand what is happening when the conceptual foundation allows new perspective to form? New theories to make their way into challenging the very foundations of our reality?

Every step in the production of the "conceptual framework" is an exercise in how perception is being changed. Can be changed.

There are moderators of all sorts who govern the information that is being written. How one view can be portrayed and sits in contradiction to the way String theory uses E8 is not the reason one might of suspected problems with acceptance here or there.

It s a organizational method on how to respond and place it accordingly. Peter is being paranoid? :)

Where Spacetime is flat?

......A Condensative Result exists. Where "energy concentrates" and expresses outward.

I mean if I were to put on my eyeglasses, and these glasses were given to a way of seeing this universe, why not look at the whole universe bathed in such spacetime fabric?

This a opportunity to get "two birds" with one stone?

I was thinking of Garrett's E8 Theory article and Stefan's here.

On March 31, 2006 the high-resolution gravity field model EIGEN-GL04C has been released. This model is a combination of GRACE and LAGEOS mission plus 0.5 x 0.5 degrees gravimetry and altimetry surface data and is complete to degree and order 360 in terms of spherical harmonic coefficients.

High-resolution combination gravity models are essential for all applications where a precise knowledge of the static gravity potential and its gradients is needed in the medium and short wavelength spectrum. Typical examples are precise orbit determination of geodetic and altimeter satellites or the study of the Earth's crust and mantle mass distribution.

But, various geodetic and altimeter applications request also a pure satellite-only gravity model. As an example, the ocean dynamic topography and the derived geostrophic surface currents, both derived from altimeter measurements and an oceanic geoid, would be strongly correlated with the mean sea surface height model used to derive terrestrial gravity data for the combination model.

Therefore, the satellite-only part of EIGEN-GL04C is provided here as EIGEN-GL04S1. The contributing GRACE and Lageos data are already described in the EIGEN-GL04C description. The satellite-only model has been derived from EIGEN-GL04C by reduction of the terrestrial normal equation system and is complete up to degree and order 150.

How many really understand/see the production of gravitational waves in regards to Taylor and Hulse?

To see Stefan's correlation in terms of "wave production" is a dynamical quality to what is still be experimentally looked for by LIGO?

As scientists, do you know this?

6:41 AM, November 11, 2007

See here

Thus the binary pulsar PSR1913+16 provides a powerful test of the predictions of the behavior of time perceived by a distant observer according to Einstein's Theory of Relativity.

Since we know the theory of Relativity is about Gravity, then how is it the applications can be extended to the way we see "anew" in our world?

A sphere, our earth, not so round anymore.

Uncle has tried to correct me on "isostatic adjustment."

Derek Sears, professor of cosmochemistry at the University of Arkansas, explains. See here

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. K. Shumacker. Scientific America

Do not have time to follow up at this moment.

7:02 AM, November 11, 2007

.....and here.


In context of the post and differences, I may not have pointed to the substance of the post, yet I would have dealt with my problem in seeing.

In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like a spinning, expanding or contracting sphere) or cylindrically symmetric (like a spinning disk).

A simple example is the spinning dumbbell. Set upon one end, so that one side of the dumbell is on the ground and the other end is pointing up, the dumbbell will not radiate when it spins around its vertical axis but will radiate if it tumbles end-over-end. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. If we imagine an extreme case in which the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.

Given the context of the "whole universe" what is actually pervading, if one did not include gravity?



So singularities are pointing to the beginning(i), yet, we do not know if we should just say, the Big Bang, because, one would had to have calculated the energy used and where did it come from "previous" to manifest?

So some will have this philosophical position about "nothing(?)," and "everything as already existing."

Wherever there are no gravitational waves the space time is flat. One would have to define these two variances. One from understanding the relation to "radiation" and the other "perfectly spherically symmetric."

Dark Matter Issue

We’re faced with the same choices today, with galaxies and clusters playing the role of the Solar System. Except that the question has basically been answered, by observations such as the Bullet Cluster. If you modify gravity, it’s fairly straightforward (although harder than you might guess, if you’re careful about it) to change the strength of gravity as a function of distance. So you can mock up “dark matter” by imagining that gravity at very large distances is just a bit stronger than Newton (or Einstein) would have predicted — as long as the hypothetical dark matter is in the same place as the ordinary matter is.

In Dark Matter Still Existing, Sean Carroll of Cosmic Variance lays the topic out for readers to understand his position on this issue.

An intergalactic collision is providing astronomers with a giant payoff: the first direct evidence of the invisible material that theorists say holds galaxies together and accounts for most of the universe's mass.

CRASH COURSE. This composite image from several observatories and telescopes shows where two clusters of galaxies collided 100 million years ago. The ordinary matter, shown in pink, from the two galaxies collided, whereas the dark matter from each galaxy, shown in purple, passed straight through.
Markevitch, et al., Clowe, et al., Magellan, Univ. of Arizona, CXC, CfA, STScI, ESO WFI, NASA

What is Dark Matter? How Can We Make It in the LaboratoryConclusions

Particle physics is in the midst of a great revolution. Modern data and ideas have challenged long-held beliefs about matter, energy, space and time. Observations have confirmed that 95 percent of the universe is made of dark energy and dark matter unlike any we have seen or touched in our most advanced experiments. Theorists have found a way to reconcile gravity with quantum physics, but at the price of postulating extra dimensions beyond the familiar four dimensions of space and time. As the magnitude of the current revolution becomes apparent, the science of particle physics has a clear path forward. The new data and ideas have not only challenged the old ways of thinking, they have also pointed to the steps required to make progress. Many advances are within reach of our current program; others are close at hand. We are extraordinarily fortunate to live in a time when the great questions are yielding a whole new level of understanding. We should seize the moment and embrace the challenges.

See:What is Dark Matter/Energy?

Minature Satellites in Space

KC-135 Flight Experiments

The Reduced Gravity Program at NASA's Johnson Space Center provides the unique "weightless" or "zero-g" environment of space flight using a specially modified KC-135A. The KC flies parabolic arcs to produce weightless periods of 20 to 25 seconds. This capability is ideal for the development and verification of space hardware, experiments, crew training and basic research.

Flight tests of the SPHERES testbed onboard NASA's KC-135 accomplished two objectives: (1) establish the functionality of testbed systems and subsystems and (2) perform limited formation flight experiments. Flight experiments were conducted over two separate weeks in early 2000, once in mid-February and again in late March. The time between flights was used to refine operations protocols, improve testbed systems, and develop more complicated experiments using lessons learned from the first week of flights.

There is a always a history to such developments and to me not ever knowing of this process about the spheres, I find it very satisfying to have some "correlation of cognition."

Synchronized Position Hold, Engage, Reorient Experimental Satellites


The MIT Space Systems Laboratory developed the SPHERES (Synchronized Position Hold Engage and Reorient Experimental Satellites) laboratory environment to provide DARPA, NASA, and other researchers with a long term, replenishable, and upgradable testbed for the validation of high risk metrology, control, and autonomy technologies for use in formation flight and autnomous docking, rendezvous and reconfiguration algorithms. These technologies are critical to the operation of distributed satellite and docking missions such as Terrestrial Planet Finder and Orbital Express.



Most would not understand the significance of this posting.

For me it is the correlation of insight that I had in a dream sometime ago, about my future. Most would not of thought that we would be capable as human beings to have this ability, to be able to project people we will become, to a people who are working in relation to what is developing and was developed in this post.

In my dream I am releasing a satellite, a Christmas tree design one without all the bells and lights.

Beauty and Asymmetry

BEHOLDING beauty with the eye of the mind, he will be enabled to bring forth, not images of beauty, but realities, for he has hold not of an image but of a reality, and bringing forth and nourishing true virtue to become the friend of God and be immortal, if mortal man may. Would that be an ignoble life? PLATO

One would have had to understand the idea "behind symmetry realizations" to understand that "asymmetry" could have been found of value?

Over time, people will get a sense of the thinking of Plato, and the way in which I have used his work. "The work of others" to see the way in which Plato may of saw.

This gives one a "starting point" in contrast to today's science, and what is evident from such developments.

I do include the spiritual basis as my "postulate about reality."

Plato said: I then postulate that all things have existed forever. It is only our ignorance of what actually exists in reality that prevents us from understanding the full scope of our understanding of God within context of this reality.

While one may of thought it is purely "abstractedly and mathematical by design," I am saying such thinking is not without the understanding of the asymmetrical relation of symmetry. As well as, "a relationship" to the way in which we deal with this reality.

It is not so unlikely that shadows are cast of all the things in the beginning, that there is a "truer point of expression" still within the context of this universe. Yet, it is universal, that it lies at the basis of reality.

See: Craftsman of Plato