Showing posts with label School of Athens. Show all posts
Showing posts with label School of Athens. Show all posts

Tuesday, February 11, 2014

The Monochord


Pythagoras in School of Athens
  
A monochord is an ancient musical and scientific laboratory instrument. It is also the class-name for any musical stringed instrument having only one string (such as the Vietnamese Đàn bầu). The word "monochord" comes from the Greek and means literally "one string." In a true monochord, a single string is stretched over a sound box. The string is fixed at both ends while one or many movable bridges are manipulated to demonstrate mathematical relationships between sounds.
[Slide 3-3: Closeup of Tablet, Bouleau. Janson, H. W. History of Art. (Fifth Edition.NY: Abrams, 1995). p.497
Raphael’s School of Athens shows Pythagoras is explaining the musical ratios to a pupil. Notice the tablet. It shows the words diatessaron, diapente, diapason. The roman numerals for 6, 8, 9, and 12, showing the ratio of the intervals, same as in the music book frontispiece.The word for the tone, ΕΠΟΓΛΟΩΝΕΠΟΓΛΟΩΝΕΠΟΓΛΟΩΝΕΠΟΓΛΟΩΝ, at the top. Under the tablet is a triangular number 10 called the sacred tetractys]

 The monochord can be used to illustrate the mathematical properties of musical pitch. For example, when a monochord's string is open it vibrates at a particular frequency and produces a pitch. When the length of the string is halved, and plucked, it produces a pitch an octave higher and the string vibrates at twice the frequency of the original (2:1) About this sound Play . Half of this length will produce a pitch two octaves higher than the original—four times the initial frequency (4:1)—and so on. Standard diatonic Pythagorean tuning (Ptolemy’s Diatonic Ditonic) is easily derived starting from superparticular ratios, (n+1)/n, constructed from the first four counting numbers, the tetractys, measured out on a monochord.[citation needed]


The Divine Monochord, from Fludd’s Utriusque Cosmi Maioris Scilicet et Minoris Metaphysica (1617)

The name "monochord" is sometimes incorrectly applied to an instrument with one open string and a second string with a movable bridge; however, such a two-string instrument is properly called a bichord. With two strings you can easily demonstrate how various musical intervals sound. Both open strings are tuned to the same pitch, and then the movable bridge is put in a mathematical position to demonstrate, for instance, the major third (at 4/5th of the string length) About this sound Play  or the minor third (at 5/6th of the string length) About this sound Play .

***



SEE:Infinite Fire Webinar II - The Emblemata of the Atalanta Fugiens by Dr. Peter J. Forshaw



See: Atalanta fugiens

Saturday, February 08, 2014

Friday, August 08, 2008

William Thurston

Xianfeng David Gu and Shing-Tung Yau
To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude and latitude lines on the rabbit allow mathematicians to map it onto different forms while preserving information.


William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.

“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Elusive Proof, Elusive Prover: A New Mathematical Mystery

Some of us are of course interested in how we can assign the relevance to perceptions the deeper recognition of the processes of nature. How we get there and where we believe they come from. As a layman I am always interested in this process, and of course, life's mysteries can indeed be a motivating factor. Motivating my interest about the nature of things that go unanswered and how we get there.


William Paul Thurston
(born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics. He is currently a professor of mathematics and computer science at Cornell University (since 2003).


There are reasons with which I present this biography, as I did in relation to Poincaré and Klein. The basis of the question remains a philosophical one for me that I question the basis of proof and intuition while considering the mathematics.

Mathematical Induction

Mathematical Induction at a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.

Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see Problem of induction for more information). In fact, mathematical induction is a form of deductive reasoning and is fully rigorous
.


Deductive reasoning

Deductive reasoning is reasoning which uses deductive arguments to move from given statements (premises), which are assumed to be true, to conclusions, which must be true if the premises are true.[1]

The classic example of deductive reasoning, given by Aristotle, is

* All men are mortal. (major premise)
* Socrates is a man. (minor premise)
* Socrates is mortal. (conclusion)

For a detailed treatment of deduction as it is understood in philosophy, see Logic. For a technical treatment of deduction as it is understood in mathematics, see mathematical logic.

Deductive reasoning is often contrasted with inductive reasoning, which reasons from a large number of particular examples to a general rule.

Alternative to deductive reasoning is inductive reasoning. The basic difference between the two can be summarized in the deductive dynamic of logically progressing from general evidence to a particular truth or conclusion; whereas with induction the logical dynamic is precisely the reverse. Inductive reasoning starts with a particular observation that is believed to be a demonstrative model for a truth or principle that is assumed to apply generally.

Deductive reasoning applies general principles to reach specific conclusions, whereas inductive reasoning examines specific information, perhaps many pieces of specific information, to impute a general principle. By thinking about phenomena such as how apples fall and how the planets move, Isaac Newton induced his theory of gravity. In the 19th century, Adams and LeVerrier applied Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune (specific conclusions) from perturbations in the observed orbit of Uranus (specific data).


Deduction and Induction



Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.


Back to the lumping in of theology alongside of Atlantis. Rebel dreams, it is hard to remove one's colour once they work from a certain premise. Atheistic, or not.

Seeking such clarity would be the attempt for me, with which to approach a point of limitation in our knowledge, as we may try to explain the process of the current state of the universe, and it's shape. Such warnings are indeed appropriate to me about what we are offering for views from a theoretical standpoint.

The basis presented here is from a layman standpoint while in context of Plato's work, brings some perspective to Raphael's painting, "The School of Athens." It is a central theme for me about what the basis of Inductive and deductive processes reveals about the "infinite regress of mathematics to the point of proof."

Such clarity seeking would in my mind contrast a theoretical technician with a philosopher who had such a background. Raises the philosophical question about where such information is derived from. If ,from a Platonic standpoint, then all knowledge already exists. We just have to become aware of this knowledge? How so?

Lawrence Crowell:
The ball on the Mexican hat peak will under the smallest perturbation or fluctuation begin to fall off the peak, roll into the trough and the universe tunnels out of the vacuum or nothing to become a “something.”


Whether I attach a indication of God to this knowledge does not in any way relegate the process to such a contention of theological significance. The question remains a inductive/deductive process?

I would think philosophers should weight in on the point of inductive/deductive processes as it relates to the search for new mathematics?

Allegory of the Cave

For me this was a difficult task with which to cypher the greater contextual meaning of where such mathematics arose from. That I should implore such methods would seem to be, to me, in standing with the problems and ultimates searches for meaning about our place in the universe. Whether I believe in the "God nature of that light" should hold no atheistic interpretation to my quest for the explanations about the talk on the origins of the universe.

See:

  • The Sound of Billiard Balls

  • Mathematical Structure of the Universe
  • Monday, February 11, 2008

    Inside Out

    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.




    I am trying to "create a image" that will use the one above. It is important that the select quoted comment below is understood. This can't be done without some reference.

    So while the exercise may be going on "inside" things are happening on the outside. Scientists have never been completely honest with themselves, while some may concern themselves with whose name said what?


    I use Plato as a namesake obviously, because of what I saw of some of our influential minds speaking, all the while making inferences to Plato. When ever you read something that resonates with you, it is of value because it correlates to something that you already know. This is what I tried to get across in the previous post, about what is "self evident." Little do some people recognize that while I may have inferred the point of some philosophical foundations, it is not without recognizing that the "qualitative phrases" have to be reduced as well to a logic. To reason.

    How do you do that? Well I'll tell you what I found and then you can think whether I understood reason in it's proper format. Whether I understood the "shadows of Plato" to mean something other then what could have been interpreted as being wrong. What is that analogy of the Cave really mean?

    Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.


    Yes I did not enter the halls of higher learning in the traditional ways. You can converse for many years, does not mean you become devoid of the lessons that spoken amongst the commentors. How is it you can think that while listening to scientists you cannot uncover the the processes they use? If I had given thirty years to study, what exactly had I studied? I am a doctor of nothing.:)

    This is a torus (like a doughnut) on which several circles are located. Unlike on a Euclidean plane, on this surface it is impossible to determine which circle is inside of which, since if you go from the black circle to the blue, to the red, and to the grey, you can continuously come back to the initial black, and likewise if you go from the black to the grey, to the red, and to the blue, you can also come back to the black.

    My quote at Backreaction on this and that, reveals not only part of the understanding gained through this "infinite regress," but also the understanding we have with the world around us. Some would be better served to see the image of the Klein bottle, but I wanted to show what is going on in a "abstract way" to what is happening inside of us, and at the same time, what is happening outside.



    I had used the brain and head as a place of our conscious awareness within context of our environment, our bodies. The topological explanations of the numbers above, and used them in the next paragraph. There will be confusion with the colour lines, please disregard that.

    While I talked of the emotive and mental realities. I included the spiritual development in the end. The way this interaction takes place, is sometimes just as the mental function(yellow). Other times, it is the emotive realization of the experience. It is coloured by our emotion(red).

    While we interact with our environment, there is this turning inside out, continuously. Sometimes we may say that "1" is the emotive realization, while the number 2 is seen as a mental extension of the situation. While the areas overlap each other, an outward progression may mean that the spiritual progress is numbered 4, while the interaction of the emotive, mental and spiritual progression may be number 3. Ultimately the spiritual progression is 4 (Violet). All these colours can mix and are significant in themself. They reveal something about our very constitution.

    While some may wonder how could any conceptualization ever integrate the "Synesthesia views" of the world when it sees itself presented with such a comparison? The journey of course leads to the "Colour of Gravity." Discard your body, and one will wonder about the "clear light." What it means, in the "perceptive state of existence." If one is prepared, then one shall not have "to much time on their hands" getting lost in the fog.

    Plato and Aristotle, Up and Down by Kelley L. Ross, Ph.D.

    Rafael has Plato pointing up and Aristotle gesturing down to indicate the difference in their metaphysics. For Plato, true existence is in the World of Forms, in relation to which this world (of Becoming) is a kind of shadow or image of the higher reality. Aristotle, on the other hand, regards individual objects in this world as "primary substance" and dismisses Plato's Forms -- except for God as a pure actuality, without matter.

    However, when it comes to ethics and politics, the gestures should be reversed. Plato, like Socrates, believed that to do the good without error, one must know what the good is. Thus, we get the dramatic moment in the Republic where Plato says that philosophers, who have escaped from the Cave and come to understand the higher reality, must be forced to return to this world and rule, so that their wisdom can benefit the state. Aristotle, on the other hand, says that the "good" is simply the goal of various particular activities, without one meaning in Plato's sense. The particular activities of most human affairs involve phronésis, "practical wisdom." This is not sophía, true wisdom, for Aristotle, which involves the theoretical knowledge of the highest things, i.e. the gods, the heavens, and God.

    Thus, for philosophy, Aristotle should point up and would represent a contemplative attitude that was certainly more congenial to religious practices in the Middle Ages. By the same token, Aristotle's contribution to what we now think of as science was hampered by his lack of interest in mathematics. Although Aristotle in general had a more empirical and experimental attitude than Plato, modern science did not come into its own until Plato's Pythagorean confidence in the mathematical nature of the world returned with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a theory of opposites that is now only of historical interest, rejected Plato's attempt to match the Platonic Solids with the elements -- while Plato's expectations are realized in mineralogy and crystallography, where the Platonic Solids occur naturally.

    Therefore, caution is in order when comparing the meaning of the metaphysics of Plato and Aristotle with its significance for their attitudes towards ethics, politics, and science. Indeed, if the opposite of wisdom is, not ignorance, but folly, then Socrates and Plato certainly started off with the better insight.


    It is good that you go to the top of the page of the linked quotes of Kelley L. Ross. You must know that I developed this site without really understanding the extent Mr. Ross had taken this issue. There is much that is familiar, and with him, an opposing view too.

    See:

  • Induction and Deduction
    Intuitively Balanced: Induction and Deduction
  • Friday, April 13, 2007

    Housebuilding



    It all start off as "a dream" or "an idea." Where do these come from? Dialogos of Eide


    This is the house similar to what we will be constructing, with some modifications of course.

    Most know of my time helping my son last year constructing his home. The journey of pictures that I have here within this bloggery. It has also some "dimensional aspect" in it's development, so I thought this might help those who are working Euclidean coordinates, may help to seal this process in some way, by being introduced to house construction.

    This is the home that my wife and I had built in 1998. It was built on ten acres of land with a wide sweeping view of the mountains in the background. Although not seen here, you may have seen some of my rainbow pictures that I had put up over the years to help with the scenery we had.

    Well the time has come for my wife and I to be entering into the venture ourselves. You will notice that the model we choose above is one floor. We thought this suitable for the coming years as when we move into retirement.

    Here is a picture of my daughter-in-law and son's house in the winter of this last year. He still has some work to do, but as per our agreement, I help him, he is helping me.

    I think I am getting the better of the deal, as he has taken the time to write me a 17 page step procedure with which I must follow. I thought this will become part of the journey for my wife and myself, so that everyone may see the process unfolding and maybe learn something about home construction. The plans of course change from country to country, while this plan is unfolding in Canada.

    We purchased a 2 acre parcel of land with which to build the new home up top. I went into the bush with the camera and with about 2 feet of snow. It was not to easy to get around, so as time progresses,and as I put in the roadway and cleared site, you will get a better idea of what it looks like.



    We had to contend with where we will live. We wanted the freedom and space to be close to where we will be building, so we bought a 19' foot travel trailer and will be putting it on the acreage while we build our new home. We thought of "renting" and our son of course offered for a time to let us live with him. We thought all around with the new baby Maley, we would leave them have their space as well.

    Laying the Foundation

    Articles on Euclid

    See No Royal Road to Geometry?

    I would like people to take note of the image supplied on the website of Euclides.Org, as it is one that I have used showing Plato and Aristotle. The larger picture of course is one done by Raphael and is painted on the wall in the "Signatores room in the Vatican."

    The Room of the Segnatura contains Raphael's most famous frescoes. Besides being the first work executed by the great artist in the Vatican they mark the beginning of the high Renaissance. The room takes its name from the highest court of the Holy See, the "Segnatura Gratiae et Iustitiae", which was presided over by the pontiff and used to meet in this room around the middle of the 16th century. Originally the room was used by Julius II (pontiff from 1503 to 1513) as a library and private office. The iconographic programme of the frescoes, which were painted between 1508 and 1511, is related to this function. See Raphael Rooms

    While one may of talked abut the past, or use a name like Plato of the past does not mean that what is being supplied from that position is not dealing with information for the 21st century. I would like you to think that while speaking about models that what the house is doing in "a psychological sense" is giving you a method by which all that you do in your life will materialize in consciousness and digs deep into the unconscious.

    How often had you seen yourself in dream time, doing something or other, in the living room, kitchen, or anything that deals with the current state of mind, that you of course will see in this house? They are the many rooms of the mind.

    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)


    See also Laying the Foundation with Respect While one indeed had to start somewhere I thought I would start here with, "Foundational Perspectives."

    I choose this as an introduction, whilst I will be starting from the ground up. This will include the planning of road way and building site. Since I have this interest about physics and where science is going these days, how could I not incorporate these things into what I am doing currently with my life now? So while I speak about the science end, I am encapsulating "this process" with regard to how I will construct my home.

    Is this possible?

    Well having spoken of the "Euclidean reference" one would have to know how one departs form such a scheme of Euclid, to know that this graduation to Non-Euclidean geometries was somehow related to the "fifth postulate" written by Euclid.

    So of course, we had those who were involved in this development historically, which serve to remind us about where someone like Dali may of been as a visionary, in terms of Time. Or "geometrically inclined" to higher dimensional figures.

    It definitely had it's connotations to "points of view." I mentioned religion, but for the nature of Salvador Dali, and his lifestyle, one would have to wonder where he was going with the Tesserack and his painting of Jesus on the Cross?

    While I do not subscribe to any religion per say, I do subscribe to the finger of Plato pointing up. Have you for one moment you thought to roll your eyes up in your head, and think of what is up their in your mind? Assign our highest values to goodness. Surely you would enlist the "Colour of gravity" in all situations as you choose to live your life? It's there for the choosing.

    Surely, that if you wore a hat on your head, or thought, to think of the roof of your house, you may indeed think of the highest ideals with which you choose to live your life. It's not my job to tell you what that is, that is yours alone.

    You will be involved with aspects of the "universal language" that knows no boundaries, no matter your race, gender, or nationality. Yet, it will be specific to you. It will have "probabilistic outcomes" according to the life you are living regardless.

    The Secret of the Golden Flower

    When ever you walk the pathways in your mind of what ever model, you are laying the road work for that which you will travel through. Why, I may have referred to the title of the "Golden Flower in the Bee story," is a result, that the probabilistic outcome of life calls upon this "chance meeting" to come to what is held in mind. So what's new having the honey of the Bee community?

    Do the Bee dance, and you learnt from others what this model is doing. So you travel. You get the benefits of the honey sometimes in new thoughts? There had to be a point "like the blank slate, glass room, a pen and paper ready" in order for the mind to be receptive to what already exists out there in the "form of ideas." How will these manifest? So indeed, it came from deep inside/outside you?

    I never thought this inductive/deductive method while thinking it topological smooth in it's orientation, was not the exchange going on with our environment. That if you live your life according to your principles, then the principles would become part of your life. That on a level not understood to clearly, the "colour of gravity" was what we could evolve too? What is our own dynamical makeup, to become part of the ideals we had set for ourselves. We set our own ship in life. The boat or vehicle, becomes part of the way we will travel in our dream time. The airplane we ride.

    Tuesday, January 30, 2007

    Hermetic Ties: Art to Esoteric Form

    The father of all perfection in the whole world is here. Its force or power is entire if it be converted into Earth. Separate the Earth from the Fire, the subtle from the gross, sweetly with great industry. It ascends from the Earth to the Heavens and again it descends to the Earth and receives the force of things superior and inferior. By this means you shall have the glory of the whole world and thereby all obscurity shall fly from you. Its force is above all force, for it vanquishes every subtle thing and penetrates every solid thing. So was the world created. From this are and do come admirable adaptations, whereof the process is here in this. Hence am I called Hermes Trismegistus, having the three parts of the philosophy of the whole world. That which I have said of the operation of the Sun is accomplished and ended.Sir Isaac Newton-Translation of the Emerald Tablet
    See: Newton on Chymistry

    Again I open this blog post with the understanding that what an artist like Raphael may try to do? May include, much of the philosophy of the times, and have these things descriptively enclosing processes indicative of what they had known, but also of what these things could hide within the self.


    In center, while Plato - with the philosophy of the ideas and theoretical models, he indicates the sky, Aristotle - considered the father of Science, with the philosophy of the forms and the observation of the nature indicates the Earth. Many historians of the Art in the face correspondence of Plato with Leonardo, Heraclitus with Miguel Angel, and Euclides with Twine agree.

    If we watched of distant spot, of century XX aC emphasizes Hermes Trismegisto, - tri three, megisto megas, three times great; perhaps the perception of infinite older than we have and takes by Mercurio name - for Greek and the Toth - for the Egyptians. Considered Father of the Wisdom and Sciences in Greece, in the cult to Osiris it presided over the ceremonies as priest and he was Masterful in Egypt like legislator, philosopher and alchemist during the reign of Ninus in the 2270 aC.

    Etimológicamente speaking, of Hermes, the gr. hermenéuiein, “hermetic” - closed, “hermenéutica” - tie art to the reading of old sacred texts talks about so much to the dark as to which it is included/understood in esoteric form. Part of saberes that it accumulated transmitted through the Hermetic Books that only to the chosen ones between the chosen ones could be revealed. As much Pitágoras and Plato as Aristotle and Euclides were initiated in the knowledge of the Hermetic School.


    In Man looking into Space, I wanted to show how casual our science has used these images and not realized the context to which the greater meaning had laid hidden, all the while it is used to "describe cosmology" and the science thereof.

    A banner has been been written across these times to which scientists hold to all that is true. In this, the reasons to dismiss any implications of history assigned along side, is asking "what validation" can be given to anything that is spoken from our times now.

    I went on in that post, "man looking into space," to explain something about the woodcuts. The art form produced, grabbed my thinking in relation to the "alchemical art forms" and grabs my thinking in regards to the "School of Athens picture."


    The Yorck Project: 10.000 Meisterwerke der Malerei. DVD-ROM, 2002. ISBN 3936122202. Distributed by DIRECTMEDIA Publishing GmbH.


    I just wanted to say that the essence of this blog post is about "the arches," and I am moving toward that description, and what is happening when we take a picture of them. Look at the "design inherent" and "dynamics" as held to gravity in it's construction. Look at what it can signify in it's "internal expression" about our contact with the world around us. The bridging that it can signify.

    I would apologize for leaving this post undone, while views pass by the essence of this post. I am indeed busy with life. So I wanted to clarify this push toward the internal dynamics, while speaking to the psychology of this work.

    A scientist may side step this look, while quoting the hermetical values of what may be said by the previous first lady Hillary Clinton. In itself, an empty page, only leaves room for what had to be expressed if it was not gotten the first time? Her attempts at humour, are the attempts to break the "rigidity of the personality?"

    The Psychology



    Myths and metaphors, like dreams, are powerful tools that draw the listener, dreamer, or reader to a character, symbol, or situation, as if in recognition of something deeply known. Myth's bypass the mind's efforts to divorce information. They make an impression, are remembered, and nudge us to find out what they mean, accounting for the avid interest that Ring audiences have in the meaning of the story.1


    Who has been so colourful in your journeys across the internet to include a wonderful language that takes you into this world of discovery of self? You had to know something about the "psychology of people" in order to give a story by nature, it's mythic description, and "most artful" to draw attention to what lies underneath.

    The Alchemists attempted to perfect the One Thing of Hermes, what they called the First Matter, by using specific physical, psychological, and spiritual techniques that they describe in chemical terms and demonstrated in laboratory experiements. However, while the alchemists spoke in terms of chemcials, furaces , flasks, and beakers, they were really talking about the changes taking place within their own bodies, minds, and souls.2


    Thus I have given two examples that I had promised sometime back to illustrate some of the "compelling work" that while ancient indeed, is not without it's efforts in todays world. It is the attempt to cross all boundaries, race, gender, and help one to recognize the diversity of the soul with out it's jacket. Shall we call your soul male or female, black or white?

    So I am bypassing this, and that has been my message, while the efforts to climb out of the constraints that we have come to recognize within the boundaries of self. Are the realization of the diversity of "all souls" and their time in expression.

    Shall we find the excuse to hold ourselves to the thoughts, that while overcoming, the constraints which still exist "within" had to be continually challenged? We have to break the "chains that bind us."

    The Arches


    Golden Rectangle
    I took the picture at a time of day when the tide was at exactly the right place to create this image: when the surface of the water reflected the underside of the bridge and they combined, together they produced what I named the Golden Rectangle as a nod to Pythagoras (my hero). The sensation I experienced at the time was of balancing consciousness and feeling.


    It probably seems that it is taking time to get to the essence of this post. IN order to get to the "psychological effect" that I am getting too it important to think of the images of these arches. It is about "each of us" and how we relate to the world. How, the "teacher and student" can exist within the same person.

    I point to the Heaven's in the case of the "school of Athens, while Aristotle points to what is on Earth?" Shall we leave no doubt of the "physical things" while we understand that there are more ephemeral qualities to these matter states? That we move continuously between them?



    The Inner/Outer World

    The drawn of our focus is the external world, but, if we were to connect the internal world with that "external view" how shall we do that. How shall we describe the whole being in this exercise?

    Part of this "exchange with reality," is that we can know by continually moving this information "through us" and creating "the space around us," we add to the total view "beyond what was apparent" with just the brain's condensible qualities in neurological display?

    By 'dilating' and 'expanding' the scope of our attention we not only discover that 'form is emptiness' (the donut has a hole), but also that 'emptiness is form' (objects precipitate out of the larger 'space') - to use Buddhist terminology. The emptiness that we arrive at by narrowing our focus on the innermost is identical to the emptiness that we arrive at by expanding our focus to the outermost. The 'infinitely large' is identical to the 'infinitesimally small'.The Structure of Consciousness John Fudjack - September, 1999


    While I quote above, the second part of the quote adds directly to the understanding. Not only are we "crossing the wires here," we are identifying "a aspect of consciousness" that is continuous.

    In this metaphor, when we are seeing the donut as solid object in space, this is like ordinary everyday consciousness. When we see the donut and the hole at its center, this is like a stage of realization in which 'form' is recognized as 'empty'. When we zoom in extremely closely and inspect the 'emptiness' at the center, or zoom out an extreme distance away from the object and the donut seems to disappear and we have only empty space - this is like certain 'objectless' states of awareness that can occur in meditation. But the final goal is not to achieve the undifferentiated state itself; it is to come to the special perspective that allows us to continue to see all three aspects at once - the donut, the whole in the middle, and the space surrounding it - this is like the 'enlightened' state, in this analogy. 10 The innermost and outermost psychological 'space' (which is here a metaphor for 'concentrated attention' and 'diffused attention') are recognized as indeed the same, continuous.


    So given "this relationship" on what we can build within self, then what use all this knowledge if we cannot grow with it? What of Plato's and Aristotle, as figures within the "centre of" Raphael's painting. Their perspective, "as positions in relation too," the "questioning stance" about this "unity of the circle" in our exchange with reality?



    So how would you exemplify "this exchange" with reality while "below the surface" all these "probable outcomes" are the manifestation of that which is real? You extend yourself "out there" while you also extend yourself inside? The "infinite regress," is to find oneself, with all that is "past" in front of you, can allow you to stand on what of, "the future" will pass through?

    First Principle saids that you acknowledge your place in the scheme of things as you "stretch" the thinking of the mind? Increase the "neurological frontier" in those neurological connections? Increase, the fluttering of the egg's feature, of that condensible brain/body.

    Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.

    But, Aristotle thinks that knowledge begins with experience. We get to first principles through induction. But there is no certainty to the generalizations of induction. The "Problem of Induction" is the question How we know when we have examined enough individual cases to make an inductive generalization. Usually we can't know. Thus, to get from the uncertainty of inductive generalizations to the certainty of self-evident first principles, there must be an intuitive "leap," through what Aristotle calls "Mind." This ties the system together. A deductive system from first principles (like Euclidean geometry) is then what Aristotle calls "knowledge" ("epistemê" in Greek or "scientia" in Latin).


    From here it would not be to unlikely that such dealings with the "reality of the world" would ask that we experiment and from such experiment, we learn the truth of the reality. While what the past is "in front" of us, to what goes beyond to it's future would be like asking the very nature of expression to manifest as this universe and laws of thermodynamics that the arrow of time only moves one way.

    "The future" arises from within then? We'll move forward by what choices we make? About our conclusions, about reality?


    1 Ring of Power, Jean Shinoda Bolen, M.D. Page 3

    2The Emerald Tablet, Dennis William Hauck, Chapter 10, Page 151

    Monday, January 29, 2007

    Whose who, in the School of Athens

    I was over visiting Clifford's blog called Asymptotia this morning and notice a blog entry called, Heretics of Alexandria. Of course, what first came to mind is the "Library of Alexandria."



    Clifford writes and paraphrases:
    This full length drama, set in Alexandria Egypt, 415 A.D. features the infamous Philosopher Hypatia, who has come into possession of a document that threatens the very basis of the new religion called Christianity; a document that some would do anything to destroy. Hypatia and a powerful Christian Bishop wage a fierce struggle for the soul of a young priest and for a document which tells a very different version of the life — and death — of Jesus. A true story.
    The writing was excellent as was the cast, and Bastian should be extremely proud of himself. (It is a mistake to call it “a true story”, though. It is a story based around historical events, which should absolutely not be confused with being a “true story”. Writers of synopses should not encouarge people to mix up the two.


    So I started to do some research on the link offered by Clifford. All of a sudden I could see the many connections bringing "Hypatia of Alexandria" into the fold.


    Hypatia of Alexandria (Greek: Υπατία; c. 370–415) was an ancient philosopher, who taught in the fields of mathematics, astronomy and astrology. She lived in Alexandria, in Hellenistic Egypt.
    Hypatia was the daughter of Theon, who was also her teacher and the last fellow of the Musaeum of Alexandria. Hypatia did not teach in the Musaeum, but received her pupils in her own home. Hypatia became head of the Platonist school at Alexandria in about 400. There she lectured on mathematics and philosophy, and counted many prominent Christians among her pupils. No images of her exist, but nineteenth century writers and artists envisioned her as an Athene-like beauty.


    Many of you who visit here know how much the "School of Athens" picture means to me?

    That there was only one woman here named "Hypatia of Alexandria" of course sent me off to have a look. AS well, "more of the meaning" with regards to the Library of Alexandria.


    9.Francesco Maria I della Rovere or Hypatia of Alexandria and Parmenides


    The frescoe of the "School of Athens" has been a haunting reminder of the many things that Raphael "enclosed in meaning."


    School of Athens by Raphael


    That I could then give numbers and names to person's within the picture was equally exciting. I started to dissect parts of this picture quite a while back, opening of course with the "very centre of that painting." The labels supplied on this post entry should give links to farther posts about this.


    1: Zeno of Citium or Zeno of Elea? – 2: Epicurus – 3: Frederik II of Mantua? – 4: Anicius Manlius Severinus Boethius or Anaximander or Empedocles? – 5: Averroes – 6: Pythagoras – 7: Alcibiades or Alexander the Great? – 8: Antisthenes or Xenophon? – 9: Hypatia or the young Francesco Maria della Rovere? – 10: Aeschines or Xenophon? – 11: Parmenides? – 12: Socrates – 13: Heraclitus (painted as Michelangelo) – 14: Plato holding the Timaeus (painted as Leonardo da Vinci) – 15: Aristotle holding the Ethics – 16: Diogenes of Sinope – 17: Plotinus? – 18: Euclid or Archimedes with students (painted as Bramante)? – 19: Strabo or Zoroaster? – 20: Ptolemy – R: Raphael as Apelles – 21: Il Sodoma as Protogenes


    I now realize that with one comment entry gone( maybe both) that I really was not so out of tune. What was Plato's influence on Hypatia of Alexandria?

    Letters written to Hypatia by her pupil Synesius give an idea of her intellectual milieu. She was of the Platonic school, although her adherence to the writings of Plotinus, the 3rd century follower of Plato and principal of the neo-Platonic school, is merely assumed.


    See also:
  • No Royal Road to Geometry?

  • 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)

    Sunday, January 07, 2007

    PLATO:Mathematician or Mystic ?

    Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.--BERTRAND RUSSELL, Study of Mathematics


    One should not conclude that such a bloggery as this is not without a heartfelt devotion to learning. That I had made no great claims to what science should be. other then what a layman point of view in learning has become excited about. What may be a natural conclusion to one who has spent a long time in science. Do not think me so wanting to knock on your door to enforce the asking of education that may be sent my way was truly as a student waiting for some teacher to appear.

    This did not mean I should not engage the world of science. Not become enamoured with it. Or, that seeing the teachers at their bloggeries, were "as if" that teacher did appear many times. This is what is good about it.

    I did not care how young you were, or that I, "too old" to listen to what scientists knew, or were theoretically endowed with in certain model selections.

    More from the Heart?


    "Let no one destitute of geometry enter my doors."


    You know that by the very namesake of Plato used here, that I am indeed interested how Plato thought and his eventual conclusions about what "ideas" mean. So, of course there is this learning that has to take place with mathematics.

    If I may, and if I were allowed to fast forward any thought in this regard, it would be to say, that the evolution of the human being is much appreciated in what can transfer very quickly "between minds" while a dialogue takes place. Hence the title of this bloggery.

    Science demands clarity, and being deficient in this transference of "pure thought" would be less then ideal speaking amongst those scientists without that mathematics. Yet, I do espouse that such intuitiveness can be gained from the simple experiment, by distilling information, from the "general concepts" which have been mention many times now by scientists.

    So it is of interest to me that the roads to mathematical understanding through it's development would be quick to point out this immediate working in the "world of the abstract imaging" is to know that such methods are deduced by it's numbers and their greater meaning.

    That such meaning can be assign to a "natural objector function" and still unbeknownst to the thinking and learning individual "a numerical pattern that lies underneath it. A "schematics" if you like, of what can become the form in reality.

    No reader of Plato can fail to recognize the important role which mathematics plays in his writing, as would indeed be expected for an author about whom the ancient tradition maintains that he had hung over the entry to his school the words "Let No One Un-versed in Geometry Enter". Presumably it was the level of ability to work with abstract concepts that Plato was interested in primarily, but if the student really had never studied Greek geometric materials there would be many passages in the lectures which would be scarcely intelligible to him. Modern readers, versed in a much higher level of mathematical abstraction which our society can offer, have sometimes felt that Plato's famous "mathematical examples'" were illustrations rather than central to his arguments, and some of Plato's mathematical excursuses have remained obscure to the present time.


    A Musical Interlude



    Plato's Academy-Academy was a suburb of Athens, named after the hero Academos or Ecademos.

    I can't help but say that I am indeed affected by the views of our universe. In a way that encompasses some very intriguing nodal points about our universe in the way that I see it.

    While I may not have shown the distinct lines of the Platonic solids, it is within context of a balloon with dye around it, that it could be so expressive of the Chaldni plate, that I couldn't resist that "harmonics flavour" as to how one might see the patterns underneath reality. How some gaussian coordinates interpretation of the "uv" lines, that were distinctive of an image in abstract spaces.

    Tuesday, October 24, 2006

    Raphael the Painter



    By 'dilating' and 'expanding' the scope of our attention we not only discover that 'form is emptiness' (the donut has a hole), but also that 'emptiness is form' (objects precipitate out of the larger 'space') - to use Buddhist terminology. The emptiness that we arrive at by narrowing our focus on the innermost is identical to the emptiness that we arrive at by expanding our focus to the outermost. The 'infinitely large' is identical to the 'infinitesimally small'.The Structure of Consciousness John Fudjack - September, 1999




    Self-portrait by Raphael


    While I am no great philosopher, the idea of truth was very important one to me. Finding some method by which to proceed was very difficult without the teachers handy. So I learned to trust my intuition as I was lead from one place to another. By it's own design, the correlation I termed in relation to cognition were very important discover about my own potential. I had to symbolically discribe the very actions of what goes in, and what comes out, turns through that channel in much the same way a electromagnetic field governs by analogy the principle of life around the human body, as information passes through the center.

    If conceived as a series of ever-wider experiential contexts, nested one within the other like a set of Chinese boxes, consciousness can be thought of as wrapping back around on itself in such a way that the outermost 'context' is indistinguishable from the innermost 'content' - a structure for which we coined the term 'liminocentric'.


    Will this become part of the greater complexity of the life form, as information becomes part of the larger context of the souls growth? How is that measured? How is t external world brought back in and then turned outward, and the "colors change" as the truth begins to dawn?

    For me the story here starts with a painter and from the very painting itself, one can imagine a larger story unfolding, as one peers into the center of the School of Athens.

    For now, the music is set aside, for the "foundational perspective" that issues forth from this blog.



    I added this biography of the artist himself and "crunched" behind him is a speculation of a kind that becomes the basis of this bloggery. It is about observation and the search for truth as we look at the work of Raphael and the following information that I hold in consideration of this painting.

    Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.


    Do we know what Raphael was trying to impart through these images?

    Inductive and Deductive

    While holding the School of Athens by Raphael then picture in mind and consider the following?

    Aristotle from a a posteriori leads perspective in one way, and Plato a prior?

    PLato saids, "Look to the perfection of the heavens for truth," while Aristotle saids "look around you at what is, if you would know the truth"


    So from that basis look at what is portrayed in the opening statement above with regards to Plato finger pointing up and Aristotle's hand sweeping pervasively?

    So while I lead one through a vast maze of links here it is not without doing my own research that I could now point you to wikipedia for examination of the many things that we could learn of Plato. Imagine Plato continues to live through all this information?

    Without Plato a a personification of the some of the ideals I have, I know who I am. The sun as a symbol of enlightenement? Then following, Plato's Cave Analogy?

    As a beginning, you see I started to point out some of the more important features of the leadng perspective of Aristotle, and the link I see to Robert Laughlins building blocks of matter?

    But before I jump so far ahead, maybe it is indeed useful to link wiki here so one gets the jest of what may be implied by an example?

    Epistemology or theory of knowledge is the branch of philosophy that studies the nature and scope of knowledge. The term "epistemology" is based on the Greek words "επιστημη or episteme" (knowledge) and "λόγος or logos" (account/explanation); it is thought to have been coined by the Scottish philosopher James Frederick Ferrier.

    Much of the debate in this field has focused on analyzing the nature of knowledge and how it relates to similar notions such as truth, belief, and justification. It also deals with the means of production of knowledge, as well as skepticism about different knowledge claims. In other words, epistemology primarily addresses the following questions: "What is knowledge?", "How is knowledge acquired?", and "What do people know?". Although approaches to answering any one of these questions frequently involve theories that are connected to others, there is enough particular to each that they may be examined separately.

    There are many different topics, stances, and arguments in the field of epistemology. Recent studies have dramatically challenged centuries-old assumptions, and the discipline therefore continues to be vibrant and dynamic.



    So while some would point to the very functions of perceiving aspects of the higher self, if there is such a thing accept in one conceptual framework, or messages from God, as Ramanujan received the equations in dream time. I think of this as a very dynamical process, that each of us possesses. If without the teacher to guide us, then the teacher most certainly makes it's way into the mind for observation?

    Innatism

    Innatism is a philosophical doctrine introduced by Plato in the socratic dialogue Meno which holds that the mind is born with ideas/knowledge, and that therefore the mind is not a tabula rasa at birth. It asserts therefore that not all knowledge is obtained from experience and the senses. Innatism is the opposite of empiricism.

    Plato claimed that humans are born with ideas/forms in the mind that are in a dormant state. He claimed that we have acquired these ideas prior to our birth when we existed as souls in the world of Forms. To access these, humans need to be reminded of them through proper education and experience.


    While it is referred to the young born into this world, what said that any person could not become that "blank slate" that would allow the wider perspective of what has been lived, is not confined to this life, but is exposed as that channel is opened for the wider perspective about life?

    I say,"
    I mean really, if, each of us is born into this world with such a blank slate, then how is an idea incorporated into such a design of our blank slate. Especially, if there had not been some influence predisposed, to draw ideas into the appropriate environment for consideration?


    While we provide for the nurturing aspect of creativity to express itself, we find that such freedoms are encouraged by observation of the introspective attitude we gain by learning about ourselves.

    The Medicine Wheel as a Mandala



    It is not so much that we learn about the very "drawing here for you" but that it is circular in nature, and by the very discription the mandala is pretty "clear cut" as to what manifested from a deeper level in my own mind.

    Now what you do not understand is that the center is very important feature on what we focus on. While the "purity of thought" is presented here. It is the idea that the closer to the source you get, the purer the thought/idea that manifests into the theoretical world.

    While I attempt to explain the process this does not disavow you from experimenting and testing, so that the advancement of knowledge and understanding reawakens you to the "nature" of one's being? What is this?

    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.

    Thursday, December 30, 2004

    Where to Now?



    Once you see parts of the picture, belonging to the whole, then it becomes clear what a nice picture we will have?:) I used it originally for the question of the idea of a royal road to geometry, but have since progressed.

    If you look dead center Plato reveals this one thing for us to consider, and to Aristotle, the question contained in the heading of this Blog.

    It is beyond me sometimes to wonder how minds who are involved in the approaches of physics and mathematics might have never understood the world Gauss and Reimann revealled to us. The same imaging that moves such a mind for consideration, would have also seen how the dimensional values would have been very discriptive tool for understanding the dynamics at the quantum level?

    As part of this process of comprehension for me, was trying to see this evolution of ordering of geometries and the topological integration we are lead too, in our apprehension of the dynamics of high energy considerations. If you follow Gr you understand the evolution too what became inclusive of the geometry developement, to know the physics must be further extended as a basis of our developing comprehension of the small and the large. It is such a easy deduction to understand that if you are facing energy problems in terms of what can be used in terms of our experimentation, that it must be moved to the cosmological pallette for determinations.

    As much as we are lead to understand Gr and its cyclical rotation of Taylor and hulse, Mercuries orbits set our mind on how we shall perceive this quantum harmonic oscillator on such a grand scale,that such relevance between the quantum and cosmological world are really never to far apart?

    As I have speculated in previous links and bringing to a fruitation, the methods of apprehension in euclidean determinations classically lead the mind into the further dynamcis brought into reality by saccheri was incorporated into Einsteins model of GR. Had Grossman not have shown Einstein of these geoemtrical tendencies would Einstein completed the comprehsive picture that we now see of what is signified as Gravity?

    So lets assume then, that brane world is a very dynamcial understanding that hold many visual apparatus for consideration. For instance, how would three sphere might evolve from this?

    Proper understanding of three sphere is essential in understanding how this would arise in what I understood of brane considerations.

    Spherical considerations to higher dimensions.

    Spheres can be generalized to higher dimensions. For any natural number n, an n-sphere is the set of points in n-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number.

    a 1-sphere is a pair of points ( - r,r)
    a 2-sphere is a circle of radius r
    a 3-sphere is an ordinary sphere
    a 4-sphere is a sphere in 4-dimensional Euclidean space
    However, see the note above about the ambiguity of n-sphere.
    Spheres for n ≥ 5 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere.


    INtegration of geometry with topological consideration then would have found this continuance in how we percieve the road leading to topolgical considerations of this sphere. Thus we would find the definition of sphere extended to higher in dimensions and value in brane world considerations as thus:



    In topology, an n-sphere is defined as the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere described above under Geometry, but perhaps lacking its metric. It is denoted Sn and is an n-manifold. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.

    a 0-sphere is a pair of points with the discrete topology
    a 1-sphere is a circle
    a 2-sphere is an ordinary sphere
    An n-sphere is an example of a compact n-manifold without boundary.

    The Heine-Borel theorem is used in a short proof that an n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. Sn is also bounded. Therefore it is compact.


    Sometimes it is very hard not to imagine this sphere would have these closed strings that would issue from its poles and expand to its circumference, as in some poincare projection of a radius value seen in 1r. It is troubling to me that the exchange from energy to matter considerations would have seen this topological expression turn itself inside/out only after collapsing, that pre definition of expression would have found the evoltuion to this sphere necessary.

    Escher's imaging is very interesting here. The tree structure of these strings going along the length of the cylinder would vary in the structure of its cosmic string length based on this energy determination of the KK tower. The imaging of this closed string is very powerful when seen in the context of how it moves along the length of that cylinder. Along the cosmic string.

    To get to this point:) and having shown a Platonic expression of simplices of the sphere, also integration of higher dimension values determined from a monte carlo effect determnation of quantum gravity. John Baez migh have been proud of such a model with such discrete functions?:) But how the heck would you determine the toplogical function of that sphere in higher dimensional vaues other then in nodal point flippings of energy concentration, revealled in that monte carlo model?

    Topological consideration would need to be smooth, and without this structure how would you define such collpases in our universe, if you did not consider the blackhole?

    So part of the developement here was to understand where I should go with the physics, to point out the evolving consideration in experimentation that would move our minds to consider how such supersymmetrical realities would have been realized in the models of the early universe understanding. How such views would have been revealled in our understanding within that cosmo?

    One needed to be able to understand the scale feature of gravity from the very strong to the very weak in order to explain this developing concept of geometry and topological consideration no less then what Einstein did for us, we must do again in some comprehensive model of application.



    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