Torii that Coalesce in a Shockwave Horn of Negative Curvature

This bit of outstanding lab work demonstrates a principle that I believe is even more far reaching than its authors imagine. It demonstrates a form of action which has a unique and universal application. Viz: a cylindrically formed  plasma torii whose succession  amounts to the characteristic shockwave horn formation over time. This bears a remarkable equivalence with this author's model for the metastable shockwave spiral pathway along the horn tore or anchor ring. This same tore that Archytas put to such good use in his proof for doubling the cube. This model hints at a property of the evolvability of universal action that Cusa perfectly illuminated in his work the Non Other.

A plume driven by buoyancy which is being created by a chemical reaction. The plume accelerates and sheds accelerating smoke ring vortices. The process is analogous to the detonation of a type Ia supernova. Experiment by Michael C. Rogers

Universal Relative Motion and Potential Energy Propagation as Subsumed By Creative Imagination

If you imagine the layerings of mere rotational (and thereby chiral) motions starting from the infinitesimal and continuously amalgamating toward the transcendent velocities of galactic groups, what you come away with should not be a path of a point. That would be to miss the continuously embedded toroidal subsumption of potential energy of which that point partakes. That toroidal form as a singularly deformed sphere propagates material waves externally as allowed by the surrounding medium through which it moves. However, on its surface, there is an orbital which at its extremum represents an alternation from negative to positive surface curvature. This is the chiral (or spiral) pathway from the toroidal pole through its inner equator, which is a point connecting two horns at center then continuing along the opposite horn and through the outer equator and back to the pole. For the mathematical imagination, the velocity of this rotation goes from zero to infinity. Thus providing a model for a metastable alternation of propagation of potential energy which anneals the shock wave velocity which is ever present here as the path along the negatively curved horn with the continuation along the positively curved outer spheroidal shell. And doesn't this model suggest a more general means for the universal annealing of the psycho-physical continuum, as distinct from the merely number line mathematical one?
Because as you move onto the domain of truly human cognition as the ability to create analogous singularities via revolutionary technological innovations, it becomes apparent that there exists an unending process of a potential chiral pathway through the propagation of the adoption of these technologies by society as it is able in a spherically propagated wavelike analogous way and onto the next such breakthrough.
Therefore, without the understanding that the noetic, creative impulse of humanity subsumes all we can know about merely physical subordinated systems we go off in a puerile, masturbatory quest like some theoretical physicist Don Juan's multiverse raised to the power of one thousand and three .
This metastatic model is akin to Leibniz' concept of monad, Cusa's microcosm macrocosm principle, and Plato's unspoken resolution of the Parmenedes paradox. It is what Sancho Panza solves in the governor bridge riddle: the power of mercy. In any epoch of human history, inevitably there is crisis. Why? Because creativity cannot sit still. The technological means of society's reducing power are necessarily limited and evanescent. Every culture thereby confronts its own demons. Zeus as the oligarch who punishes Aeschylus' Prometheus the fire bearer. That is to say, the creative spark latent in each and every one of us. This is the secret of the tragedy in Shakespeare's works. The imposition of a self destructive impulse upon the governance of men that would like Cronus eat its own children is the antithesis of this model. The principle is above the here and now: as scientific and artistic creativity it is truly immortal. Either you choose to embrace it or shun it at humanity's peril. You are at once free and yet constrained, dear reader. How else would you have it?

Gauss and Riemann Understood from the Standpoint of Vernadsky, Leibniz, and LaRouche: Updated

 In nearly all descriptions of the principle conformal mapping one sees displayed a simple globe whose south pole is situated at the origin of a Cartesian x,y two dimensional plain and rays from the surface projected through the north pole. This represents a transformation from the flat plain to the positively curved sphere. 

By Original: Mark.Howison at English Wikipedia This version: CheChe - This file was derived from:  Stereographic projection in 3D.png:, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=56357179

However, the great Carl Friedrich Gauss developed a different type of mapping of irregular extended curvature onto a unit sphere: the "Gauss map." Here the normal from the curved surface that is given by the differential curvature (but not its magnitude) is mapped by a parallel projection onto the unit sphere. It is the tendency for change at that "point" that is being mapped which is something altogether different than a mere point in the above mapping.

Now, in an extended surface the map will show an orientation as one moves across the surface. Also, importantly since on any surface the same normal can be repeated there can be many sheets or layers. This could for instance pertain to the Einstein gravitational tensor directional locus of curvature mapped from the known universe. Also, it has been shown that the visual perceptual apparatus operates in an analogous manner of mapping. There exist "grid cells"in the entorhinal cortex of the mammalian brain  that are activated based upon location, direction and distance orientation.

The great student of Gauss, Bernhard Riemann developed this principle yet further with his surface function. It is a surface that combines different sheets of a multi-valued function through singular branch points of that function. Here, our attention is turned from curvature per se to how a function that can take on many values. Viewed from the standpoint of an n dimensional manifold, a branch point might provide a pathway to a new degree of functional freedom to an n+1 dimensional manifold as Riemann detailed in his doctoral habilitation dissertation. 

The conception of a singularity where a function branches becomes especially important in physical terms beyond the mathematical formalism. This can be shown in systems that undergo a critical condition and emerge into another regime or phase change such as in plasma physics. It can also be thought of as supplying a means of representing a multi-valued function of proteins in biophysics. There are a whole host of examples where proteins may have more than one role in disparate biological functions and sometimes wholly contrary ones. 

Indeed, the most important case of this principle comes into play for the human economy. If we view an invention as a kind of creative singularity for the functional "space" of the physical economy. That is, due to the development of a new degree of freedom the economy "goes over" into a new functional set of relations via the branchpoint of that invention. 

No matter, which type of extended space  we have to deal with the principle of Leibniz' monadology comes into play. For as Leibniz clearly proved, the mind, or soul if you will, is not an extended substance. That is, it has no parts. Now this is what Riemann is referring to as its "compactness" in his statement "The mind is a compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop." Likewise, all true infinitesimals or monads have no parts. Leibniz called the non-mechanical force arising therefrom "vis viva." If one looks at the human economy as an extended functional system of physical relations one can perform a type of mapping of its distinct elements that operate together in much the same way as a Gauss map. The normal to the "curvature" is the inter-relationship of types of physical goods produced and operatives for the entirety of the reproductive cycle of the economy. In this way, we have a mapping characterized by distinct categories that were developed by Lyndon LaRouche's correction of Marxian economics.

This type of mapping of physical inputs and outputs can be carried out as energy throughput for the entire planet's biosphere for instance. Which brings us to back to the issue of "vis viva" and types of lower and higher monads. Firstly, it is clear that Leibniz qualified the human mind as the highest type of monad. Also that it, the substance of mind or soul is imperishable. How is this so? To show this, we must investigate the concept of pre-established harmony of the monads. In all "spheres" of reality as defined by Vernadsky, i.e. Lithosphere, Biosphere, and Noosphere there is a type of connatus that bounds it, hence its compactness. (Here one should adduce an Astrosphere to Vernadsky's as mankind enters the age of space colonization.) Now Leibniz proposed that not all monads have a kind of window upon the world.  Some, however operate as a window for the higher monads. Such are the cilia signaling cells and the place cells, for instance. I would contend that this idea represents a type of mapping of the particular frame of reference of the monad to the universal. (It is of no small import that Einstein's principle of relativity is just such a kind of distinctly Riemannian mapping.)  It is here that the harmonics comes into play and it is also in that sense that this enduring non perishable lawful relationship governing all reality makes this the "best of all possible worlds." (So much for ill begotten theories of stochastic and lawless chaos of the rule of irrationality.) 

To conclude this survey, next take the physics of pressure waves that Riemann presented in his 1857 paper "On the propagation of planar airwaves of finite amplitude." In this paper Riemann in essence has predicted so called sonar shockwaves. For in any medium there is a governing and limiting characteristic rate of transmission. At the point of going beyond that, a singularity occurs where the curvature goes from a positive spherical (or technically for the case of spacetime hyperspherical) to a negative Mach cone type. 

Now this is representative of the manner in which a nonlinear progression is pre-established in the ongoing composition of external and internal reality is distinctly cognized by human noetic or creative mentation in the arts and science. The searching out of such anomalous singularities is the true method for the production of necessary human progress. And here I will end this brief overview to remind my audience that whether these principles come to the fore in human relations will determine either the pathway to a new Renaissance or a new Dark Age.

Since I wrote this post there has been dramatic progress in research on the nature of grid cells. I am happy to report that my intuition that toroidal action is indicative of a principle of metastability across many facets of physical reality has been confirmed yet again. It turns out that the topology of grid cells in the brain take the form of a torus. This confirms to me that in the domain of animal life, there exists a wonderfully built in capacity for a sort of problem solving skill that relies upon a surface that has both positive and negative curvature embedded within it. Recalling that Riemann pioneered non Euclidean curved surfaces and the propagation of shockwaves (that result in negative curved horn surface in a series of Mach cones) as a student of Gauss, it is only lawful that this particular geometry would be in the words of their great predecessor Leibniz be pre-established. See this wonderful linked video. 

Have You Eyes But Cannot See?

The drama being displayed here is the alternative to the clock-maker's silly surmise of a "big bang." This marvel is the shockwave form of horn torii that is being created anew continuously in this our well composed and indescribably beautiful and singular universe. Here are the waters of life that move upon the heavens. Here too is the noetic principle that moves humanity beyond its infant cradle upon earth.

Can you see beyond the circumscribed orb of your mere senses, oh child of this, thy destiny? Will you at last put aside these mere baubles that weigh you down even unto oblivion? "Tis time, aye and well past time to waken. The universal voyage upon this awesome sea of potential harmony awaits.

Shockwaves of Galactic Clusters and Implications of the Microcosm and Macrocosm Interrelationship

 I have repeatedly brought attention to the seminal work of Bernard Riemann on the principle of shockwave propagation. Keeping in mind his insistence as well that the physical geometry outside the range of our limited human or better yet perhaps mammalian perceptions may bring into play the alternative curvatures of of non-Euclidean "space." 

In this regard, the discovery of the presence of shockwave formations of merging galaxy clusters in the radio wave spectrum as so called radio relics is very important research for moving the frontiers of scientific progress forward. This is especially due to their almost unimaginably vast structure at the same time propelling the very smallest particles' acceleration to relativistic speed of light velocities.

 This is the indissoluble connection that is always apparent of the macrocosm with the microcosm as Nicholas of Cusa demonstrated in his "triune" principle. The very large and the very small macrocosm/microcosm connectedness as an analogue of the human relationship of the universal creator with the individual person via creative reason, in non religious terms.

The curvature as the shockwave front nears the limit of transmissibility for the medium of electromagnetic propagation proceeds naturally from a positive spheroid type to a negative horn. Indeed, Riemann is again proven correct that the physical system precedes choice of mathematical geometry. The various types of further such potential particle accelerations and scatterings, etc. are the platform upon which we can be confident scientific progress accruing to the future benefit of humanity will necessarily proceed as both Riemann and Cusa would surely agree. 

MeerKAT radio contours (magenta) on A2384 XMM-Newton image. Credit: Parekh et al., 2020

" The origin of the radio relic is puzzling.(...) This radio relic could be the result of shock wave propagation during the passage of the low-mass A2384(S) cluster through the massive A2384(N) cluster, creating a trail appearing as a hot X-ray filament." V. Parekh, et al.: https://arxiv.org/abs/2009.02724

Spiral Paths from Atomic Phase Change to Black Hole Star Systems

Illustration from:

Field Ionization of Cold Atoms near the Wall of a Single Carbon Nanotube

Anne Goodsell, Trygve Ristroph, J. A. Golovchenko, and Lene Vestergaard Hau

I have maintained consistently for some time now the primacy of spiral orbitals along the surface of horn torii as a toy model for Riemannian metastable shockwave transformations through the singularity at its center. This toy model serves not only to adumbrate physical transformations, but also the historical impact of revolutionary ideas upon the functioning of society. The research illustrated above where a super cooled rubidium atom spirals toward a highly charged carbon nanowire and is subsequently stripped of an electron that tunnels into the tube while the resulting ion is repulsed is, I believe, an elegant demonstration of this principle.

Spirals occur as radiative processes move through spacetime. Imagine the orbit of the earth around the sun as the sun orbits the galaxy as the galaxy orbits its own group, etc. Embedded spirals. Here is an illustration of a spiral from a joint black hole star system whose anomalous physics is not well understood.

However, there are few attempts to model these radiative pathways as a field. Penrose's Twistor Theory is one such. However, his attempt to locate consciousness as a quantum effect is as misguided as Descartes' the pineal gland is the location of the soul. 

The Quixotic quest for a physical location of mind or soul is a result of a teleology or search for a final cause which is simply wrong. The assumption that that there is a frankly rather magical yet undiscovered physical explanation for human consciousness is due to a mode of thinking that assumes that substance is equivalent to mere physical sensation. This should have been abandoned once and for all after Leibniz devastatingly proved that the mind is of a different order that physical substance. That is, the mind is an indissoluble whole having no parts. It cannot be divided. Physical substance, meaning everything other than mind that may be conceived on the other hand may be divided. (The seeming paradox of individual quantum states, notwithstanding. For the condition of superposition shows that these states too are divisible via decoherence because of the very fact they exist as divisible quanta.) The mind in this sense participates as a kind of non-other to use Nicholas of Cusa's term. As Leibniz determined the mind is a sort of miniature creator reflecting the universal creative process of becoming. Or what is in Christian terms called Imago Viva Dei. The problem is that the materialists like Penrose can't stomach these truths, the which are not self evident, they are proven. 

What Prevents Science from Achieving Accelerated Breakthroughs

The edifice of assumptions regarding the DNA code as the program that is determinate for life is fatally flawed. The reality is that life is of a higher order of transfinite than the non living. Likewise human creative reason is of a higher such order than the domain of lower life forms. The question is why do the vast majority of scientists not know what transfinite ordering is?

Edgar Allan Poe, whence the name of this blog, in his Eureka presaged this lack of epistemological clarity thus:

"Now, my dear friend," continues the letter-writer, "it cannot be maintained that by the crawling system, exclusively adopted, men would arrive at the maximum amount of truth, even in any long series of ages; for the repression of imagination was an evil not to be counterbalanced even by absolute certainty in the snail processes. But their certainty was very far from absolute. The error of our progenitors was quite analogous with that of the wiseacre who fancies he must necessarily see an object the more distinctly, the more closely he holds it to his eyes. They blinded themselves, too, with the impalpable, titillating Scotch snuff of detail; and thus the boasted facts of the Hog-ites were by no means always facts -- a point of little importance but for the assumption that they always were. The vital taint, however, in Baconianism -- its most lamentable fount of error -- lay in its tendency to throw power and consideration into the hands of merely perceptive men -- of those inter-Tritonic minnows, the microscopical savans -- the diggers and pedlers of minute facts, for the most part in physical science -- facts all of which they retailed at the same price upon the highway; their value depending, it was supposed, simply upon the fact of their fact, without reference to their applicability or inapplicability in the development of those ultimate and only legitimate facts, called Law.



"Than the persons" -- the letter goes on to say -- "than the persons thus suddenly elevated by the Hog-ian philosophy into a station for which they were unfitted -- thus transferred from the sculleries into the parlors of Science -- from its pantries into its pulpits -- than these individuals a more intolerant -- a more intolerable set of bigots and tyrants never existed on the face of the earth. Their creed, their text and their sermon were, alike, the one word 'fact' -- but, for the most part, even of this one word, they knew not even the meaning. On those who ventured to disturb their facts with the view of putting them in order and to use, the disciples of Hog had no mercy whatever. All attempts at generalization were met at once by the words 'theoretical,' 'theory,' 'theorist' -- all thought, to be brief, was very properly resented as a personal affront to themselves. Cultivating the natural sciences to the exclusion of Metaphysics, the Mathematics, and Logic, many of these Bacon-engendered philosophers -- one-idead, one-sided and lame of a leg -- were more wretchedly helpless -- more miserably ignorant, in view of all the comprehensible objects of knowledge, than the veriest unlettered hind who proves that he knows something at least, in admitting that he knows absolutely nothing.



"Nor had our forefathers any better right to talk about certainty, when pursuing, in blind confidence, the a priori path of axioms, or of the Ram. At innumerable points this path was scarcely as straight as a ram's-horn. The simple truth is, that the Aristotelians erected their castles upon a basis far less reliable than air; for no such things as axioms ever existed or can possibly exist at all. This they must have been very blind, indeed, not to see, or at least to suspect; for, even in their own day, many of their long-admitted 'axioms' had been abandoned: -- 'ex nihilo nihil fit,' for example, and a 'thing cannot act where it is not,' and 'there cannot be antipodes,' and 'darkness cannot proceed from light.' These and numerous similar propositions formerly accepted, without hesitation, as axioms, or undeniable truths, were, even at the period of which I speak, seen to be altogether untenable: -- how absurd in these people, then, to persist in relying upon a basis, as immutable, whose mutability had become so repeatedly manifest!



"But, even through evidence afforded by themselves against themselves, it is easy to convict these a priori reasoners of the grossest unreason -- it is easy to show the futility -- the impalpability of their axioms in general. I have now lying before me" -- it will be observed that we still proceed with the letter -- "I have now lying before me a book printed about a thousand years ago. Pundit assures me that it is decidedly the cleverest ancient work on its topic, which is 'Logic.' The author, who was much esteemed in his day, was one Miller or Mill; and we find it recorded of him, as a point of some importance, that he rode a mill-horse whom he called Jeremy Bentham: -- but let us glance at the volume itself!



"Ah! -- 'Ability or inability to conceive,' says Mr. Mill very properly, 'is in no case to be received as a criterion of axiomatic truth.' Now, that this is a palpable truism no one in his senses will deny. Not to admit the proposition, is to insinuate a charge of variability in Truth itself, whose very title is a synonym of the Steadfast. If ability to conceive be taken as a criterion of Truth, then a truth to David Hume would very seldom be a truth to Joe; and ninety-nine hundredths of what is undeniable in Heaven would be demonstrable falsity upon Earth. The proposition of Mr. Mill, then, is sustained. I will not grant it to be an axiom; and this merely because I am showing that no axioms exist; but, with a distinction which could not have been cavilled at even by Mr. Mill himself, I am ready to grant that, if an axiom there be, then the proposition of which we speak has the fullest right to be considered an axiom -- that no more absolute axiom is -- and, consequently, that any subsequent proposition which shall conflict with this one primarily advanced, must be either a falsity in itself -- that is to say no axiom -- or, if admitted axiomatic, must at once neutralize both itself and its predecessor.



"And now, by the logic of their own propounder, let us proceed to test any one of the axioms propounded. Let us give Mr. Mill the fairest of play. We will bring the point to no ordinary issue. We will select for investigation no common-place axiom -- no axiom of what, not the less preposterously because only impliedly, he terms his secondary class -- as if a positive truth by definition could be either more or less positively a truth: -- we will select, I say, no axiom of an unquestionability so questionable as is to be found in Euclid. We will not talk, for example, about such propositions as that two straight lines cannot enclose a space, or that the whole is greater than any one of its parts. We will afford the logician every advantage. We will come at once to a proposition which he regards as the acme of the unquestionable -- as the quintessence of axiomatic undeniability. Here it is: -- 'Contradictions cannot both be true -- that is, cannot coexist in nature.' Here Mr. Mill means, for instance, -- and I give the most forcible instance conceivable -- that a tree must be either a tree or not a tree -- that it cannot be at the same time a tree and not a tree: -- all which is quite reasonable of itself and will answer remarkably well as an axiom, until we bring it into collation with an axiom insisted upon a few pages before -- in other words -- words which I have previously employed -- until we test it by the logic of its own propounder. 'A tree,' Mr. Mill asserts, 'must be either a tree or not a tree.' Very well: -- and now let me ask him, why. To this little query there is but one response: -- I defy any man living to invent a second. The sole answer is this: -- 'Because we find it impossible to conceive that a tree can be anything else than a tree or not a tree.' This, I repeat, is Mr. Mill's sole answer: -- he will not pretend to suggest another: -- and yet, by his own showing, his answer is clearly no answer at all; for has he not already required us to admit, as an axiom, that ability or inability to conceive is in no case to be taken as a criterion of axiomatic truth? Thus all -- absolutely his argumentation is at sea without a rudder. Let it not be urged that an exception from the general rule is to be made, in cases where the 'impossibility to conceive' is so peculiarly great as when we are called upon to conceive a tree both a tree and not a tree. Let no attempt, I say, be made at urging this sotticism; for, in the first place, there are no degrees of 'impossibility,' and thus no one impossible conception can be more peculiarly impossible than another impossible conception: -- in the second place, Mr. Mill himself, no doubt after thorough deliberation, has most distinctly, and most rationally, excluded all opportunity for exception, by the emphasis of his proposition, that, in no case, is ability or inability to conceive, to be taken as a criterion of axiomatic truth: -- in the third place, even were exceptions admissible at all, it remains to be shown how any exception is admissible here. That a tree can be both a tree and not a tree, is an idea which the angels, or the devils, may entertain, and which no doubt many an earthly Bedlamite, or Transcendentalist, does.



"Now I do not quarrel with these ancients," continues the letter-writer, "so much on account of the transparent frivolity of their logic -- which, to be plain, was baseless, worthless and fantastic altogether -- as on account of their pompous and infatuate proscription of all other roads to Truth than the two narrow and crooked paths -- the one of creeping and the other of crawling -- to which, in their ignorant perversity, they have dared to confine the Soul -- the Soul which loves nothing so well as to soar in those regions of illimitable intuition which are utterly incognizant of 'path.'



"By the bye, my dear friend, is it not an evidence of the mental slavery entailed upon those bigoted people by their Hogs and Rams, that in spite of the eternal prating of their savans about roads to Truth, none of them fell, even by accident, into what we now so distinctly perceive to be the broadest, the straightest and most available of all mere roads -- the great thoroughfare -- the majestic highway of the Consistent? Is it not wonderful that they should have failed to deduce from the works of God the vitally momentous consideration that a perfect consistency can be nothing but an absolute truth? How plain -- how rapid our progress since the late announcement of this proposition! By its means, investigation has been taken out of the hands of the ground-moles, and given as a duty, rather than as a task, to the true -- to the only true thinkers -- to the generally-educated men of ardent imagination. These latter -- our Keplers -- our Laplaces -- 'speculate' -- 'theorize' -- these are the terms -- can you not fancy the shout of scorn with which they would be received by our progenitors, were it possible for them to be looking over my shoulders as I write? The Keplers, I repeat, speculate -- theorize -- and their theories are merely corrected -- reduced -- sifted -- cleared, little by little, of their chaff of inconsistency -- until at length there stands apparent an unencumbered Consistency -- a consistency which the most stolid admit -- because it is a consistency -- to be an absolute and unquestionable Truth.



"I have often thought, my friend, that it must have puzzled these dogmaticians of a thousand years ago, to determine, even, by which of their two boasted roads it is that the cryptographist attains the solution of the more complicated cyphers -- or by which of them Champollion guided mankind to those important and innumerable truths which, for so many centuries, have lain entombed amid the phonetical hieroglyphics of Egypt. In especial, would it not have given these bigots some trouble to determine by which of their two roads was reached the most momentous and sublime of their truths -- the truth -- the fact of gravitation? Newton deduced it from the laws of Kepler. Kepler admitted that these laws he guessed -- these laws whose investigation disclosed to the greatest of British astronomers that principle, the basis of all (existing) physical principle, in going behind which we enter at once the nebulous kingdom of Metaphysics. Yes! -- these vital laws Kepler guessed -- that it is to say, he imagined them. Had he been asked to point out either the de ductive or in ductive route by which he attained them, his reply might have been -- 'I know nothing about routes -- but I do know the machinery of the Universe. Here it is. I grasped it with my soul -- I reached it through mere dint of intuition.' Alas, poor ignorant old man! Could not any metaphysician have told him that what he called 'intuition' was but the conviction resulting from de ductions or in ductions of which the processes were so shadowy as to have escaped his consciousness, eluded his reason, or bidden defiance to his capacity of expression? How great a pity it is that some 'moral philosopher' had not enlightened him about all this! How it would have comforted him on his death-bed to know that, instead of having gone intuitively and thus unbecomingly, he had, in fact, proceeded decorously and legitimately -- that is to say Hog-ishly, or at least Ram-ishly -- into the vast halls where lay gleaming, untended, and hitherto untouched by mortal hand -- unseen by mortal eye -- the imperishable and priceless secrets of the Universe!



"Yes, Kepler was essentially a theorist; but this title, now of so much sanctity, was, in those ancient days, a designation of supreme contempt. It is only now that men begin to appreciate that divine old man -- to sympathize with the prophetical and poetical rhapsody of his ever-memorable words. For my part," continues the unknown correspondent, "I glow with a sacred fire when I even think of them, and feel that I shall never grow weary of their repetition: -- in concluding this letter, let me have the real pleasure of transcribing them once again: -- 'I care not whether my work be read now or by posterity. I can afford to wait a century for readers when God himself has waited six thousand years for an observer. I triumph. I have stolen the golden secret of the Egyptians. I will indulge my sacred fury.'"

The Ubiquity of Positive to Negative Curvature Transformations

Yesterday, three examples confirming Bernard Riemann's truly beautiful hypothesis on the function of the curvature of space were brought to my attention. First is a remarkable and outstanding confirmation of what appears to be the playful "creativity" of plant life in problem solving in order to reproduce. In this case, sphagnum moss overcomes its limits in fostering offspring:

(PhysOrg.com) -- Scientists in the US have solved the mystery of how peat mosses manage to get their spores high enough to catch the wind, discovering that they produce vortex rings of air, like miniature "mushroom clouds" to boost the spores along -- the first time plants have been shown to be capable of creating such rings.

The tip of each stalk contains from 20-250,000 spores in a spherical, waterlogged capsule that dehydrates on sunny days. As it dries out the capsule collapses in on itself to form a cylinder, with the air compressed inside it. When the air pressure reaches a critical point the capsule blows its top off, shooting spores and air upwards to reach the turbulent layer at about 10 cm high. The process takes under one hundredth of a millisecond.

The vortex ring is produced as the air trapped in the capsule explodes outwards. The leading edge pushes against the atmosphere but then rolls back to form a donut-shaped ring. Several animals are known to make vortex rings, but until now no plant has been known to do so. Dolphins produce water vortex rings; jellyfish and squid use water rings to push themselves along, the human heart pushes blood from one chamber to another using vortex rings, and of course humans who smoke can produce smoke rings. 

Explosion of a Sphagnum palustre capsule. (Recorded at 10,000 fps with a 0.097 ms exposure; displayed at 15 fps.) A clear "mushroom cloud" vortex ring forms and reaches a nearly fixed width within 0.5 ms. Spores are carried upward within the vortex ring and its wake. Credit: Science, doi:10.1126/science.1190179

A vortex ring is a very efficient way to move through air or water because it moves as a single unit. The vortex ring formed by the exploding spore capsule enables the spores to keep together as they rise. Without the ring individual spores would slow down and disperse before reaching the turbulent layer, and most would fall back to the ground. Edwards said they had witnessed spore clouds reaching as high as 17 cm, but even the average height of 11 cm gets the spores well into the turbulence.The findings are published in today’s edition of the journal Science.Science journal Podcast: an interview with Dwight Whitaker on how Sphagnum mosses use "air guns" and vortex rings to aid reproductive success. Credit: Science, DOI:10.1126/science.1190179  More information: Sphagnum Moss Disperses Spores with Vortex Rings, Dwight L. Whitaker and Joan Edwards, Science 23 July 2010: Vol. 329. no. 5990, p. 406, DOI: 10.1126/science.1190179 

And here is an astronomical example on the web from the very same day of publication the preceding item:

Messier 76 
Credit & Copyright: Ken Crawford (Rancho Del Sol Obs.)

Explanation: "Nebula at the right foot of Andromeda ... " begins the description for the 76th object in Charles Messier's 18th century Catalog of Nebulae and Star Clusters. In fact, M76 is one of the fainter objects on the Messier list and is also known by the popular name of the "Little Dumbbell Nebula". Like its brighter namesake M27 (the Dumbbell Nebula), M76 is recognized as a planetary nebula - a gaseous shroud cast off by a dying sunlike star. The nebula itself is thought to be shaped more like a donut, while the box-like appearance of its brighter central region is due to our nearly edge-on view. Gas expanding more rapidly away from the donut hole produces the fainter loops of far flung material. The fainter material is emphasized in this composite image, highlighted by showing emission from hydrogen atoms in orange and oxygen atoms in complementary blue hues. The nebula's dying star can be picked out in the sharp false-color imageas the blue-tinted star near the center of the box-like shape. Distance estimates place M76 about 3 to 5 thousand light-years away, making the nebula over a light-year in diameter.

The final example from  yesterday came from investigations of cellular geometry. It establishes how protein signalling uses negative curvature to "decide" when to proceed with meiosis.
As I have so often pointed out here, it is the alternation of positive spherical curvature to a negative horn shockwave form that necessarily serves as a defining overriding characteristic of physical, biophysical, and noetic "space."

Proof of Spiral Wave in Neocortex

It is with great satisfaction that I welcome the news of this spectacular research. Over the course of this blog's history, I have maintained that there is an analogy with a form of spiral shockwave in nature with its correlative acting upon human society via technological advance. It is therefore absolutely lawful that these researchers show that so called traveling brainwaves are also spiral. This advances Bernardt Riemann's insight that new ideas create actual massive objects or in today's parlance neurogenesis, coupled with his work on shockwave propagation. It is the case that a spiral form of pathway along a horn results in an acceleration of a shockwave dynamic up to a singularity. This is what I have attributed to how society passes in a nonlinear fashion from one technological mode of functioning to an ever higher manifold of functioning.

A Search for the Higher Dimensional Non Linear Manifold or Aesthetic Intelligibility

The tool that Bernhard Riemann adopted of n+1 multiply connected manifold affording an instance of a kind of phase space degree of freedom is really but a mathematical metaphor for the underlying reality of ongoing non linear development of the universe. It is therefore quite disheartening to see the gruesome distortion that the Royal Society's so-called M Theory makes of supposed extra dimensional strings. These folks seem to have gone down the proverbial rabbit hole of Lewis Carroll and never come back. The very same Lewis Carroll who was in reality as Charles Dodgson a dedicated hater of the revolution that Riemann and collaborators created.

Today I came across a remarkable work of mathematical biophysics that for most laymen would seem utterly incomprehensible. It is the development of method for mapping the non linear molecular motion of protein "metadynamics in the conformational space nonlinearly dimensionally reduced by Isomap."

However it is in essence no more mysterious than Brunelleschi's building of Il Doumo in Florence or his revolutionary development of perspectiva in the fifteenth century.

I have for some time now on this blog remarked that the quality of multi-dimensional connectedness of biophysics is uniquely suited for a treatment by Riemannian methods. The manner in which biophysical processes known as proteins interact on the level of the "proteome" wonderfully illustrates this. It has further been my keen interest in certain anomalous types of phases of living, non living, and intellectual substance that has led me to share new examples in these pages with you. The model of the way in which in Renaissance emerges from a dark age in the history of humanity I have likened to a shockwave that is meta-stable in acting upon economy and culture. It is analogous to a long lived soliton in its power to transform the practice and thinking of humanity. The creation of the radically new and singular occurs as a culmination of a lawful development of preceding ideas. The closest we can come to the emotion associated with this is in the music of Beethoven, especially his Fidelio Overture No. 3 conducted by Furtwangler, late quartets and piano works.

I find the fact that the network of molecular motions illustrated from the referenced paper is in the form of a horn torii is lawful. This is precisely what I have hypothesized as a model for non linear development  in these pages. If this direction is synthesized with the class of higher least action Abelian physical functions, one of which called the catenary that the aforementioned Brunelleschi used to solve the construction, then humanity may once again create a Renaissance out of the current nightmarish abyss into which we seem to be hurtling like a somnambulist.

 Three-dimensional Isomap embeddings of trans,trans-1,2,4-trifluorocyclooctane. Each ball represents one of 8375 conformations of the molecule. Each conformation was connected to ten nearest neighbors (measured as RMSD) to form a graph, which is shown as sticks. Threedimensional Isomap embeddings were obtained by multidimensional scaling of pairwise geodesic distances between points measured as the shortest paths on the graph. Isomap embedings were rotated by angle (0.00, 2.40, –0.55 rad) for better clarity

Gustav Theodor Fechner

Recently, in examining Bernard Riemann's influences from his philosophical fragments, I came upon the work of Fechner. Today, Fechner is almost exclusively remembered as the father of the field of "psychophysics." In particular, his logarithmic function for the threshold of conscious perceptions. However, Riemann was very much captivated by Fechner's writing on Zend Avesta. It is quite remarkable that Riemann's work on sonic shockwave propagation predated the actual demonstration of the said phenomena by some decades. The reason for this is, I believe that the principle involved actually transcends the narrow application that Riemann gave it in that instance, i.e. of pressure and the speed of sound . That is that given a wave being propagated beyond the speed at which the medium is capable of accepting a shockwave aggregates perforce. As this continues over time the shockwave can be thought of as a negatively curved horn surface. This idea immediately links to Riemann's habilitation thesis wherein he details the non Euclidian geometries as the basis of choice for metric for physical space. It is the moving over from positively curved propagation to a negatively curved time series that is there in germ form in that thesis of Riemann. And I believe that likewise Fechner in the Zend Avesta evinces the germ of autowave phenomenology when he states (in English translation from the German): "Our future spheres of existence, though all incorporated in the same great body, the earth, will not disturb, confuse or efface each other. Even here our spheres of existence necessarily cross and intersect each other, as the means of mutual intercourse, which in the here after will only increase in intimacy, variety and consciousness; and in our brain the material changes connected with our reminiscences cross and intersect each other, leaving them nevertheless undisturbed and uneffaced." (my emphasis added) I believe that this is a remarkable instance of how a method for viewing interconnectedness of physical and psychological phenomena leads to a correct hypothesis of neurological characteristics before any experimental apparatus is available.

Motivation for a New Direction in Mathematical Modeling

(DNA Topology as a Dual Function of Homotopy Induced Solitonic and Discrete Resonance)

This quality of duality cannot be resolved merely by assuming a stochastic frame of reference for the unique problem inherent in phase space transformations. This is as true for the field quantum physics; and in general self organization theory that induces the formation of discrete attractor and emergent states from any “arbitrary” random preconditions. The toroidal dynamics (hence torsion, chirality, asymmetry, as well as orientation) of n dimensional Riemannian space as it undergoes homotopy to Costa Surface of n degree embedded in such space, is the equivalence of subsumption of transfinite ordinals extended outside the simple number theoretical domain of the “continuum.” The Vernadskyian definition of a modular continuum, delimited by irreducible boundary conditions into interoperating domains, i.e., the inert, biotic, and noosphere conforms to this conception. Further, the Hopf positively curved minimal surface’s connection with Costa in the sense of phase transformation allows for the orientability via Cartan Repere Mobile construction of Kiehn to evolve naturally, rather than stochastically. (It is of no small note that Vernadsky references Cartan’s geometry within his text explicating the Noosphere.) The model for the positively curved space wave function transformation to the hyperbolic horn (Mach cone) is Riemann’s paper on shock wave propagation. Toroidal action is a metastable configuration of Alfven’s plasma cosmogony in the macrocosmic scale and is invariant down to the infinitesimal. In this context, the metastable formation of solitons of determinate orderings is properly located. The connection these have with catalysis, DNA replication, and far from equilibrium systems in general is instructive. (See Davia) The fact that autowave dynamics are differentia specifica in the biotic domain as in contrast with the inert is not negligible and is suggestive of a further refinement of possible wave dynamics within the higher transfinite dimension of the noosphere. The application of Hausdorff fractional dimension and its fractal sequelae does not conform with my use of term transfinite dimension in the foregoing. However, it does properly situate the question of induced singularities from a continuous manifold.

Neurogenesis and conformality resonance

It is proven that an enriched environment is directly related to adult neurogenesis in certain cortical regions. As classical music is the semi-mediated expression of human creativity, the case for sonic resonance in this instance is not a trivial topological problem. Again, the connection at the cosmological scale of star formation and topological invariance is intriguing. This, if you will, is the true locus of “connectionism.” It is decidedly not the theory of “encoding” prevalent in academia today. (As a matter of fact the craze of trying to undo the relevance of Goedel’s proof among logicians and information theorists has led to a rather degenerate embracing of non-reality as substance. This has its approximate correlative in the attempt at ergodic modeling and distribution theory as a replacement for causality.) In this light, it is intriguing that current studies show that the orientation of the plane of cleavage in neurogenesis that regulates brain tissue growth. The Gbetagamma subunits of heterotrimeric G proteins are likely causal elements. Additionally, the vortical-like chiral nature of the enzyme ATP synthase is well understood. It is further evidence of a topological refinement for the biological equivalent of the model of toroidal torsion. It is also astounding that this enzyme exhibits nearly 100% mechanical chemical efficiency as an engine that makes possible the cellular consumption of free energy for the biosphere, including both aerobic and anaerobic types. (Also, separation in these two domains is defined as counter- or clockwise rotation on the mitochondria, exclusively (!)) All of which further reflects on the need for a model that can account for the formation of solitonic singularities within this transfinite hierarchical domain. Another qualifying refinement of in the arena of bioenergetics is that the quantum mechanical “laws” governing regimes of possible spin topologies are violated by biophysical macromolecular states.

Dynamics of Human Thought

Media of transmission for human thought are certainly of the most complex and excitable types imaginable. The first question to be addressed is that of the quality of hylozoic monism of this subject matter. We can view many different aspects or dimensions of human thinking and their impact upon various portions or manifolds of relationships of the continuum of reality. (This notion of continuum is not restricted to the idea of linear extension and types of number, as will be developed further on.) However, it is essential to comprehend that there exists a world line of development whereby ideas are characterized as ordered in the sense of Cantor’s transfinite. In this sense, the entirety of human history of advancement of thought and praxis is invariantly self similar. (By the way, in this regard the principle of fractal generation is simply a convenient mathematical device for producing discreteness out of such self similarity. It is not the generating principle but merely a relic of that generation, in reality the generation of singularities rarely proceeds by such a uniformly simple manner.)
For humanity, it is the ability of scientific discovery to overcome the relatively fixed limits or boundary conditions of any particular mode of functioning of society from the standpoint of thermodynamic disequilibrium, i.e. negative entropy. By this, it must be understood that due to the necessary nature of directedness or hylozoism, if you will, one mode must be superceded by another. This quality of the vectoring of society is not merely accidental, but is built into the very ordering of the inorganic and biological substrates. For example, the fact that there exists a birefringent axis or anisotropy for electromagnetic propagation in the universe, which at the most macroscopic regime exhibits precisely this quality. Similarly, the anti-racemic and optical chirality distinguishing the dynamics of living systems is a further higher ordered transfinite development of this same anisotropy that characterizes the inorganic. Electromagnetic propagation is now ordered once again according to a new lawfulness. Now, human thought or what Vernadsky labeled the Noosphere, is of a yet further transfinite such ordering distinguished by its own particular power over the lower two subsumed realms. It is only here, that an awareness for consciousness of these ordering principles has any meaning.

As to the question of interaction of wave functions and the excitable media: firstly the approach of Riemann in his groundbreaking paper on the principle of the dynamics of propagation of sound and pressure in a physical medium. In this, he established the principle that the medium itself possesses a boundary of rate of transmissibility. The resulting shockwave formation induces a change in the quality of curvature of wave propagation. (For astrobiology compare galactic ram pressure.)The so-called Mach cone, taken as an accelerated series as the speed of sound (pressure) continuously surpassed, represents in its entirety a negatively curved hyperbolic horn. That is, what was a propagation of a radiative spherical kind now has gone over through a shockwave formation into a different species of curvature. In the instance of the propagation of electromagnetic at the speed of light, the medium is not able to rectify itself as for sound and instead, a transformation of a different sort occurs via Einstein’s equation. This being a different transfinite species of boundary.

This same principle certainly applies to the two remaining substrates, i.e. the biosphere and the noosphere. Mutagenic transformations, I contend, are nothing but the same shockwave principle applied to the biological realm. When an organization of life develops an ability to reduce energy through a new quality hitherto unknown, this is nothing but the surpassing of the previously fixed rate of transmissibility of the medium. In the same way that the speed of light represents a higher kind of boundary condition for the inorganic, the boundary separating humanity from all other biology is a sort of speed limit, if you will, for mutagenic transformations. This again, of course, is of a higher ordered transfinite than Einstein’s principle was for the inorganic.

Now, you are probably asking yourself, how might this same shockwave principle apply to human thought? In any mode of functioning of society there exists an active “media/manifold” of a set of relationships created by the successive discovery of principles of underlying organization of the three substrates, i.e. inert, biota, and human cognition and interrelationships among these (see for example the role of Deinococcus radiodurans). The human individual’s cognitive development occurs, of course in an historically concrete/discrete context. The previous associated development of society has made the medium of cognition transparent for the mental development of this individual. For example as a thought experiment, consider the following: (along the same line of dialectic of mnemonics from Plato’s Meno) how is it possible that a gifted young child in the modern era is able to seemingly intuitively grasp the principles of projective geometry and readily apply them in drawing? View the historical artifacts of the lengthy and arduous development of these principles; compare Massacio’s nearly completed perspectival artwork with that of DaVinci. So, I posit that the substrate of the mind is also a developing organic “compact” type of n + m dimensional manifold, a multi-sheeted hyper-surface, connected through topological singularities, (unfortunately sometimes referred to analytically as defects) and exhibiting negative and positive regions of curvature. Now, crucial scientific discoveries necessarily create for human practice free energy that allows society to overcome the bounding conditions of this thermodynamically relatively fixed reactive medium. This signifies the movement from n dimensions to n + m (topological cobordism.) Further, the geometrical transformation of such a succession, I posit, also acts upon the principle of least actional, minimal surface, topological constraints. For example, the latest in a centuries’ long mathematical development in this arena is the remarkable discovery of the Brazilian geometer Celsoe Costa of a principle of generation of such surface by way of homotopic deformation of toroidal types. Here, from this perspective, are properly situated considerations of self similar, relatively “force free” species of action, found in the subsumed substrates we have been considering, i.e. solitons and cellular automata or autowaves. Such species of hypertori and their deformations are intimately connected with what has come before as the geometrical analysis situs of instantiations of “compact” manifolds as a substrate capable of resolving and emerging whole the principle of shockwave transformations. Topological singularities that are either discovered in nature, e.g. Falaco, morning glories, hurricanes, spiral nebula, Davidov, aorta autowaves etc. or produced in the laboratory under extreme circumstances, e.g. plasma kink instabilities, light bullets, BEC breathers, sonoluminescence etc. are representative of anomalous conditions, which when their principles of organization are made explicit through scientific investigation act to transform the domain of human practice and create new thermodynamic boundary conditions.

Such qualities of human thinking themselves may be cognized as having a wavelike nature in terms of propagation through the medium of human practice. Consider for example, how the historical personage of Plato was directly and mediately responsible for sparking off a methodological renaissance centuries after his death. His opus may be viewed as propagating itself historically in two senses from the standpoint of curved space. In his own epoch, his attempts to win over the Syracusan tyrant to his system, create an academy, etc. represent a radiative propagation of his method analogical to the previously instanced propagation of plane waves below the limits that the medium is capable of supporting. Why? Primarily because his efforts failed to transform his own society. That is to say, even though the ideas were effectively transmitted in the medium of language and otherwise of his day, the power (potential energy) inhering in them failed to catalyze the desired transformation of society. However, centuries later Plato’s class of ideas were able to transform the practice of society very much in principle as a shockwave. So then, human thought, although it is insubstantial from the perspective of spatial extension, except as it adheres in some way to the nervous system, cortical manifold is able to act in a particular sense to transform the geometry of human practice. In the same oriented/vectored characteristic of action that birefringence has for electromagnetic inert propagation, chiral or helical enantiomorphisms for the biotic, thought has but on a transfinitely higher ordering for human practice.