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Showing posts with label Non Euclidean. Show all posts
Showing posts with label Non Euclidean. Show all posts

Friday, May 24, 2013

Who is the Clockmaker?

Crucifixion (Corpus Hypercubus) - oil painting by Salvador Dalí
I see a clock, but I cannot envision the clockmaker. The human mind is unable to conceive of the four dimensions, so how can it conceive of a God, before whom a thousand years and a thousand dimensions are as one?
  • From Cosmic religion: with other opinions and aphorisms (1931), Albert Einstein, pub. Covici-Friede. Quoted in The Expanded Quotable Einstein, Princeton University Press; 2nd edition (May 30, 2000); Page 208, ISBN 0691070210
The phrase of course stuck in my mind. Who is the clockmaker. I was more at ease with what Einstein quote spoke about with regards to the fourth dimension and here, thoughts of Dali made their way into my head.

The watchmaker analogy, watchmaker fallacy, or watchmaker argument, is a teleological argument. By way of an analogy, the argument states that design implies a designer. The analogy has played a prominent role in natural theology and the "argument from design," where it was used to support arguments for the existence of God and for the intelligent design of the universe.

The most famous statement of the teleological argument using the watchmaker analogy was given by William Paley in his 1802 book. The 1859 publication of Charles Darwin's theory of natural selection put forward an alternative explanation for complexity and adaptation, and so provided a counter-argument to the watchmaker analogy. Richard Dawkins referred to the analogy in his 1986 book The Blind Watchmaker giving his explanation of evolution.

In the United States, starting in the 1960s, creationists revived versions of the argument to dispute the concepts of evolution and natural selection, and there was renewed interest in the watchmaker argument.
I have always shied away from the argument based on the analogy, fallacy and argument, as I wanted to show my thoughts here regardless of what had been transmitted and exposed on an objective level argument. Can I do this without incurring the wrought of a perspective in society and share my own?

I mean even Dali covered the Tesseract by placing Jesus on the cross in a sense Dali was exposing something that such dimensional significance may have been implied as some degree of Einstein's quote above? Of course I speculate but it always being held to some idea of a dimensional constraint that no other words can speak of it other then it's science. Which brings me back to Einstein's quote.

The construction of a hypercube can be imagined the following way:
  • 1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
  • 2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
  • 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
  • 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

So for me it is about what lays at the basis of reality as to question that all our experiences, in some way masks the inevitable design at a deeper level of perceptions so as to say that such a diagram is revealing.

I operate from this principal given the understanding that all experience is part of the diagram of the logic of a visual reasoning in which such examples are dispersed upon our assessments of the day. While Einstein spoke, he had a reason from which such quote espoused the picture he had in his head?

Also too if I were to deal with the subjectivity of our perceptions then how could I ever be clear as I muddy the waters of such straight lines and such with all the pictures of a dream by Pauli?  I ask that however you look at the plainness of the dream expanded by Jung, that one consider the pattern underneath it all.  I provide 2 links below for examination.



This page lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Clicking on any picture will magnify it.

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.



See Also:

  • Pauli's World Clock

  • Saturday, April 20, 2013

    Geometrical Underpinnings

    On the Hypotheses which lie at the Bases of Geometry.
    Bernhard Riemann
    Translated by William Kingdon Clifford

    [Nature, Vol. VIII. Nos. 183, 184, pp. 14--17, 36, 37.]


    It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.

    From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space - the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.



    Click the image to open in full size.
    "We all are of the citizens of the Sky" Camille Flammarion


    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.


    There is some question here as to what signifies a liberation of a kind and how this may have affected your perceptions. How is it so easy for what you may have read of one page to come back to it later and see and read something different? So how had you changed?

    Wigner's friend is a thought experiment proposed by the physicist Eugene Wigner; it is an extension of the Schrödinger's cat experiment designed as a point of departure for discussing the Quantum mind/body problem. See: WIGNER'S FRIEND
    Conclusion: *The state of mind of the observer plays a crucial role in the perception of time.* On the Effects of External Sensory Input on Time Dilation." A. Einstein, Institute for Advanced Study, Princeton, N.J.
    Einstein:Since there exist in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.

    The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean. Bernhard Riemann

    Riemannian Geometry, also known as elliptical geometry, is the geometry of the surface of a sphere. It replaces Euclid's Parallel Postulate with, "Through any point in the plane, there exists no line parallel to a given line." A line in this geometry is a great circle. The sum of the angles of a triangle in Riemannian Geometry is > 180°.

    While this may seem abstract in term of it's mathematical underpinnings, it allows us to see in ways that we might ever have been privileged to see before. So you turn your head to everything you have observed before and a whole new light has been thrown on the world. By consensus, this new view allows you to see deeper into the universe in ways that we had only taken from a standpoint of a man looking into outer space.

    The Binary Pulsar PSR 1913+16:

    So while being lead through the circumstance of historical individual pursuers to solving the Parallel postulate, liberation was found in order to move a geometrical proposition forward in time. Some may say that time is a illusion then?

    So as a new paradigmatic change that has been initiated it's application and is pushed into the world so as to ascertain it's functionality. Does it then become real?




     "...underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements-- and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism'-- then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name...)* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled 'The Kaleidoscope'....)"

    Thursday, July 19, 2012

    Process Fractal vs Geometry Fractals

    Let proportion be found not only in numbers, but also in sounds, weights, times and positions, and whatever force there is.Leonardo Da Vinci
    The Mandelbrot set, seen here in an image generated by NOVA, epitomizes the fractal. Photo credit: © WGBH Educational Foundation

     "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." So writes acclaimed mathematician Benoit Mandelbrot in his path-breaking book The Fractal Geometry of Nature. Instead, such natural forms, and many man-made creations as well, are "rough," he says. To study and learn from such roughness, for which he invented the term fractal, Mandelbrot devised a new kind of visual mathematics based on such irregular shapes. Fractal geometry, as he called this new math, is worlds apart from the Euclidean variety we all learn in school, and it has sparked discoveries in myriad fields, from finance to metallurgy, cosmology to medicine. In this interview, hear from the father of fractals about why he disdains rules, why he considers himself a philosopher, and why he abandons work on any given advance in fractals as soon as it becomes popular. A Radical Mind

    As I watch the dialogue between Bruce Lipton and Tom Campbell here, there were many things that helped my perspective understand the virtual world in relation to how the biology subject was presented. It is obvious then why Bruce Lipton likes the analogies Tom Campbell has to offer. The epiphanies Bruce is having along the road to his developing biological work is very important. It is how each time a person makes the leap that one must understand how individuals change, how societies change.



    Okay so for one,  the subject of fractals presents itself and the idea of process fractals and Geometry Fractals were presented in relation to each other. Now the talk moved onto the very thought of geometry presented in context sort of raised by ire even though I couldn't distinguish the differences. The virtual world analogy is still very unsettling to me.

    So ya I have something to learn here.

    I think my problem was with how such iteration may be schematically driven so as toidentify the pattern. Is to see this process reveal itself on a much larger scale. So when I looked at the Euclidean basis as a Newtonian expression the evolution toward relativity had to include the idea of Non Euclidean geometries. This was the natural evolution of the math that lies at the basis of graduating from a Euclidean world. It is the natural expression of understanding how this geometry can move into  a dynamical world.

    So yes the developing perspective for me is that even though we are talking abut mathematical structures here we see some correspondence in nature . This has been my thing so as to discover the starting point?

    A schematic of a transmembrane receptor


    It the truest sense I had already these questions in my mind as  I was going through the talk. The starting point for Bruce is his biology and the cell. For Tom, he has not been explicit here other then to say that it is his studies with Monroe that he developed his thoughts around the virtual world as it relates to the idea of what he found working with Monroe.

    So it is an exploration I feel of the work he encountered and has not so far as I seen made a public statement to that effect. It needs to be said and he needs to go back and look over how he had his epiphanies. For me this is about the process of discovery and creativity that I have found in my own life. Can one feel so full as to have found ones wealth in being that you can look everywhere and see the beginnings of many things?

    This wealth is not monetary for me although I recognized we had to take care of or families and made sure they were ready to be off on their own. To be productive.

    The Blind Men and the Elephant
    John Godfrey Saxe (1816-1887)
     So for me the quest for that starting point is to identify the pattern that exists in nature as much as many have tried various perspective in terms of quantum gravity. Yes, we are all sort of like blind men trying to explain the reality of the world in our own way and in the process we may come up with our epiphanies.

    These epiphanies help us to the next level of understanding as if we moved outside of our skeletal frame to allow the membrane of the cell to allow receptivity of what exist in the world around as information. We are not limited then to the frame of the skeleton hardened too, that we cannot progress further. The surface area of the membrane then becomes a request to open the channels toward expansion of the limitations we had applied to ourselves maintaining a frame of reference.

    Friday, June 10, 2011

    Donald Coxeter


    Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.




    "I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Harold Scott Macdonald (H. S. M.) Coxeter
    While a layman in my pursuance and understanding of the nature of geometry, it is along the way we meet some educators who fire up our excitement. For me it is about the truth of what lies so close to the soul's ideal.

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

    As if searching for a foundation principle, and highly subjective one in my case, I have been touched by example, as if to direct my attention to the early geometer.



    Georg Friedrich Bernhard Riemann 1826 – 1866

    Riemannian Geometry, also known as elliptical geometry, is the geometry of the surface of a sphere. It replaces Euclid's Parallel Postulate with, "Through any point in the plane, there exists no line parallel to a given line." A line in this geometry is a great circle. The sum of the angles of a triangle in Riemannian Geometry is > 180°.

    To me this is one of the greatest achievements of mathematical structures that one could encounter, It revolutionize many a view, that been held to classical discriptions of reality.

    In the quiet achievement of Riemann’s tutorial teacher Gauss, recognized the great potential in his student. On the curvature parameters, we recognize in Gauss’s work, what would soon became apparent? That we were being lead into another world for consideration?





    XXV.  Gaussian Co-ordinates-click on Picture

    Albert Einstein (1879–1955).  Relativity: The Special and General Theory.  1920.

    So here we are, that we might in our considerations go beyond the global perspectives, to another world that Einstein would so methodically reveal in the geometry and physics, that it would include the electromagnetic considerations of Maxwell into a cohesive whole and beyond.


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


    The intuitive development that we are lead through geometrically asks us to consider again, how Riemann moved to a positive aspect of the universe?


    See:Donald Coxeter: The Man Who Saved Geometry

    Sunday, February 25, 2007

    The Colour of Gravity

    I am not sure how this post is to unfold, yet in my mind different exercises were unfolding as to how I should explain it. Can I come from an artist's perspective I wondered? Say "by chance" anything that seems relevant here in writing, and any relation to science "is" metaphorical by nature?

    Yellow, Red, Blue
    1925; Oil on canvas, 127x200cm; Centre Georges Pompidou, Paris


    These free, wild raptures are not the only form abstraction can take, and in his later, sadder years, Kandinsky became much more severely constrained, all trace of his original inspiration lost in magnificent patternings. Accent in Pink (1926; 101 x 81 cm (39 1/2 x 31 3/4 in)) exists solely as an object in its own right: the ``pink'' and the ``accent'' are purely visual. The only meaning to be found lies in what the experience of the pictures provides, and that demands prolonged contemplation. What some find hard about abstract art is the very demanding, time-consuming labour that is implicitly required. Yet if we do not look long and with an open heart, we shall see nothing but superior wallpaper.
    I underlined for emphasis.

    Does one want to gleam only what is coming across in geometrical form as a painting without understanding the depth of the artist in expression? Some may say, why any association at all, and just leave science to what it knows best without implicating any theoretical positions with the thought pertaining to gravity here.

    Yes that's why I selected the title of this post as thus, and why I am going to give perspective to what I may, "as artist in writing" see with these words, and then you decide whether it is useful to you.

    The Field as the Plane

    An ancient thought penetrated my thinking as I thought of "the field" that a society can work in agriculture, and yet, by definition it was the plane, "length and width" that was also appealing here. I did not want to loose it's "origination" while I moved any thinking to the "abstract of brane" and the like, without firmly attaching it to the ground.

    But who was to know that this plane could be moved to any "fifth dimensional understanding" without having studied the relationship to dimensional thinking and the like. The physics elevated.

    I allow this one time escapism to "other thinking" to demonstrate what use the colour of gravity implies while at the same time "theoretical positions" talk about it's place in the universe. If one did not accept the moves in science and the way it expressed itself to allow geometrical inclination, then how the heck could non-euclidean thinking ever make it's way into how we will discuss "the fields" about us?

    It meant that a perspective "on height" be adopted? As an observer I was watching from a position. While in that sleeping/dosing state, I wondered how else to express myself as these concepts were amalgamating themselves into a "conceptual frame of reference?"

    The picture of the field(I am referring to the ancient interpretation) continued in my mind, and "by abstract" I thought to introduce a line extend from the centre of this field upward. So here I am looking at this field before me. Now I had wondered off previous by bring "the brane" in here, yet is not without that sight I thought how the heck could any idealization so ancient make sense to what the colour of gravity to mean.

    Title page of Opticks .... by Sir Isaac Newton, 1642-1727. Fourth edition corrected by the author's own hand, and left before his death with the bookseller. Published in 1730. Library call number QC353 .N48 1730.

    So "an idea" came to mind.

    While correlating Newton's work here and the "extra dimensional thinking," I also wanted to include the work of the "Alchemist Newton". "To expand" the current thinking of our "emotive states" as a "vital expression of the biological being."

    Draw into any further discussion of the "philosophical or other wise," these views of mine which are a necessary part of what was only held to a "religious and uneducated evolutionary aspect of the human being."

    A cosmologist may still say that such thoughts of Einstein used in this vain is wrong, but I could never tear myself away from the views of "durations of time."

    Colour Space and Colour Theory

    The CIE 1931 colour space chromaticity diagram with wavelengths in nanometers. Note that the colors depicted depend on the color space of the device on which you are viewing the image.

    So by having defined the "frame of reference," and by introducing "Colour of gravity" I thought it important and consistent with the science to reveal how dynamical any point within that reference can become expressive. The history in association also important.

    In the arts and of painting, graphic design, and photography, color theory is a body of practical guidance to color mixing and the visual impact of specific color combinations. Although color theory principles first appear in the writings of Alberti (c.1435) and the notebooks of Leonardo da Vinci (c.1490), a tradition of "colory theory" begins in the 18th century, initially within a partisan controversy around Isaac Newton's theory of color (Opticks, 1704) and the nature of so-called primary colors. From there it developed as an independent artistic tradition with only sporadic or superficial reference to colorimetry and vision science.


    So you tend to draw on your reserves for such comparatives while thinking about this. I knew to apply "chemical relations" to this idea, and the consequential evidenced, by the resulting shadings by adding. I wanted to show "this point" moving within this colour space and all the time it's shading was describing the "nature of the gravity."

    Adding a certain mapping function between the color model and a certain reference color space results in a definite "footprint" within the reference color space


    By adding that vertical line in the field, the perimeter of my field of vision had to some how be drawn to an apex, while all kinds of thoughts about symmetry and perfection arose in my pyramidal mind.

    All these colours, infinite in their ability to express the human emotive state, as a consequence of philosophical and expressed as function of the emotive being?

    CIE 1976 L*, a*, b* Color Space (CIELAB)

    CIE L*a*b* (CIELAB) is the most complete color model used conventionally to describe all the colors visible to the human eye. It was developed for this specific purpose by the International Commission on Illumination (Commission Internationale d'Eclairage, hence its CIE initialism). The * after L, a and b are part of the full name, since they represent L*, a* and b*, derived from L, a and b. CIELAB is an Adams Chromatic Value Space.

    The three parameters in the model represent the lightness of the color (L*, L*=0 yields black and L*=100 indicates white), its position between magenta and green (a*, negative values indicate green while positive values indicate magenta) and its position between yellow and blue (b*, negative values indicate blue and positive values indicate yellow).

    The Lab color model has been created to serve as a device independent model to be used as a reference. Therefore it is crucial to realize that the visual representations of the full gamut of colors in this model are never accurate. They are there just to help in understanding the concept, but they are inherently inaccurate.

    Since the Lab model is a three dimensional model, it can only be represented properly in a three dimensional space.


    Entanglement

    the quantum entanglement would become so spread out through these interactions with the environment that it would become virtually impossible to detect. For all intents and purposes, the original entanglement between photons would have been erased.

    Never the less it is truly amazing that these connections do exist, and that carefully arranged laboratory conditions they can be observed over significant distances. They show us, fundamentally, that space is not what we once thought it was. What about time?
    Page 123, The Fabric of the Cosmo, by Brian Greene


    So many factors to include here, yet it is with the "idea of science" that I am compelled to see how things can get all mixed up, while I say emotive state, or Colours of gravity?

    It gets a little complicated for me here, yet the "Fuzzy logic" introduced or "John Venn's logic" is not without some association here. Or, the psychology I had adopted as I learnt to read of models and methods in psychology that could reveal the thinking we have developed, and what it included.

    Least I forget the "real entanglement" issues here, I have painted one more aspect with the "Colour of Gravity" to be included in this dimensional perspective, as we look to the models in science as well?

    Working from basic principles and the history of spooky has made this subject tenable in today's world. A scientist may not like all the comparisons I have made based on it, I could never see how the emotive and mental statements of the expressive human being could not have been included in the making of the reality.

    That I may of thought the "perfection of the human being" as some quality of the God in us all, would have granted sanction to some developing view of "religious virtuosity," against the goals of the scientist. So as ancient the views painted, there was something that may have been missed of the "Sensorium," and goes toward the basis of the philosophy shared currently by Lee Smolin.

    This entanglement to me is a vital addition to our exploration of the universe. Our place and observation within it? It did not mean to discount our inclusion within it, within a larger "oscillatory perspective."

    Sunday, January 28, 2007

    The Mathematikoi had Synesthesia?


    Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism.

    Later resurgence of ideas similar to those held by the early Pythagoreans are collected under the term Neopythagoreanism.

    The Pythagoreans were called mathematikoi, which means "those that study all1"


    To say it is easy in knowing where to begin, is a understatement of what has been an enormous struggle to define the world around me. Indicative of the complications of how one may have seen this world in regards to the "views of a Synesthesist," would have taxed most "science minds" if they had "this inkling" of the complexity this brings to science. Think about what is implied here when one refers to "studying it all?"

    So as I lay in the twilight hours of the mind's rest period, there are these things that I am asking of myself, as to how I may point to what is comparative in the "geometric views of science" and what is comparative to the views of that science in relation to examples given of the Synesthesist who sees from a certain position.

    Again, my mind falls back in the history of humanities evolution and while the distinctiveness of sectors of that past history, it would not be unkind to draw from that history and present the question of what a Synesthesist might have seen in relation to the numbers?


    Create and play with the most beautiful, hypnotic light illusions you have ever seen.


    I seen the above in relation to Lubos's post. It would be nice to offer the "equation correlations" to these "colour displays" in string theory?:)

    Numbers

    Are you quicker then I then to see that numbers may have had the colour attached to their very nature, that "all things" then my have had this basis of "music" and "colour association" thrown "into the mix/cross over points"" to call it the Pythagorean?

    So imagine being strapped with the job to start from some place, and move any mind to consider the complexity of "departing euclidean views" to meld with the "non-euclidean reality" assigned our everyday species to "what is natural" from straight lines and such. Has now moved to a dynamical world of "Faraday lines" Gauss's role as "teacher of Gaussian Co-ordinates" to views of his student, "Riemann?"


    This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found.


    Should one be so crude as to see that straight lines can have a "greater implication of design" that one would not have seen, had they not understood Gauss's work? That if you moved yourself to natures's domain, how many lines are really that straight?

    Ask your self then what is natural and what was man-made? That these straight lines are indeed an order to mankind's "ode to building and living," while there are these "other worldly visions" supplied in the "non euclidean realm" existed free from man's definition of nature.

    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. 2


    The Interior Probabilities Manifests as Colour

    How foolish would I be then to tell you that "Heaven' Ephemeral Qualities," are coloured to the degrees that "gravity defines itself in time?" That "model building" had to take place, so that the understanding of where this gravity explains itself, could find correlations to humans experiencing "durations of time" within in the living of day to day.

    Again I move one back to what this "egg of fluttering does" as of physiological consequent, as the correlations of those same colours manifest in the qualities of those same thought patterns. Those experiences mapped to MRI imaging are condensible features "in the physical" do not explain the "Ephemeral Quality" assigned to each of these regions. Had one knew how to switch around the "value of consciousness" to the condensible feature as brain matter, one would have known about the happenings taking place "outside" of our bodies.

    It is here to then that I take from the "metaphysical realm" and bring it into the relations of what is happening in the physical brain. While history has shown groups who gathered to see what was happening, saw "human experiencing" as they went through these colour modes.

    1 Hemmenway, Pryia – Divine Proportion pp66, Sterling Publishing, ISBN 1-4027-3522-7
    2 Reflections on Relativity8.6 On Guass's Mountain

    Friday, January 05, 2007

    Images or Numbers By Themself

    “Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate” (cited by Ivars Peterson in Science News, 5/4/2002).


    I have an idea in mind here that will be slow to show because I am not sure how it is supposed to be laid out. So maybe by showing these numbers by them self? What use, if one did not, or was not able to see in another way?


    Figure 22.10: Double slit diffraction


    I looked at the "straight lines" of Thomas Young's trajectories of photon emission and while quite understandably shown to be of consequence in this post "Interference." I was more interested in how something could start off in one place and do this rotation of sorts, and then come back for examination again in the real world. The Spectrum

    Plato:
    What a novel idea to have the methods used by the predecessors like Maxwell, to have been united from Faraday's principals? To have Maxwell's equation Gaussian in interpretation of Riemann geometry, somehow, united by the geometries of Einstein and defined as gravity?


    But it is also in mind "that the image" has to be put here also before the numbers can show them self. What use these numbers if I do not transcend them to what they can imply in images, to know that the thinking here has to be orientated in such a way that what was simple and straight forward, could have non-euclidean orientations about it?


    Michael Faraday (September 22, 1791 – August 25, 1867) was a British scientist (a physicist and chemist) who contributed significantly to the fields of electromagnetism and electrochemistry.


    So one reads history in a lot of ways to learn of what has manifested into todays thinking. What lead from "Gaussian coordinates in an "non-euclidean way" to know that it had it's relation in today's physics. To have it included in how we see the consequences of GR in the world. It had been brought together for our eyes in what the photon can do in the gravitational field.

    Our Evolution to Images


    The Albrecht Durer's Magic Square



    Ulam's Spiral



    Pascal's Triangle


    Evolve to What?

    Who was to know what Leonard Susskind was thinking when his mathematical mind was engaged in seeing this "rubber band" had some other comparative abstraction, as something of consequence in our world. Yet, people focus on what they like to focus on, other then what "lead the mind" to think the way they do?


    Poincaré Conjecture
    If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut......


    I have to rest now.

    Wednesday, December 27, 2006

    The Geometrics Behind the Supernova and it's History



    It is not always easy for people to see what lies behind the wonderful beauty of images that we take from the satellite measures of space, and it's dynamical events illustrated in Cassiopeia A. There before you is this majestic image of beauty, as we wonder about it's dynamics.


    These Spitzer Space Telescope images, taken one year apart, show the supernova remnant Cassiopeia A (yellow ball) and surrounding clouds of dust (reddish orange). The pictures illustrate that a blast of light from Cassiopeia A is waltzing outward through the dusty skies. This dance, called an "infrared echo," began when the remnant erupted about 50 years ago. Image credit: NASA/JPL-Caltech/Univ. of Ariz.
    An enormous light echo etched in the sky by a fitful dead star was spotted by the infrared eyes of NASA's Spitzer Space Telescope.

    The surprising finding indicates Cassiopeia A, the remnant of a star that died in a supernova explosion 325 years ago, is not resting peacefully. Instead, this dead star likely shot out at least one burst of energy as recently as 50 years ago.



    How is it such information arrives to us, and we would have to consider the impulse's behind such geometrical explanations. Which we are lucky to see in other ways. So, of course we needed to see the impulse as dynamically driven by the geometrical inclinations of that collapse, and all it's information spread outward by the description in images painted.


    Credit: Weiqun Zhang and Stan Woosley
    This image is from a computer simulation of the beginning of a gamma-ray burst. Here we see the jet 9 seconds after its creation at the center of a Wolf Rayet star by the newly formed, accreting black hole within. The jet is now just erupting through the surface of the Wolf Rayet star, which has a radius comparable to that of the sun. Blue represents regions of low mass concentration, red is denser, and yellow denser still. Note the blue and red striations behind the head of the jet. These are bounded by internal shocks.


    If I had approached you early on and suggested that you look at "bubble geometrodynamics" would it have seemed so real that I would have presented a experiment to you, that would help "by analogies" to see what is happening? Might I then be called the one spreading such information that it was not of value to scientists to consider, that I was seeing in ways that I can only now give to you as example? What science has done so far with using the physics with cosmological views?


    Image Credit: NASA/JPL-Caltech/STScI/CXC/SAO
    This stunning false-color picture shows off the many sides of the supernova remnant Cassiopeia A, which is made up of images taken by three of NASA's Great Observatories, using three different wavebands of light. Infrared data from the Spitzer Space Telescope are colored red; visible data from the Hubble Space Telescope are yellow; and X-ray data from the Chandra X-ray Observatory are green and blue.

    Located 10,000 light-years away in the northern constellation Cassiopeia, Cassiopeia A is the remnant of a once massive star that died in a violent supernova explosion 325 years ago. It consists of a dead star, called a neutron star, and a surrounding shell of material that was blasted off as the star died. The neutron star can be seen in the Chandra data as a sharp turquoise dot in the center of the shimmering shell.


    In this image above we learn of what manifests in "jet production lines," and such examples are beautiful examples to me of what the geometrics are doing. You needed some way to be able to explain this within context of the universe's incidences "as events." We say this action is one with which we may speak to this "corner of the universe." Yet it is very dynamical in it's expression as we see it multiplied from various perspectives.


    The structure of Model J32 as the jet nears the surface 7820 seconds after core collapse.


    So by experiment(?) I saw such relations, but what use such analogies if they are laid waste to speculation that what was initiated such ideas had been the inclination of geometrics detailed as underlying the basis of all expression as an example of some non euclidean views of Riemann perspectives leading shapes and dynamics of our universe by comparison within the local actions of stars and galaxies?

    Gamma Rays?



    So we get this information in one way or another and it was from such geometrical impulse that such examples are spread throughout the universe in ways that were not understood to well.


    X-ray image of the gamma-ray burst GRB 060614 taken by the XRT instrument on Swift. The burst glowed in X-ray light for more than a week following the gamma-ray burst. This so-called "afterglow" gave an accurate position of the burst on the sky and enabled the deep optical observations made by ground-based observatories and the Hubble Space Telescope. Credit: NASA/Swift Team
    A year ago scientists thought they had figured out the nature of gamma-ray bursts. They signal the birth of black holes and traditionally, fall into one of two categories: long or short. A newly discovered hybrid burst has properties of both known classes of gamma-ray bursts yet possesses features that remain unexplained.

    The long bursts are those that last more than two seconds. It is believed that they are ejected by massive stars at the furthest edge of the universe as they collapse to form black holes.


    So looking back to this timeline it is important to locate the ideas spread out before us. Have "some place" inclusive in the reality of that distance from the origins of the stars of our earliest times. 13.7 billions years imagine!


    Fig. 1: Sketchy supernova classification scheme
    A supernova is the most luminous event known. Its luminosity matches those of whole galaxies. The name derives from the works of Walter Baade and Fritz Zwicky who studied supernovae intensively in the early 1930s and used the term supernova therein.
    Nowadays supernova is a collective term for different classes of objects, that exhibit a sudden rise in luminosity that drops again on a timescale of weeks.
    Those objects are subdivided into two classes, supernovae of type I or II (SNe I and SNe II). The distinguishing feature is the absence or the presence of spectral lines of hydrogen. SNe I show no such lines as SNe II do. The class of SNe I is further subdivided in the classes a, b and c. This time the distinguishing feature are spectral features of helium and silicon. SN Ia show silicon features, SN Ib show helium but no silicon features and SN Ic show both no silicon and no helium spectral features.
    The class of SN II is further subdivided in two classes. Those are distinguished by the decline of the lightcurve. Those SN II that show a linear decline are named SN II-L and those that pass through a plateau-phase are referred to as SN II-P.



    So given the standard information one would have to postulate something different then what is currently classified?

    A new Type III (what ever one shall attribute this to definition, versus Type I, Type IIa?


    ssc2006-22b: Brief History of the Universe
    Credit: NASA/JPL-Caltech/A. Kashlinsky (GSFC)
    This artist's timeline chronicles the history of the universe, from its explosive beginning to its mature, present-day state.

    Our universe began in a tremendous explosion known as the Big Bang about 13.7 billion years ago (left side of strip). Observations by NASA's Cosmic Background Explorer and Wilkinson Anisotropy Microwave Probe revealed microwave light from this very early epoch, about 400,000 years after the Big Bang, providing strong evidence that our universe did blast into existence. Results from the Cosmic Background Explorer were honored with the 2006 Nobel Prize for Physics.

    A period of darkness ensued, until about a few hundred million years later, when the first objects flooded the universe with light. This first light is believed to have been captured in data from NASA's Spitzer Space Telescope. The light detected by Spitzer would have originated as visible and ultraviolet light, then stretched, or redshifted, to lower-energy infrared wavelengths during its long voyage to reach us across expanding space. The light detected by the Cosmic Background Explorer and the Wilkinson Anisotropy Microwave Probe from our very young universe traveled farther to reach us, and stretched to even lower-energy microwave wavelengths.

    Astronomers do not know if the very first objects were either stars or quasars. The first stars, called Population III stars (our star is a Population I star), were much bigger and brighter than any in our nearby universe, with masses about 1,000 times that of our sun. These stars first grouped together into mini-galaxies. By about a few billion years after the Big Bang, the mini-galaxies had merged to form mature galaxies, including spiral galaxies like our own Milky Way. The first quasars ultimately became the centers of powerful galaxies that are more common in the distant universe.

    NASA's Hubble Space Telescope has captured stunning pictures of earlier galaxies, as far back as ten billion light-years away.


    Would sort of set up the challenge?

    Wednesday, December 13, 2006

    Visual Abstraction to Equations

    Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.


    Some of you might have noticed the reference to the Ashmolean Museum?


    Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.


    It seems only the good scientist John Baez had epitomes the construction of the Platonic solids? A revision then, of the "time line of history" and the correction he himself had to make? Let's not be to arrogant to know that once we understand more and look at "the anomalies" it forces us to revise our assessments.

    The Art form

    I relayed this image and quote below on Clifford's site to encourage the thinking of young people into an art form that is truly amazing to me. Yes I get excited about it after having learnt of Gauss and Reimann's exceptional abilities to move into the non euclidean world.

    Some think me a crackpot here? If you did not follow the history then how would you know to also include the "physics of approach," as well? Also, some might ask what use "this ability to see the visual abstraction" and I think this art form is in a way destined, to what was kept in glass cabinets and such, even while the glass cabinet in analogy is held in the brain/space of them) who have developed such artistic abilities.

    It's as if you move past the layers of the evolution of the human being(brain casings) and it evolution and the field that surrounds them. Having accomplished the intellect( your equations and such), has now moved into the world of imagery. Closet to this is the emotive field which circumvents our perspective on the greater potential of the world in the amazing thought forms of imagery. This move outward, varies for each of us from time to time. Some who are focused in which ever area can move beyond them. This paragraph just written is what would be considered crackpot(I dislike that word)because of the long years of research I had gone through to arrive at this point.

    Of course, those views above are different.

    Mapping



    Is it illusionary or delusional, and having looked at the Clebsch's Diagonal Surface below, how is it that "abstraction" written?



    The enthusiasm that characterized such collections was captured by Francis Bacon [1, p. 247], who ironically advised "learned gentlemen" of the era to assemble within "a small compass a model of the universal made private", building

    ... a goodly, huge cabinet, wherein whatsoever the hand of man by exquisite art or engine has made rare in stuff, form or motion; whatsoever singularity, chance, and the shuffle of things hath produced; whatsoever Nature has wrought in things that want life and may be kept; shall be sorted and included.


    There is no doubt that the long road to understanding science is the prerequisite to mapping the images from an equation's signs and symbols. While not sitting in the classroom of the teachers it was necessary to try and move into the fifth dimensional referencing of our computer screen to see what is being extolled here not just in image development, but of what the physics is doing in relation.

    In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27



    So of course to be the historical journey was established like most things, Mandelstam current and what is happening there as an interlude, as well as helping to establish some understanding of the abstractions that had been developed.



    But yes, before moving to current day imagery and abstraction, I had to understand how these developments were being tackled in today's theoretical sciences.

    Wednesday, December 06, 2006

    Reaching for the Stars


    Mars in 6 weeks? And back in a total of four months? That's the prediction of a design team working on antimatter rocket concepts at Pennsylvania State University. But first, you have to get the stuff - and store it. (PSU)
    The popular belief is that an antimatter particle coming in contact with its matter counterpart yields energy. That's true for electrons and positrons (anti-electrons). They'll produce gamma rays at 511,000 electron volts.

    But heavier particles like protons and anti-protons are somewhat messier, making gamma rays and leaving a spray of secondary particles that eventually decay into neutrinos and low-energy gamma rays.

    And that is partly what Schmidt and others want in an antimatter engine. The gamma rays from a perfect reaction would escape immediately, unless the ship had thick shielding, and serve no purpose. But the charged debris from a proton/anti-proton annihilation can push a ship.

    "We want to get as close as possible to the initial annihilation event," Schmidt explained. What's important is intercepting some of the pions and other charged particles that are produced and using the energy to produce thrust."


    So our history here in this blog has detailed how we see the issues of "collision processes developed(Cern), that we may now see the cosmological playground teaming with the opportunities to produce this "stuff" that would send our spaceships to Mars?

    The extension of the thinking of experimental development, has allowed us to think of "what is possible" and what this propulsion system can do, as we make our way into the new territories? As we set sail our ships, searching for those new lands.


    A Penn State artist's concept of n antimatter-powered Mars ship with equipment and crew landers at the right, and the engine, with magnetic nozzles, at left.


    Of course "storage" is always a troubling issue here so they developed what is call the Penning Trap. But it is not without some insight that our geometrical understanding developed in the events in the cosmos, could not be transformed in that same geometrical sense to propel those ships?


    This "Penning trap" developed at Penn State University stores antiprotons.
    It sounds like science fiction, but researchers are learning to create and store small amounts of antimatter in real-life labs. A portable electromagnetic antimatter trap at Penn State University, for example, can hold 10 billion antiprotons. If we could learn how to use such antimatter safely, we could impinge some on a thin stream of hydrogen gas to create thrust. Alternatively, a little antimatter could be injected into a fusion reactor to lower the temperatures needed to trigger a fusion reaction.


    So you ask how is that possible?

    The gravitational collapse sets up the very ideas for us as we make use of that "propulsion system" to move that space ship. So in a sense, "the collider process" at Cern is a gigantic model of what we want in the developmental process as the new engine of our spaceship.


    A schematic of the heart of a Penning trap where a cloud of antiprotons (the fuzzy bluish spot) is kept cold and quiet by liquid nitrogen and helium and a stable magnetic field. (PSU)
    Anti-protons, explained Dr. Gerald Smith of Pennsylvania State University, can be obtained in modest quantities from high-energy accelerators slamming particles into solid targets. The anti-protons are then collected and held in a magnetic bottle
    .


    While previously here I have spoken about how we may use Susskind's thought experiment as a monitoring system of gravitational considerations, it is also this thought process that helps us adjust the ship according to how much thrust is needed in face of the lagrangian views we encounter in star systems?

    However, by using "matter/antimatter annihilation", velocities just below the speed of light could be reached, making it possible to reach the next star in about six years.


    I think Stephen Hawking is going to have to work faster, in order to elucidate his thoughts on this travel. That while I may have started this lesson from the idea of 1999, it is much more advanced then many had understood. The "experimental process" of Cern is much greater then most of us had realized.

    Also there is a developmental "thought pattern" that needs to be understood as we speak about how such a geometrics could have been seen underneath the very structures of our realities. Not only within the cosmos at large but in the dynamical processes of the quantum world.

    Angels and Demons



    Cern IMagery takes a "dramatic position" on what it is saying about itself? :) I would like to think that the fun is in how "mirror world" has somehow been transposed into what we know of the develpmental processes we are given as we now lok at what may help us move into the cosmos.

    If as a society we were "uncultured" we might have thought the tribal influence of the "bad side" of all things? But in that exploratory sense al the tidbits had to add up to something, yet without our understanding of what lies beneath, one might have never gone "past" Robert Mclaughlin, to realize, the geometrical nature that imbues the process we are developing.



    This was Riemann lesson to Gauss in his thesis, who like his student had thought for sure "vision capable now," would also have been transferred into a "whole new world" of understanding of the non euclidean geometries.

    What do they say about the devil being in the details?

    Plato:
    This image had horns drawn on it, with a tail attached. Something about “angels and demons?” I don’t think we should take the “anti” too literal in face of an outcome, or should we?


    It's about how we can take a legitimate process and build ideas on it, according to the very nature of the "negative and positive expressions" of what Riemann set out to do.

    ON a large scale, we see the dynamics of this process, yet failed to see it work at a microcosmic sense as we deal with the colliders? As we move forward in the propulsion systems, it is importance how we see this developmental process take on dynamic views.