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

Thursday, August 22, 2013

Euclid


Socratica Studios



Maybe you can identify the section from which this picture is taken from. The picture seen in the Fresco painted by Raphael, is entitled the School of Athens in  heading above?:)
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)

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

    Wednesday, May 06, 2009

    Galileo Galilei's Balance

    Let proportion be found not only in numbers, but also in sounds, weights, times and positions, and whatever force there is.Leonardo Da Vinci


    Can it be said that measure has ever left the mind of those whose age has come and gone, and what appears as "no traces left behind" might indeed be the passage of age of one thought to the next? Progressive?

    (This illustration and text are from a magazine advertisement for NBC, probably dating from the 1940’s. It was found among the files of the Print and Picture Department of the Free Library of Philadelphia. The word “eureka!” is coming out of Archimedes’ mouthSee:Archimedes


    ***


    So it is ever the attempt for ingenuity to come forward from the deep recesses of the mind, for daylight to shine, and the measures revealed in as natural setting as possible? Who are these inventors then that help us to begin with one step and move forward with another? Is it their genius that we shall capitalize on, or shall it be the ignited pathways that are set "by ignition," that we say that brain has taken a new course in thinking? Blazing new trails?

    These "ideas" always existed? But where?

    De Architectura, VitruviusFirst Milanese period 1481/2 - 1499


    The first printed edition of the De Architectura (On Architecture) by the ancient Roman architect, Vitruvius, appears.


    Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC) was a Roman writer, architect and engineer (possibly praefectus fabrum during military service or praefect architectus armamentarius of the apparitor status group), active in the 1st century BC. By his own description[1] Vitruvius served as a Ballista (artilleryman), the third class of arms in the military offices. He likely served as chief of the ballista (senior officer of artillery) in charge of doctores ballistarum (artillery experts) and libratores who actually operated the machines.[2]



    ***


    So in today's world what value would Google have to produce a library and recording of all books that exists so that the route to knowledge shall be forever enshrined in some medium that we use now. If all the institutions and architecture of our civilization were to be gone tomorrow, shall such records be readily available to the "searching mind seeking knowledge" if there be survivors, that it will easily dial up the internet, or by wireless, gain access?

    My job more or less has been to enshrine "such knowledge" in the way we record information, that it is different then what Google seeks to do. Yes, some day by such shadows as again from one of Stefan's earlier link supplied below as "Euclidean Geometry and the Shadows," there will be an alignment with such geometries, which shall point the way to a vault where all things that man has gone through, that one day we will know our true history from the beginning of time till now.

    This vault to me was about our access to the ideas that exist out there in the ethers and how we might attain by touch of our finger upon it's pulse, that such connection is more then what exists in the Sistine Chapel above, that we had connected to that same resource as Jung's world of the collective unconscious.

    How so if we did not pursue the very notion of ideas that are attained at "some center" and found connected outside of our self?

    Galileo Galilei

    In 1586 at the age of 22, Galileo (1564-1642) wrote a short treatise entitled La Bilancetta (“The Little Balance”). He was skeptical of Vitruvius’s account of how Archimedes determined the fraud in Hiero's crown and in this treatise presented his own theory based on Archimedes’ Law of the Lever and Law of Buoyancy. He also included a description of a hydrostatic balance that determined the precise composition of an alloy of two metals.

    Just as it is well known to anyone who takes the care to read ancient authors that Archimedes discovered the jeweler’s theft in Hiero’s crown, it seems to me the method which this great man must have followed in this discovery has up to now remained unknown. Some authors have written that he proceeded by immersing the crown in water, having previously and separately immersed equal amounts [in weight] of very pure gold and silver, and, from the differences in their making the water rise or spill over, he came to recognize the mixture of gold and silver of which the crown was made. But this seems, so to say, a crude thing, far from scientific precision; and it will seem even more so to those who have read and understood the very subtle inventions of this divine man in his own writings; from which one most clearly realizes how inferior all other minds are to Archimedes’s and what small hope is left to anyone of ever discovering things similar to his [discoveries]. I may well believe that, a rumor having spread that Archimedes had discovered the said theft by means of water, some author of that time may have then left a written record of this fact; and that the same [author], in order to add something to the little that he had heard, may have said that Archimedes used the water in that way which was universally believed. But my knowing that this way was altogether false and lacking that precision which is needed in mathematical questions made me think several times how, by means of water, one could exactly determine the mixture of two metals. And at last, after having carefully gone over all that Archimedes demonstrates in his books On Floating Bodies and Equilibrium, a method came to my mind which very accurately solves our problem. I think it probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself. Galileo and the Scientific Revolution by Laura Fermi and Gilberto Bernardini (Translated with the assistance of Cyril Stanley Smith) Basic Books, Inc., New York, 1961 Library of Congress Catalog Card Number 61-7486
    SEE:The Golden Crown Sources

    It had not escaped me that the shadows and stories of the cave become fodder for thought, even amongst our own scientists, that they began the process one way philosophically and thought, what is capable in us to see that any dimensional significance would be an higher portal for some movement of the abstract in mind, is realized in topological dance?


    ***


    See:

  • God the Geometer




  • Euclidean Geometry and the Shadows



  • Update:

    The Truth

    Sunday, June 08, 2008

    Who said it?

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


    "No Royal Road to Geometry?"

    Click on the Picture

    Are you an observant person? Look at the above picture. Why ask such a question as to, "No Royal Road to Geometry?" This presupposes that a logic is formulated that leads not only one by the "phenomenological values" but by the very principal of logic itself.

    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)


    I don't think I could of made it any easier for one, but to reveal the answer in the quote. Now you must remember how the logic is introduced here, and what came before Euclid. The postulates are self evident in his analysis but, little did he know that there would be a "Royal Road indeed" to geometry that was much more complex and beautiful then the dry implication logic would reveal of itself.

    It's done for a reason and all the geometries had to be leading in this progressive view to demonstrate that a "projective geometry" is the final destination, although, still evolving?

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

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

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


    If you are "ever the artist" it is good to know in which direction you will use the sun, in order to demonstrate the shadowing that will go on into your picture. While you might of thought there was everything to know about Plato's cave and it's implication I am telling you indeed that the logic is a formative apparatus concealed in the geometries that are used to explain such questions about, "the shape of space."

    The Material World

    There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

    The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.


    Polytopes and allotrope are examples to me of "shapes in their formative compulsions" that while very very small in their continuing expression, "below planck length" in our analysis of the world, has an "formative structure" in the case of the allotrope in the material world. The polytopes, as an abstract structure of math thinking about the world. As if in nature's other ways.



    This illustration depicts eight of the allotropes (different molecular configurations) that pure carbon can take:

    a) Diamond
    b) Graphite
    c) Lonsdaleite
    d) Buckminsterfullerene (C60)
    e) C540
    f) C70
    g) Amorphous carbon
    h) single-walled carbon nanotube


    Review of experiments

    Graphite exhibits elastic behaviour and even improves its mechanical strength up to the temperature of about 2500 K. Measured changes in ultrasonic velocity in graphite after high temperature creep shows marked plasticity at temperatures above 2200 K [16]. From the standpoint of thermodynamics, melting is a phase transition of the first kind, with an abrupt enthalpy change constituting the heat of melting. Therefore, any experimental proof of melting is associated with direct recording of the temperature dependence of enthalpy in the neighbourhood of a melting point. Pulsed heating of carbon materials was studied experimentally by transient electrical resistance and arc discharge techniques, in millisecond and microsecond time regime (see, e.g., [17, 18]), and by pulsed laser heating, in microsecond, nanosecond and picosecond time regime (see, e.g., [11, 19, 20]). Both kind of experiments recorded significant changes in the material properties (density, electrical and thermal conductivity, reflectivity, etc. ) within the range 4000-5000 K, interpreted as a phase change to a liquid state. The results of graphite irradiation by lasers suggest [11] that there is at least a small range of temperatures for which liquid carbon can exist at pressure as low as 0.01 GPa. The phase boundaries between graphite and liquid were investigated experimentally and defined fairly well.


    Sean Carroll:But if you peer closely, you will see that the bottom one is the lopsided one — the overall contrast (representing temperature fluctuations) is a bit higher on the left than on the right, while in the untilted image at the top they are (statistically) equal. (The lower image exaggerates the claimed effect in the real universe by a factor of two, just to make it easier to see by eye.)
    See The Lopsided Universe-.

    #36.Plato on Jun 12th, 2008 at 10:17 am

    Lawrence,

    Thanks again.

    “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

    Moving to polytopes or allotrope seem to have values in science? Buckminister Fuller and Richard Smalley in terms of allotrope.

    I was looking at Sylvestor surfaces and the Clebsch diagram. Cayley too. These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope[E8]“(where Coxeter[I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)

    As the furthest extent of the Conjecture , how shall we place the dynamics of Sylvestor surfaces and B Fields in relation to the timeline of these geometries? Historically this would seem in order, but under the advancement of thinking in theoretics does it serve a purpose? Going beyond “planck length” what is a person to do?

    Thanks for the clarifications on Lagrange points. This is how I see the WMAP.

    Diagram of the Lagrange Point gravitational forces associated with the Sun-Earth system. WMAP orbits around L2, which is about 1.5 million km from the Earth. Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. The forces at L2 tend to keep WMAP aligned on the Sun-Earth axis, but requires course correction to keep the spacecraft from moving toward or away from the Earth.


    Such concentration in the view of Sean’s group of the total WMAP while finding such a concentration would be revealing would it not of this geometrical instance in relation to gravitational gathering or views of the bulk tendency? Another example to show this fascinating elevation to non-euclidean, gravitational lensing, could be seen in this same light.

    Such mapping would be important to the context of “seeing in the whole universe.”


    See:No Royal Road to Geometry
    Allotropes and the Ray of Creation
    Pasquale Del Pezzo and E8 Origination?
    Projective Geometries

    Wednesday, January 23, 2008

    Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

    As I pounder the very basis of my thoughts about geometry based on the very fabric of our thinking minds, it has alway been a reductionist one in my mind, that the truth of the reality would a geometrical one.



    The emergence of Maxwell's equations had to be included in the development of GR? Any Gaussian interpretation necessary, so that the the UV coordinates were well understood from that perspective as well. This would be inclusive in the approach to the developments of GR. As a hobbyist myself of the history of science, along with the developments of today, I might seem less then adequate in the adventure, I persevere.




    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.



    For me the education comes, when I myself am lured by interest into a history spoken to by Stefan and Bee of Backreaction. The "way of thought" that preceded the advent of General Relativity.


    Einstein urged astronomers to measure the effect of gravity on starlight, as in this 1913 letter to the American G.E. Hale. They could not respond until the First World War ended.

    Translation of letter from Einstein's to the American G.E. Hale by Stefan of BACKREACTION

    Zurich, 14 October 1913

    Highly esteemed colleague,

    a simple theoretical consideration makes it plausible to assume that light rays will experience a deviation in a gravitational field.

    [Grav. field] [Light ray]

    At the rim of the Sun, this deflection should amount to 0.84" and decrease as 1/R (R = [strike]Sonnenradius[/strike] distance from the centre of the Sun).

    [Earth] [Sun]

    Thus, it would be of utter interest to know up to which proximity to the Sun bright fixed stars can be seen using the strongest magnification in plain daylight (without eclipse).


    Fast Forward to an Effect

    Bending light around a massive object from a distant source. The orange arrows show the apparent position of the background source. The white arrows show the path of the light from the true position of the source.

    The fact that this does not happen when gravitational lensing applies is due to the distinction between the straight lines imagined by Euclidean intuition and the geodesics of space-time. In fact, just as distances and lengths in special relativity can be defined in terms of the motion of electromagnetic radiation in a vacuum, so can the notion of a straight geodesic in general relativity.



    To me, gravitational lensing is a cumulative affair that such a geometry borne into mind, could have passed the postulates of Euclid, and found their way to leaving a "indelible impression" that the resources of the mind in a simple system intuits.

    Einstein, in the paragraph below makes this clear as he ponders his relationship with Newton and the move to thinking about Poincaré.

    The move to non-euclidean geometries assumes where Euclid leaves off, the basis of Spacetime begins. So such a statement as, where there is no gravitational field, the spacetime is flat should be followed by, an euclidean, physical constant of a straight line=C?

    Einstein:

    I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity. ... In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry. ... If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincare:--Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity. Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable ... to retain Euclidean geometry. For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. (33-4)


    It is never easy for me to see how I could have moved from what was Euclid's postulates, to have graduated to my "sense of things" to have adopted this, "new way of seeing" that is also accumulative to the inclusion of gravity as a concept relevant to all aspects of the way in which one can see reality.

    See:

  • On the Hypothese at the foundations of Geometry

  • Gravity and Electromagnetism?

  • "The Confrontation between General Relativity and Experiment" by Clifford M. Will
  • 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.

    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 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