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Intro to Alchemy, with comments on Leonardo
Historical credits. Individuals: Pythagoras, Musashi, Böhme, Kepler, Galileo, Leibniz, Newton, Jung, de Broglie, von Neumann, Dirac, Rife
For 2,600 years Pythagoras defines Western in Western civilization. He built mathematical and physical sciences from scratch relying on numbers and adherence to high symmetry. Pythagoras sought rules for music and by applying particular ratios established the scale of the musical octave (ratios of notes making do, re, me, ..) that is unique to the West. Some ratios were found harmonious and a claim made that planetary orbits subscribe to harmonious ratios. Some reports speak of Pythagoras applying harmony to healing. Pythagoreans' admiration of rationing resurfaced at the Rennaissance as the 'Harmony of the heavenly spheres.' Planetary orbits were measured and orbit ratios found to correspond to the notes on the musical scale.
Through the geometric relationship of numbers Pythagoras (or Pythagoreans) discovered the irrational numbers, which are constructible exactly only through the Pythagorean Theorem and are infinite in length. That is, every irrational number has infinite precision. Being infinite and issuing from geometry, Pythagoreans described irrationals as the "unspeakables," for it is not possible to describe irrationals with words. (Today, a better translation would be 'nonverbal.') The five pointed pentagram star became the Pythagorean standard as it embodies preponderance of the golden proportions -- the golden proportion being made with the irrational number of the square root of five. Pythagoreans discovered the first five perfect geometric 3D solids having the highest possible symmetry, which were the starting point for Kepler some 2000 years later. Plato was the first to actually publish these five solids and so they are at times called the Platonic solids. Pythagoras is the author of the words philosopher (a lover of knowledge) and cosmos (the order one finds in the universe of stars).
The central symbol of Pythagoras is Tetractys, which expresses the discrete steps of dimensions in geometry (0D, 1D, 2D, 3D). Tetractys also embodies the harmonious notes of vibrating musical strings having 2:1, 3:2, and 4:3 ratios -- which are all harmonious. The Pythagorean School he founded in southern Italy taught 'All is number.' While the secrecy of the school's findings makes it difficult to associate Pythagoras with particular innovations, it appears Pythagoras intended the school to operate as one whole: The intellectual property at the times could only be enforced by the school. Scholars directly attribute to Pythagoras the ideas that are "central" to the school's teaching. Pythagoreans classified numbers not only as even and odd, but also as evenly even (binary), incommensurable (irrational), incomposite (prime), and abundant (divisible by many other numbers such is the case with 360 and 24). The application of numbers also includes the idea of priority (as an unchanging characteristic, though, and that is adequate for physics but not for economics). Attributed to Pythagoras is the right angle theorem, which in its geometric form is valid for both the rational and irrational numbers -- and is the only tractable generator of irrational numbers. (Transcendentals are not constructible via the Pythagorean Theorem -- cannot say if circle squaring was resolved by Pythagoras in some fashion. There are hints on squaring by ancient Greeks.) The Pythagorean Theorem spawned the imperative pursuit of geometry lasting over 23 hundred years, until effectively lowered in priority by Kepler's orbit equations. Still, the incommensurability aspect of numbers is understood at but a gut level.
Well understood but not much publicized is the Pythagorean pursuit of salvation; by understanding the numbers and their operations, one could earn and gain admittance into the universe.
Pythagorean even-odd transformation and invariance rules among numbers led to the mathematical group theory and the discovery of symmetry invariance of the atomic even and odd functions. (This may seem a stretch but the similarities are there as Pythagoreans were the only ones working the even and the odd aspects of numbers. Pythagorean classification of some numbers as evenly-odd has tantalizing links for even-odd symmetry in matter [advanced].) Johann Balmer's integer based formula governing discrete -- that is quantized, wavelengths of light issuing from hydrogen opened atom's orbitals and the whole of quantum mechanics to the Pythagorean tradition. In the future, Pythagorean concepts will also become applicable to the pyramid geometry and to the creation/realization of the unique group of incommensurable numbers -- the transcendentals. (Also see Leibniz, below.) The Pythagorean School admitted women -- something Plato and Aristotle would not follow. Pythagoras considered Tetractys the symbol of the universal order and, based on his comprehension of dimensions of independence, he likely understood more about the universe than is known today. The meaning of Tetractys, as applicable to dimensional independence, was not picked up by Euclid and is still not understood.
Tetractys is extended and applied by this site through HyperStates. One Pythagorean application is on the golden proportion page while in the free energy article we comment on the Pythagorean discovery and the excitement over irrational numbers.
In the rendition above, Pythagoras is shown on the left reading from a book on music in a detail from a painting by Raphael that could be called "Who is Who in Magna Graecia." A student holds a black tablet in front, which shows (on right) the harmonic proportions as well as Tetractys. Pythagoras is reported to wear trousers, an unusual attire for his times.
Samurai of feudal Japan of the late 16th century, Musashi maps military strategy into the five ancient elements of Ground (earth), Water, Fire, Wind (air, culture), and Void (aether or quantum vacuum). Balancing his desires of independence and loyalty, Musashi was feared and admired. He speaks of spirit as encompassing all that can be learned and put to – in his case military, art, and agriculture – practice
In his drawings around 1620, Böhme shows, via back-to-back semicircles, the separation and a point contact between the real and the virtual domains, as well as some primary charkas and a dantien. Being a pious man, he fits it in a context of the church
The original SOW (son-of-a-witch), Johan (or Johannes) Kepler takes time from making a decent living doing horoscopes to successfully defend his 70 year old mother on charges of witchcraft. From the field data of (de) Brahe, Kepler derives two-body gravitational orbit equations that have general solutions. The mathematical solution of planetary orbits allows the application of the parameter of time to make orbit predictions. His use of a mathematical equation is the first departure from the 'angelical knowledge and power keeping planets going,' which, presumably, was out of man's purview. His equation was first applied by Newton who calculated the orbit of a certain comet, which allowed Halley to make a successful prediction of this comet's return. [Kepler's calculation of the volume of a beer barrel should be the textbook example on science-industry cooperation!] Even small satellites being launched into the Earth's orbit comply with Kepler's equation. He was the first to propose that the gravitational force is a mutual force between material bodies, including bodies of water, which led to his action-at-distance moon-causes-tides proposition. Kepler describes obtaining the orbit equation as his "War on Mars." As a mathematician, Kepler first noticed that the ratio of consequent Fibonacci numbers converge toward the golden ratio. Following the appearance of a telescope, Kepler places optics in the geometric-mathematical context. Kepler is the author of the words satellite and focus, and his optical tracing method continues to be in use.
We have book review of Kepler's Witch. In the book, the intricacies of the Rudolf II Court include the year long negotiations between Kepler, the Court, and Brahe's family for the Kepler's use of Brahe's field data. Similar protracted negotiations dealt with Kepler's royalties when publishing his findings. A particular letter Kepler wrote shows him in a unique light. At the times, the Counter-Reformation executives were actually billing Kepler for his mother's jail expenses. In the letter to the jailers of his mother Kepler points out that since his seventy year old mother was held in chains it would not be reasonable to pay for two guards keeping an eye on her.
There are two aspects that in retrospect are the most relevant, particularly as his work happened during the Counter-Reformation and contributed to the end of the Dark Ages. Galileo Galilei is, first and foremost, the prototype entrepreneur who applied discoveries in his own business. He improved and promoted the telescope to the "civilian" authorities and, while cognizant of the need for a sponsorship he leveraged the technology of the press to publish his findings. Secondly, Galileo directly challenged -- consciously or subconsciously -- the Aristotelian credo 'Nature abhors a vacuum.' Galileo worked with a vacuum in the derivation of the (cannonball's) parabolic -- that is suborbital, trajectory, and also postulated the conservation of momentum. (It was then 100 years from the fall of Constantinople where Islamic gunnery were the principal breach weapon.) Accused of heresy at seventy by the Church and unable to squash the subpoena, Galileo travels to Rome. Because the accusation at the times is equivalent to a conviction -- the "suspicion of heresy" was really the finding -- Galileo repents and draws the minimum sentence; Galileo spends the rest of his days under house arrest.
It seems there exists no complexity Gottfried Leibniz (or Leibnitz) could not structure. Leibniz did not mind trying to make sense of the most abstract ideas one finds in metaphysics such as the monad. His discovery of the integral calculus was deeply connected to his love for a continuum, which he continued to hold on to as he delved into the ever smaller world of the infinitesimal. His clarity of presentation served him well as a diplomat-for-hire and as the inventor of the integral-differential math notation that became preferable to "Newton's own." Leibniz can well be a model of an ideal professional who continued furious and prolific career of discovery in face of much criticism while managing to acquire a considerable fortune. He also stayed close to physics applications by formulating and differentiating the scientific unit of 'momentum' and 'energy' of moving bodies. Leibniz was first to realize that the direction, in addition to energy, is being conserved -- today summarily referred to as the 'vector' law. There is one unique and very cool word Leibniz coined: the Transcendental number. He defined the Transcendental number and also came up with a numerical series (Pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 . .) that converges to the most famous of all Transcendentals, the Pi.
As much as one could make him a "mad hatter" or "The Moon is falling!" kind of guy, Isaac Newton is the archetypical alchemist, identifying with the volatility of new issues he pursued. Being a man of science, Newton fits gravity and light in the real, scientific context. He formulates the gravitational force equation -- and confirms Kepler's orbit equations -- when he places force vs. distance in the inverse square relationship. Newton's physics includes the existence of absolute spatial distance and absolute time. (An object performing discontinuous movement does not violate absolute space and time. The 'absolute' variable refers to the multitude of independent observers obtaining the same value when measuring such variable.) Considering the absolute speed light has in a vacuum (in ether), as well as the latest discoveries in electron's dual slit (self)superposition that requires the absolute reference if the math is to match reality, Newton is due for another outstanding insight. Through his review of fringe (superposition) rings, Newton was first to measure the wavelength of light (1/80047 of an inch for the blue light; the node -- that is, absence, of the blue light makes yellow). Newton calculated the air gap and determined the wavelength based on light's standing wave -- that is, he calculated the wavelength without needing to know lightspeed. Newton is the author of the concept of mass inertia (vis inertiae) as a dynamic characteristic of mass, and his particular design of a reflecting telescope is scalable so well many models continue to be built.
From archetypes to synchronicity to introvert/extravert, Jung practices balancing of mental objects' associations as a way to mental health. Visualize. His description of inferior systems can be worked – in the context of this web site – as 'dependent variables capable of becoming independent.' This meshes with the Pythagorean notion of priority.
'If wave has particle properties, then particle has wave properties.' Simple. The hyperbolic reversal holds. This guy is not done yet, for he needs to generalize his wavelength equation to include virtual (complex) numbers and angular momentum in addition to his linear momentum formulation. The mechanism of momentum conservation and the non-linear inertia at high speed are to be found in de Broglie's generalized momentum-to-wave (frequency) equation. Still vaguely recognized is the advantage of reversibility that is inherent in de Broglie equation. De Broglie wave-momentum relationship (not wave-particle -- see primer on QM) is also the foundation of the proof that absolute spatial distance and absolute frame of reference can be derived and applied -- if math is to describe the reality of the experimental observation (dual slit).
Moreover, this site makes a case that a wave can have particle-like properties only at the wave's transformation, for only at transformation (at absorption) can the photon's energy manifest as real energy.
Works the quantum mechanical concepts to the conclusion that the collapse of the wavefunction is instantaneous. What appears to be a curiosity of the atomic environment, the instantaneous nature of the wavefunction reduction will turn out to be the most radical and the farthest reaching milestone of quantum mechanics -- once the quantum mechanical concepts are generally recognized as operating on any scale.
Von Neumann also contributed to computer development via the implementation of the 'program-store' where the program shares memory with data. While his support for Turing that the 'Turing machine is a universal machine' is freely quoted, such claim is based on intractability because it calls for unlimited time. (Scientists like to hop on intractability in cases where they do not have workable solutions of their own.) In time, von Neumann should be untangled from intractability, for his instant quantum reduction idea is just the answer for many intractable problems.
One of the quantum mechanical guys, Dirac quietly postulates the existence of matter-antimatter pairs. Duality of the real (formal) and the virtual (informal) begins to get mathematical shape. He is the first to introduce transformation operation into the behavior of the electron and simplifies the understanding of the electron's dual nature.
Rife
In the most remarkable confluence of five disciplines, Raymond Rife engages optics, physics, mechanical engineering, microbiology, and electronics to attack degenerative diseases. He was the first to observe live viruses, their vulnerability, and their ability to change their benign-cancerous modality. Because governments and Nobel prize committees cannot recognize discoveries that empower individuals, Rife's recognition will come only if an infrastructure is put in place that would capitalize on his accomplishments — an unlikely event because his results are complete and the delivery inexpensive. Rife had much difficulty raising funds and Timken Bearings company stands out as the most remarkable benefactor as well. Presently one must seek this technology to avail its powerful benefits and perhaps that is how it should be. A case can be made that Rife's discovery of UV resonance (point source resonance in the ultra violet range) lives on in a much more expensive nuclear magnetic resonance (NMR) imaging -- but with Rife's cure component missing.
In an interesting 2007 development, a vaccine is being marketed against a cancer-causing virus of the cervix. The vaccine emulates the virus only geometrically, however. While no credit or a reference is given to Rife, if you become familiar with Raymond's work you will recognize that his cancer cure includes -- in addition to actually destroying the virus -- the effort of keeping the virus in its bening state. Flipping the virus into its benign state or keeping the virus from morphing could well be the mechanism behind the new vaccine (presently not claimed as a cure). [To some it is apparent that a vaccine brings in more money than a cure.] We give Raymond credit inside our article on the squaring of a circle.
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And the guys who get furiously busy when there is time to move to a new level. And the guys who almost die getting there
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Alchemy
With minor exceptions, alchemy is found worldwide. It is enticing to call alchemy the predecessor to chemistry, metallurgy and medicine -- and well it may have been. Yet, a better way of saying it is that alchemy spawned these and other pursuits while alchemy continues to have a life of its own. Alchemy's utility is in its non-causal, associative qualities. When alchemy talks about transforming led to gold, the alchemist is also interested in the change from being "led," (being dependent, a follower, heavy, soft, or "sinker") to becoming gold — a leader with non-corruptible qualities.
Alchemy never has a single message; more supporting nuances there are the better. The art of alchemy, for example, is to associate the donkey and the ass, all puns intended. Alchemy in modern times is closer to public relations, marketing and advertising than it is to hard sciences. Today's alchemy can well be seen as brain storming -- for the "think outside the box" proposition is very much the alchemical way of doing business -- that is, find innovative solutions outside of the established context. The power of alchemy is in ferreting out and marshalling forces no one thought were there -- as when an unusual political campaign strategy suddenly works.
Traditionally, alchemy sets the most significant and the most difficult goals. For example, the creation of life itself is one of alchemy's goals. You could wake up one day and say, "How would I organize the spherical galaxy?" Now, that's alchemy. Alchemy is discredited along two arguments: (1) It is all too theoretical or "iffy," and (2) It is impossible.
Alchemy, almost by definition, deals with things most people would consider impossible and, therefore, alchemy's outcome is a solution and never a proof of impossibility. In the geometry of the compass and straightedge, the alchemist could deliver a formula yielding all angles that are divisible by three -- whereas the scientist delivers the proof that angles in general are not divisible by three. The alchemist works and delivers the proof that absolute spatial distance can and must be determined -- if de Broglie's wavelength is to find broader applications.
The traveling salesman problem is that of a non-polynomial class and is akin to code-breaking. While (1) we know that the solution exists and (2) we know the length of time it will take to obtain a solution, the alchemist cannot recommend a solution that calls for intractable processes. Because current processes to obtain a solution to non-polynomial problems are indeed intractable, such "solution" takes so much time it is for all practical purposes unusable and cannot be applied. If an alchemist is to figure out a solution that can be applied, he or she must come up with a tractable solution. The alchemist cannot and does not use intractable methods and does not recommend intractable solutions, for intractability is not a solution. Scientists, on the other hand, wallow in intractability and every time you hear how many millions of years it takes to get to another solar system -- that's the scientific mindset of the left brain talking -- talking about things that are not only impractical but that ignore the simple fact that the scientists' solutions are not usable. In making things worse, scientists are quick to claim that their methods are the only available methods and 'if we cannot get there from here, they (or anybody) cannot get here from there.' Not so. Tractable methods are available and intractable methods should always be filed in the dead end category. One can surely spend a lot of money pursuing a ball lightning research at a university, but -- by accepting intractable methods -- it just will never be understood there. Alchemically, difficult pursuits are not addressed with 'if-then' methods, for such methods are used by the left brain (and by the present computer) and the 'if-then' method is intractable when solving non-polynomial problems. In fact, the way to tell the alchemist apart from a scientist is that the alchemist does not use 'if' much but a scientist uses it often. (Can you call yourself a leader if you require something else to happen first?)
A good example on mathematical and thinking tools goes back to Euclid and to his proof that irrational numbers cannot be created by rationing of real numbers. The unavailability of a rationing method, however, cannot stop an alchemist from dealing with irrationals! You and I can construct the square root of two using the Pythagorean Theorem in short order. So what if it cannot be done algebraically by rationing -- it can be done geometrically. The idea is not to dwell on ways and means that show something to be impossible, for that is but one way that happens to be a dud. Euclid, it seems, is treating incommensurables as a dead end (at worst) or a loose end (possibly) or in the context of geometric superiority over arithmetic [I'd like that]. Transcendentals, being a unique subset of incommensurables, do present a challenge because the infinite addition mechanism needs to be found (think QM superposition), but the scientist would like to have you throw the boat away because he cannot move the boat with the paddle he has. Here, then, is where the insight comes in. (Self-test:-) If you think numbers are no big deal then you just walk away. If you think numbers are not the fundamental root, you will never think the stone can be shaped without hammer or heat.
Western alchemy is the only one that recognizes and speaks about the mystery of 'Three versus Four.'
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The couplex is the point of separation and the point of joining between Three and Four. Leonardo combines Three (Vitruvius Man legs together) and Four (legs apart) inside one circle but Three and Four cannot be combined within one context -- and the circle is the delimiter of context. Leonardo also takes the navel as the center of such circle, which is okay (it is a point). Leonardo thus combines '3 vs. 4' within the human body as a whole while superposing it with 'square vs. circle.' This is "okay" and reflects Leonardo's (or Vitruvius') intense search for mapping of cosmic geometries onto man. The '3 vs. 4' is the original Western alchemical mystery, which is different from the 'square vs. circle' problem. '3 vs. 4' deals with 'real vs. virtual' or 'tangible vs. intangible' or 'matter vs. spirit,' while the 'square vs. circle' deals with linear (straight, Euclidean) and curving (transcendental, Riemann) geometries that are applicable at the atomic level and, also, for the intercept of cosmic energies. '3 vs. 4', however, cannot be combined within one circle. '3 vs. 4' and 'square vs. circle' cannot be equated or mutually "mapped" but they both exist side by side and they are both existential to man. In a way of a comment, Leonardo -- or those who come after him -- need to separate '3' and '4' and then the squaring of the circle is (should be shown as) happening in both the upper (Four) and the lower (Three) individual components of the human body. Leonardo makes a significant step above Aristotle while illuminating the mysteries of numbers and geometry in the artistic (right brain) context rather than in the written (and exhaustive) textual modality. At times the power of dominating ideas allows the truth to come through even though some technical aspects might not be just right.
The reflection of
cosmos in the human geometries is shown here {April,
2006}. The joining and separation is at the
couplex (at the navel) where both circles touch. Four is above, Three
is below: When it comes to the intercept of cosmic energies, then your thoughts, your inherent body geometries, and exercise (motion that happens in a particular geometric and mental framework) come to the forefront. The basic notion is that of the 'squaring of the circle.' Technically, incoming energies (not just light), have geometric associations and they need to be folded within the circle or sphere (geometry that is symmetrical about a point) if they are to "stick around." The word 'energy' is but a rudimentary expression -- for actual knowledge is contained within cosmic energies. The '3 vs. 4' is an issue with much depth and breadth. (In the East, for example, dantien generally stops with Yin and Yang.) The idea is that the human not only emulates the cosmos but through the shapes and geometries the human becomes the receiving structure for 'all there is.' The human (and some animals') brain has similar arrangement where the 3 vs. 4 separation is at the Corpus Callosum. The book Quantum Pythagoreans derives and names the couplex and, in a way of introduction, applies it to the transformative action between both halves of our brain. The 3 vs. 4 is a construct that is applicable to many concepts such as stability, organization, and creation. |
Cosmologists maintain that the utility of stargazing is in applying the cosmic discoveries down here on earth, and that is how 'as above, so below -- as below, so above' saying formed. Alchemists, however, understand (or want to understand) infinities, and the 'above' and 'below' stuff is about gods. The 'above' and 'below' goes well beyond "heaven" and "earth" -- it is a mystery of its own and Thoth is not even subtle about it.
Occultists maintain that some relationships are virtual and cannot be observed but that the relationship's existence can be visualized, computed, and possibly realized (manifested). Alchemists studying matter find the mathematical computability foundation of atoms compelling and invigorating so much the alchemists' credo may well become 'as inside, so above.'
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QUANTUM
PYTHAGOREANS In Quantum Pythagoreans the separation is followed by joining. The separation enables all individual elements to showcase their strengths while the joining creates systems -- or monads -- that continue to improve you and your universe. The power of the Pythagorean foundation highlights the opportunities of applying new energy sources, and building atomic structures from scratch. Some applications are presented in considerable detail while others are ideas waiting to happen. Much work is in the area of the couplex, for this point has been misunderstood in the West in the past. The book explains why it is a point -- that is, why couplex must have the geometric construct of a point. Couplex can also be found in the Great Pyramid. |
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