Showing posts with label Euclid. Show all posts
Showing posts with label Euclid. Show all posts

Saturday, January 06, 2007

Mersenne Prime: One < the Power of two

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.)

This is part of the education of my learning to understand the implications of the work of Riemann in context of the Riemann Hypothesis. Part of understanding what this application can do in terms helping us to see what has developed "from abstractions of mathematics," to have us now engaged in the "real world" of computation.

In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two multiplied by itself a certain number of times. Note that one is a power (the zeroth power) of two. Written in binary, a power of two always has the form 10000...0, just like a power of ten in the decimal system.

Because two is the base of the binary system, powers of two are important to computer science. Specifically, two to the power of n is the number of ways the bits in a binary integer of length n can be arranged, and thus numbers that are one less than a power of two denote the upper bounds of integers in binary computers (one less because 0, not 1, is used as the lower bound). As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system, might limit the score or the number of items the player can hold to 255 — the result of a byte, which is 8 bits long, being used to store the number, giving a maximum value of 28−1 = 255.

I look forward to the help in terms of learning to understand this "ability of the mind" to envision the dynamical nature of the abstract. To help us develop, "the models of physics" in our thinking. To learn, about what is natural in our world, and the "mathematical patterns" that lie underneath them.

What use the mind's attempt to see mathematics in such models?

"Brane world thinking" that has a basis in Ramanujan modular forms, as a depiction of those brane surface workings? That such a diversion would "force the mind" into other "abstract realms" to ask, "what curvatures could do" in terms of a "negative expressive" state in that abstract world.

Are our minds forced to cope with the "quantum dynamical world of cosmology" while we think about what was plain in Einstein's world of GR, while we witness the large scale "curvature parameters" being demonstrated for us, on such gravitational look to the cosmological scale.

Mersenne Prime

Marin Mersenne, 1588 - 1648

In mathematics, a Mersenne number is a number that is one less than a power of two.

Mn = 2n − 1.
A Mersenne prime is a Mersenne number that is a prime number. It is necessary for n to be prime for 2n − 1 to be prime, but the converse is not true. Many mathematicians prefer the definition that n has to be a prime number.

For example, 31 = 25 − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 211 − 1 is not a prime because it is divisible by 89 and 23. And 24 -1 = 15 can be shown to be composite because 4 is not prime.

Throughout modern times, the largest known prime number has very often been a Mersenne prime. Most sources restrict the term Mersenne number to where n is prime, as all Mersenne primes must be of this form as seen below.

Mersenne primes have a close connection to perfect numbers, which are numbers equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. In the 18th century, Leonhard Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist (any that do have to belong to a significant number of special forms).

It is currently unknown whether there is an infinite number of Mersenne primes.

The binary representation of 2n − 1 is n repetitions of the digit 1, making it a base-2 repunit. For example, 25 − 1 = 11111 in binary

So while we have learnt from Ulam's Spiral, that the discussion could lead too a greater comprehension. It is by dialogue, that one can move forward, and that lack of direction seems to hold one's world to limits, not seen and known beyond what's it like apart from the safe and security of home.

Sunday, August 27, 2006

Numerical Relativity and Math Transference

Part of the advantage of looking at computer animations is knowing that the basis of this vision that is being created, is based on computerized methods and codes, devised, to help us see what Einstein's equations imply.

Now that's part of the effort isn't it, when we see the structure of math, may have also embued a Dirac, to see in ways that ony a good imagination may have that is tied to the abstractions of the math, and allows us to enter into "their portal" of the mind.

NASA scientists have reached a breakthrough in computer modeling that allows them to simulate what gravitational waves from merging black holes look like. The three-dimensional simulations, the largest astrophysical calculations ever performed on a NASA supercomputer, provide the foundation to explore the universe in an entirely new way.

According to Einstein's math, when two massive black holes merge, all of space jiggles like a bowl of Jell-O as gravitational waves race out from the collision at light speed.

Previous simulations had been plagued by computer crashes. The necessary equations, based on Einstein's theory of general relativity, were far too complex. But scientists at NASA's Goddard Space Flight Center in Greenbelt, Md., have found a method to translate Einstein's math in a way that computers can understand.

Already having this basis of knowledge availiable, it was important to see what present day research has done for us, as we look at these images and allow them to take us into the deep space as we construct measures to the basis of what GR has done for us in a our assumptions of the events in the cosmo.

But it is more then this for me, as I asked the question, on the basis of math? I have enough links here to show the diversity of experience created from mathematical structures to have one wonder how indeed is th efinite idealization of imagination as a endless resource? You can think about livers if you likeor look at the fractorialization of the beginning of anythng and wonder I am sure.

That has been the question of min in regards to a condense matter theorist who tells us about the bulding blocks of matter can be anything. Well, in this case we are using "computer codes" to simulate GR from a mathematical experience.

So you see now don't you?:)

Is Math Invented or Discovered?

The question here was one of some consideration, as I wondered, how anyone could have delved into the nature of things and come out with some mathematcial model? Taken us along with the predecessors of endowwment thinking(imagination). To develope new roads. They didn't have to be 6 0r 7 roads Lubos, just a assumation. Sort of like, taking stock of things.

So I may ask, "what are the schematics of nature" and the build up starts from some place. Way back, before the computer modeling and such. A means, by which we will give imagination the tools to carry on.

So the journey began way back and the way in which such models lead our perspectives is the "overlay" of what began here in the postulates and moved on into other worldy abstractions?

This first postulate says that given any two points such as A and B, there is a line AB which has them as endpoints. This is one of the constructions that may be done with a straightedge (the other being described in the next postulate).

Although it doesn't explicitly say so, there is a unique line between the two points. Since Euclid uses this postulate as if it includes the uniqueness as part of it, he really ought to have stated the uniqueness explicitly.

The last three books of the Elements cover solid geometry, and for those, the two points mentioned in the postulate may be any two points in space. Proposition XI.1 claims that if part of a line is contained in a plane, then the whole line is. In the books on plane geometry, it is implicitly assumed that the line AB joining A to B lies in the plane of discussion.

One would have to know that the history had been followed here to what it is today.

Where Non-euclidean geometry began, and who were the instigators of imaginitive spaces now that were to become very dynamic in the xyzt direction.

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

These picture above, belongs to a much larger picture housed in the Raphael rooms in Rome. This particular picture many are familiar with as I use part of it as my profile picture. It is called the "Room of the Segnatura."

The point is, that if you did not know of the "whole picture" you would have never recognized it's parts?

Thursday, April 06, 2006

Hyperbolic Geometry and it's Rise

Omar Khayyám the mathematician(6 april 2006 Wikipedia)

He was famous during his lifetime as a mathematician, well known for inventing the method of solving cubic equations by intersecting a parabola with a circle. Although his approach at achieving this had earlier been attempted by Menaechmus and others, Khayyám provided a generalization extending it to all cubics. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry.

Giovanni Girolamo Saccheri(6 April 2006 Wikipedia)

Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin from 1694 to 1697, and philosophy, theology, and mathematics at Pavia from 1697 until his death. He was a protege of the mathematician Tommaso Ceva and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).

Of course the question as to "Victorian" was on mind. Is non-euclidean held to a time frame, or not?

Victorian Era(wikipedia 6 April 2006)

It is often defined as the years from 1837 to 1901

Time valuations are being thought about here. In regards too, non euclidean geometry and it's rise. Shows, many correlations within that time frame. So that was suprizing, if held to a context of the victorian socialogical time frame. But we know this statement is far from the truth?

Seminar on the History of Hyperbolic Geometry, by Greg Schreiber

We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.

History (wikipedia 6 April 2006)

Hyperbolic geometry was initially explored by Giovanni Gerolamo Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by János Bolyai, Karl Friedrich Gauss, and Nikolai Ivanovich Lobachevsky, after whom it is sometimes named.

Sunday, March 26, 2006

On Gauss's Mountain

You must understand that any corrections necessary are appreciated. The geometrical process spoken too here must be understood in it's historical development to undertand, how one can see differently.

Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

These tidbits, would have been evidence as projects predceding as "towers across valleys" amd "between mountain measures," to become what they are today. Allows us to se in ways that we are not used too, had we not learnt of this progression and design that lead from one to another.

8.6 On Gauss's Mountains

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface.

One had to recognize the process that historically proceeded in our overviews "to non-euclidean perspectives," "geometrically enhanced" through to our present day headings, expeirmentallly.

Michelson interferometer(27 Mar 2006 wikipedia)

Michelson interferometer is the classic setup for optical interferometry and was invented by Albert Abraham Michelson. Michelson, along with Edward Morley, used this interferometer for the famous Michelson-Morley experiment in which this interferometer was used to prove the non-existence of the luminiferous aether. See there for a detailed discussion of its principle.

But Michelson had already used it for other purposes of interferometry, and it still has many other applications, e.g. for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. But for most purposes, the geometry of the Mach-Zehnder interferometer is more useful.

A quick summation below leads one onto the idea of what experimental validation has done for us. Very simply, the graduation of interferometer design had been taken to astronomical proportions?

Today the Count expands on this for us by showing other information on expeirmental proposals. How fitting that this historical drama has been shown here, in a quick snapshot. As well the need for understanding the "principal inherent" in the project below.

VLBI is a geometric technique: it measures the time difference between the arrival at two Earth-based antennas of a radio wavefront emitted by a distant quasar. Using large numbers of time difference measurements from many quasars observed with a global network of antennas, VLBI determines the inertial reference frame defined by the quasars and simultaneously the precise positions of the antennas. Because the time difference measurements are precise to a few picoseconds, VLBI determines the relative positions of the antennas to a few millimeters and the quasar positions to fractions of a milliarcsecond. Since the antennas are fixed to the Earth, their locations track the instantaneous orientation of the Earth in the inertial reference frame. Relative changes in the antenna locations from a series of measurements indicate tectonic plate motion, regional deformation, and local uplift or subsidence.


  • Apollo Moon Measure
  • Sunday, February 26, 2006

    Roots, and the Rings of History

    The jump from conventional field theories of point-like objects to a theory of one-dimensional objects has striking implications. The vibration spectrum of the string contains a massless spin-2 particle: the graviton. Its long wavelength interactions are described by Einstein's theory of General Relativity. Thus General Relativity may be viewed as a prediction of string theory!
    Author Unknown

    How on earth, did such geometries take us into the abstract realms that it did?

    Euclid postulate found embedded and manifested in Reimann's developing perspective and the fruitation, General Relativity? IN some imaginative space, I see Einstein gleefully sitting, eating his apple?:)It must be "a tree" kind of thing.

    On Wassily Kandinsky Musical Score

    The term "Composition" can imply a metaphor with music. Kandinsky was fascinated by music's emotional power. Because music expresses itself through sound and time, it allows the listener a freedom of imagination, interpretation, and emotional response that is not based on the literal or the descriptive, but rather on the abstract quality that painting, still dependent on representing the visible world, could not provide.

    Well okay not this old, but the idea is, that learning through history had gone through much revision. That no matter the idea, that no physics was discovered then, it is the way that a result, manifested today. One may refer to Democritius, and know that the relevance, had started back then? Platonic solids, or some Pythagorean notion of numbers underlying nature?

    A cross section of a Rocky Mountain juniper (Juniperus scopulorum) snag found in El Malpais National Monument near Grants, New Mexico (it's about 3 feet across) (photo © H.D. Grissino-Mayer and R.K. Adams). This tree had a pith date of 256 BC and an outer ring of about AD 1320, making this tree nearly 1,600 years old when it died!

    But we do not want to talk about the impurities of philosophy, while we deal with "abstract and concrete" things do we?:) That the very subject, had been adopted and shunned by our greater teacher(Feynman), is good to know this is apart and separate from ourselves? That because a teacher did it, that we shall too? Or ,was it, that the appreciation for science at it's deepest level, didn't make room for such speculation, or some defining nature of a crackpottery(without history), who had supposedly calculated the proton's mass? Yes, the crackpot might of jumped on this notion. :) Scream about the aether and said, "strings is no different."

    In May 1996, Chris Baisan and I found this tree, a Rocky Mountain juniper (Juniperus scopulorum), at El Malpais National Monument (photo © H.D. Grissino-Mayer), and currently it is the reigning oldest wood yet discovered in New Mexico - 256 B.C.!

    SidneyFest and the Parents/Teachers Before Us

    Richard Feynman's history, was entangled with Murray's and seeing what was there then in Caltech 25 years ago, and what exists today at Harvard, is a reminder of what began under Murray's Gellman's umbrella. As to how John Schwartz career was preserved. The seed bed of Murray's Gellman understanding arose from the 1950's, to Susskind and Nambu in the 70's, to where in Harvard it is today.

    Some have wide sweeping claims to this history. The illegitimacy and rights to something, being theoretical dogma? As some false God set before us? While the religiousness of institution, is to bring forth those who work the equative understanding before them. Tried and tested. Who would in their right mind, is going to denounce the fathers/teachers before them?

    Lubos Motl:
    Just one or two comments. Murray also talked about the representation theory for the hadrons. Sidney played a rather important role in these developments, too. Murray mentioned that they sometimes incorporated the same particles into different representations - one of them was wrong and I forgot who was it. During his talk, Murray's cell phone started to ring twice. Murray Gell-Mann '69 interrupted his talk and studied who was calling him. "One call missed," was the answer after one minute of research. Gell-Mann, who is a Yale graduate, admitted that Harvard had been pretty good. Also, Harvard had created a string theory group only 25 years after Gell-Mann and his friends did the same thing at Caltech, which is not bad.

    So what new fruit have to you to bear, given the "disassociated state of existance," that one would never acknowledge. As never really needing to acknowldege, "standing on the shoulder of giants"?

    “Since Buddha was enlightened under a bodhi tree, it has become a symbol of enlightenment,” said Mahajan. “The tree is more than religion, it is a symbol of peace, meditation, oneness with yourself, finding harmony with the world. Whenever there is chaos going on, people can use this to find themselves, and a oneness with themselves and the world we live in.”


    Feynman's Rainbow, by Leonard MlodinowWarner Books 2003
    Euclid's Window: The Story of Geometry from Parallel Lines to Hyerspaceby Leonard MlodinowFreePress 2001

    Friday, October 07, 2005

    Projective Geometries

    Action at a Distance

    Now ths statement might seem counterproductive to the ideas of projective geometry but please bear with me.

    In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often with early theories of gravity and electromagnetism to describe how an object could "know" the mass (in the case of gravity) or charge (in electromagnetism) of another distant object.

    According to Albert Einstein's theory of special relativity, instantaneous action-at-a-distance was seen to violate the relativistic upper limit on speed of propagation of information. If one of the interacting objects were suddenly displaced from its position, the other object would feel its influence instantaneously, meaning information had been transmitted faster than the speed of light.

    Test of the Quantenteleportation over long distances in the duct system of Vienna Working group Quantity of experiment and the Foundations OF Physics Professor Anton Zeilinger

    Quantum physics questions the classical physical conception of the world and also the everyday life understanding, which is based on our experiences, in principle. In addition, the experimental results lead to new future technologies, which a revolutionizing of communication and computer technologies, how we know them, promise.

    In order to exhaust this technical innovation potential, the project "Quantenteleportation was brought over long distances" in a co-operation between WKA and the working group by Professor Anton Zeilinger into being. In this experiment photons in the duct system "are teleportiert" of Vienna, i.e. transferred, the characteristics of a photon to another, removed far. First results are to be expected in the late summer 2002.

    One of the first indications to me came as I looked at the history in regards to Klein's Ordering of Geometries. Now I must admit as a layman I am very green at this understanding but having jumped ahead in terms of the physics involved, its seems things have been formulating in my head, all the while, this underatnding in terms of this "order" has been lacking.

    In Euclidean geometry, the basic notions are distances and angles. The transformations that preserve distances and angles are precisely the rigid motions. Effectively, Klein's idea is to reverse this argument, take the group of rigid motions as the basic object, and deduce the geometry. So a legitimate geometric concept, in Euclidean geometry, is anything that remains unchanged after a rigid motion. Right-angled triangle, for example, is such a concept; but horizontal is not, because lines can be tilted by rigid motions. Euclid's obsession with congruent triangles as a method of proof now becomes transparent, for triangles are congruent precisely when one can be placed on top of the other by a rigid motion. Euclid used them to play the same role as the transformations favored by Klein.

    In projective geometry, the permitted transformations are projections. Projections don't preserve distances, so distances are not a valid conception projective geometry. Elliptical is, however, because any projection of an ellipse is another ellipse.

    So spelt out here is one way in which this progression becomes embedded within this hisotry of geometry, while advancing in relation to this association I was somewhat lifted to question about Spooky action at a distance. WEll if such projective phase was ever considered then how would distance be irrelevant(this sets up the idea then of probabilistic pathways and Yong's expeirment)? There had to be some mechanism already there tht had not been considered? Well indeed GHZ entanglement issues are really alive now and such communication networks already in the making. this connection raised somewhat of a issue with me until I saw the the phrase of Penrose, about a "New Quantum View"? Okay we know these things work very well why would we need such a statement, so I had better give the frame that help orientate my perspective and lead to the undertanding of spin.

    Now anywhere along the line anyone can stop such erudication, so that these ideas that I am espousing do not mislead. It's basis is a geometry and why this is important is the "hidden part of dirac's mathematics" that visionization was excelled too. It is strange that he would not reveal these things, all the while building our understanding of the quantum mechanical nature of reality. Along side of and leading indications of GR, why would not similar methods be invoked as they were by Einstein? A reistance to methodology and insightfulness to hold to a way of doing things that challenegd Dirac and cuased sleepless nights?

    Have a look at previous panel to this one.

    While indeed this blog entry open with advancements in the Test in Vienna, one had to understadn this developing view from inception and by looking at Penrose this sparked quite a advancement in where we are headed and how we are looking at current days issues. Smolin and others hod to the understnding f valuation thta is expeirmentally driven and it is not to far off to se ehosuch measure sare asked fro in how we ascertain early universe, happening with Glast determinations.

    Quantum Cryptography

    Again if I fast forward here, to idealization in regards to quantum computational ideas, what value could have been assigned to photon A and B, that if such entanglement states recognize the position of one, that it would immediately adjust in B?

    Spooky At any Speed
    If a pair of fundamental particles is entangled, measuring an attribute of one particle, such as spin, can affect the second particle, no matter how far away. Entanglement can even exist between two separate properties of a single particle, such as spin and momentum. In principle, single particles or pairs can be entangled via any combination of their quantum properties. And the strength of the quantum link can vary from partial to complete. Researchers are just beginning to understand how entanglement meshes with the theory of relativity. They have learned that the degree of entanglement between spin and momentum in a single particle can be affected by changing its speed ("boosting" it into a new reference frame) but weren't sure what would happen with two particles.

    So there is this "distance measure" here that has raised a quandry in my mind about how such a projective geometry could have superceded the idea of "spooky things" and the issues Einstein held too.

    So without understanding completely I made a quantum leap into the idealization in regards to "logic gates" as issues relevant to John Venn and introduced the idea around a "relative issues" held in my mind to psychological methods initiated by such entanglement states.

    As far a one sees here this issue has burnt a hole in what could have transpired within any of us that what is held in mind, ideas about geomtires floated willy Nilly about. How would such "interactive states" have been revealled in outer coverings.

    The Perfect Fluid

    Again I am fstforwding here to help portray question insights that had been most troubling to me. If suych supersymmetrical idealizations arose as to the source and beginning of existance how shall such views implement this beginning point?

    So it was not to unlikely, that my mind engaged further problems with such a view that symmetry breaking wouldhad tohave signalled divergence from sucha state of fluid that my mind encapsulated and developed the bubble views and further idealizations, about how such things arose from Mother.

    What would signal such a thing as "phase transitions" that once gauged to the early universe, and the Planck epoch, would have revealled the developing perspective alongside of photon developement(degrees of freedom) and released information about these early cosmological events.

    So I have advance quite proportinately from the title of this Blog entry, and had not even engaged the topological variations that such a leading idea could have advanced in our theoretcical views of Gluonic perceptions using such photonic ideas about what the tragectories might have revealled.

    So indeed, I have to be careful here that all the while my concepts are developing and advanced in such leaps, the roads leading to the understanding of the measure here, was true to form and revalled issues about things unseen to our eyes.

    It held visionistic qualities to geometric phases that those who had not ventured in to such entanglement states would have never made sense of a "measure in the making." It has it's limitation, though and why such departures need to be considered were also part of my question about what had to come next.

    Friday, June 03, 2005

    Music in Plato's Academy

    Academy was a suburb of Athens, named after the hero Academos or Ecademos. The site was continuously inhabited from the prehistoric period until the 6th century A.D. During the 6th century B.C., one of the three famous Gymnasiums of Athens was founded here. Moreover, it is recorded that Hippias, the son of Peisistratos, built a circuit wall, and Cimon planted the area with trees which were destroyed by Sulla in 86 B.C. In 387 B.C. Plato founded his philosophical school, which became very famous due to the Neoplatonists, and remained in use until A.D. 526, when it was finally closed down by emperor Justinian.

    Can a different kind of thinking encase the brain's ability to "envision the abstract of space" to know that it's harmonic values can be seen as the basis of experience?

    For instance, in Plato's academy, and in contrast, and the revolution of the sixties saw the Beatlemania as subversive? It's lifestyle?

    So on the one hand our parents resisting change in the formal art of music and lyrics, might have actually had some values?:) Rap, as a fungal fractorial growth of lyric inspired, emotive rythmics dances around the fire of a most primitive kind, finds an outlet for our youth?

    If one thought of the "dissonance of thinking" that Plato saw, could it corrupt youth to it's potential? He saw "sound as instrumental" in moving youth to the farthest reaches, while "bad noise" subversive. This wouldn't have been a cosmological assertion, could it, about the nature of our universe and chaos?

    So while beating hearts and rhtymns may have moved the harmonic brain into better retention times( there is some science here), this would not have been known to the revolt against beatle mania. Just that, they wanted to resist corruption of the youth?

    I have no script, so I adlib.

    An artistic view having grokked paradigmal changes, creates possible artistic pathways for all of us. It takes as little time as asking, "what the future holds."

    Ole forms of mathematical construct is a value of mathematical height of abstraction. We common people, would have never understood this loftiness, had we not see their images? But they speak more, about the content, then what little science is known to the public mind. So those who knew better, scoff and make fun?

    Feynman as a joker, gave us toy models in which to exorcise our mind of misplaced interactive features of science's theoretical opinion?

    The Mathematics Of Plato's Academy: A New Reconstruction(Second Edition)
    by David Fowler
    Reviewed by Fernando Q. Gouvêa

    Greek geometry was not arithmetized
    . In other words, the way we automatically connect the notion of "length" or "area" to numbers is something completely foreign to Greek mathematics. This is perhaps what makes it so hard for us to think mathematics in the Greek way. The idea that a length is a number is so deeply ingrained in our thought that it takes a conscious effort to conceive of an approach to geometry that does not make such an assumption. It is such an arithmetized interpretation that led historians to describe Book II of the elements as "geometric algebra". Fowler argues that Greek geometry was completely non-arithmetized. The strongest evidence comes from his analysis of the very difficult Book X, where he shows, I think successfully, that the way Euclid (or Theaetetus?) structures the argument precludes an arithmetical approach.

    Tuesday, December 28, 2004

    The Sound of the Landscape

    Ashmolean Museum, Oxford, UK

    As you know my name is Plato (The School of Athens by Raphael:)I have lived on for many years now, in the ideas that are presented in the ideas of R Buckminister Fuller, and with the helping hands of dyes, have demonstrated, the basis of these sounds in balloon configuration worth wondering, as simplice's of these higher dimensional realizations.

    A Chladni plate consist of a flat sheet of metal, usually circular or square, mounted on a central stalk to a sturdy base. When the plate is oscillating in a particular mode of vibration, the nodes and antinodes set up form a complex but symmetrical pattern over its surface. The positions of these nodes and antinodes can be seen by sprinkling sand upon the plates;

    Now you know from the previous post, that I have taken the technical aspects of string theory, and the mathematical formulations, and moved them into a encapsulated state of existance, much as brane theory has done.

    I look at this point(3 sphere derivation from euclid point line plane), on the brane and I wonder indeed, how 1R radius of this point becomes a circle. Indeed, we find this "idea" leaving the brane into a bulk manifestation of information, that we little specks on earth look for in signs of, through our large interferometers called LIGO's

    John Baez:
    Ever make a cube out of paper? You draw six square on the paper in a cross-shaped pattern, cut the whole thing out, and then fold it up.... To do this, we take advantage of the fact that the interior angles of 3 squares don't quite add up to 360 degrees: they only add up to 270 degrees. So if we try to tile the plane with squares in such a way that only 3 meet at each vertex, the pattern naturally "curls up" into the 3rd dimension - and becomes a cube!

    The same idea applies to all the other Platonic solids. And we can understand the 4d regular polytopes in the same way!

    The Hills of M Theory

    The hills are alive with the sound of music
    With songs they have sung for a thousand years.
    The hills fill my heart with the sound of music
    My heart wants to sing every song it hears....

    It's a wonder indeed that we could talk about the spacetime fabric and the higher dimensions that settle themselves into cohesive structures(my solids) for our satisfaction? What nodal points, do we have to wonder about when a string vibrates, and one does not have to wonder to much about the measure of the Q<->Q distance, as something more then the metric field resonates for us?

    This higher dimensional value seen in this distance would speak loudly to its possiblites of shape, but it is not easily accepted that we find lattice structures could have ever settled themselves into mass configurations of my solids.

    Lenny Susskind must be very pround of this landscape interpretation, as it is shown in the picture above. But the question is, if the spacetime fabric is the place where all these higher dimensions will reveal themselves, then what structure would have been defined in this expression from it's orignation, to what we see today?

    Alas, I am taken to the principles of," Spacetime in String Theory," by Gary T. Horowitz

    If one quantizes a free relativistic (super) string in flat spacetimeone finds a infinite tower of modes of increasing mass. Let us assume the string is closed,i.e., topologically a circle

    Friday, December 03, 2004

    Inverse Fourth Power Law

    By moving our perceptions to fifth dimenisonal views of Kaluza and KLein, I looked at methods that would help me explain that strange mathematical world that I had been lead too geometrically. If such a bulk existed, then how would we percieve scalable features of the energy distributed within the cosmo?

    The angular movements needed to signal the presence of additional dimensions are incredibly small — just a millionth of a degree. In February, Adelberger and Heckel reported that they could find no evidence for extra dimensions over length scales down to 0.2 millimetres (ref. 11). But the quest goes on. The researchers are now designing an improved instrument to probe the existence of extra dimensions below 0.1 mm. Other physicists, such as John Price of the University of Colorado and Aharon Kapitulnik of Stanford University in California, are attempting to measure the gravitational influence on small test masses of tiny oscillating levers.

    In previous posts I have outline the emergence and understanding of hyperdimensional realities that we were lead too. Our early forbearers(scientifically and artistic embued with vision) as they moved through the geometrical tendencies, that if followed , made me wonder about that this strange mathematical world. How would we describe it, and how would it make sense?

    Our new picture is that the 3-D world is embedded in extra dimensions," says Savas Dimopoulos of Stanford University. "This gives us a totally new perspective for addressing theoretical and experimental problems.

    Quantitative studies of future experiments to be carried out by LHC show that any signatures of missing energy can be used to probe the nature of gravity at small distances. The predicted effects could be accessible to the Tevatron Collider at Fermilab, but the higher energy LHC has the better chance.
    These colliders are still under construction, but results also have consequences for "table-top" experiments, being carried out here at Stanford, as well as the University of Washington and the University of Colorado. Here’s the basic idea: imagine there are two extra dimensions on a scale of a millimeter. Next, take two massive particles separated by a meter, at which distance they obviously behave according to the well-known rules of 3-D space. But if you bring them very close, say closer than one millimeter, they become sensitive to the amount of extra space around. At close encounter the particles can exchange gravitons via the two extra dimensions, which changes the force law at very short distances. Instead of the Newtonian inverse square law you’ll have an inverse fourth power law. This signature is being looked for in the ongoing experiments

    As you look at the issue of two points(introduction to hyperdimensional realites of quark confinement as a example), it is well understood, by this point that such emergence had to be geometriclaly consistent on many levels. That such royal roads leading too, culminate in some realistic measure? In that mathematical realm, we had left off, and in recognition of the fifth postulate of euclid. By acceptance and creation of this extra dimension, it was well apparent, that such tendencies were developing along side with the physics as well.

    But we had to determine where this mathematical realm had taken us, in terms of measure? We are quckly reminded of the place in which such measures become the constant rallying point around important questions of these views.

    Physics at this high energy scale describes the universe as it existed during the first moments of the Big Bang. These high energy scales are completely beyond the range which can be created in the particle accelerators we currently have (or will have in the foreseeable future.) Most of the physical theories that we use to understand the universe that we live in also break down at the Planck scale. However, string theory shows unique promise in being able to describe the physics of the Planck scale and the Big Bang.

    It wasn't a game anymore, that we did not suspect that reductionism might have taken us as far as the energy we could produce could take us? So we had to realize there was limitations to what we could percieve at such microscopic levels.

    High energy particles have extremely small wavelengths and can probe subatomic distances: high energy particle accelerators serve as supermicroscopes:

    To see What?

    The structure of matter


    Faced by these limitations and newly founded conceptual views based on the quantum mechanical discription of spacetime as strings, how would we be able to look at the cosmos with such expectancy? To know, that the views energetically described, would allow further developement of the theoretcial positons now faced with in those same reductionistic views?

    What has happened as a result of considering the GR perspective of blackholes, that we had now assigned it relevance of views in cosmological considerations? Such joining of quantum mechanical views and GR, lead us to consider the sigificance of these same events on a cosmological scale. This view, had to be consistent, geometrically lead too?

    If we discover the Planck scale near the TeV scale, this will represent the most profound discovery in physics in a century, and black hole production will be the most spectacular evidence of that new discovery

    Friday, November 26, 2004

    No Royal Road to Geometry?

    All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

    It was interesting to me that I find some thread that has survived through the many centuries , that moves through the hands of individuals, to bring us to a interesting abstract world that few would recognize.

    While Euclid is not known to have made any original discoveries, and the Elements is based on the work of his predecessors, it is assumed that some of the proofs are his own and that he is responsible for the excellent arrangement. Over a thousand editions of the work have been published since the first printed version of 1482. Euclid's other works include Data, On Divisions of Figures, Phaenomena, Optics, Surface Loci, Porisms, Conics, Book of Fallacies, and Elements of Music. Only the first four of these survive.

    Of interest, is that some line of departure from the classical defintions, would have followed some road of developement, that I needed to understand how this progression became apparent. For now such links helped to stabilize this process and the essence of the departure form this classical defintion needed a culmination reached in Einstein's General Relativity. But long before this road was capture in it's essence, the predecessors in this projective road, develope conceptual realizations and moved from some point. To me, this is the fifth postulate. But before I draw attention there I wanted to show the index of this same projective geometry.

    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.

    The move from the fifth postulate had Girolamo Saccheri, S.J. (1667 - 1733) ask the question?

    What if the sum of the angles of a triangle were not equal to 180 degrees (or p radians)?" Suppose the sum of these angles was greater than or less than p. What would happen to the geometry we have come to depend on for so many things? What would happen to our buildings? to our technology? to our countries' boundaries?

    The progression through these geometries leads to global perspectives that are not limited to the thread that moves through these cultures and civilizations. The evolution dictates that having reached Einstein GR that we understand that the world we meet is a dynamical one and with Reason, we come t recognize the Self Evident Truths.

    At this point, having moved through the geometrical phases and recognitions, the physics of understanding have intertwined mathematical realms associated with Strings and loop and other means, in which to interpret that dynamical world called the Planck Length(Quantum Gravity).

    Reichenbach on Helmholtz