HyperFlight Square the circle Geometry and its magic Squaring (quadrature) of the circle divides a circle in an attempt of making the turns the same as straightaways Making the circle out of the square is about reversible transformations of the circular geometry (2D, circle, compass) and the straight line geometry (1D, straightedge). Yes, we can figure out the magic of Thoth The Pythagorean pursuit in squaring of the circle must be relevant and useful. Pythagoreans, then and now, apply math that describes the operations of nature. Squaring of the circle is about energy We put Euclid on the Pythagorean platform, too, for his fifth proposition is weak So you take the circumference of a circle and make a square having the periphery that's exactly the same. Is there more to this than an intellectual exercise? Keeping your brain in gear? Proving something others said is impossible? While it is commonly accepted that the squaring of the circle is not possible, there are ongoing attempts to work this problem. Indeed, every person should understand what is behind the squaring of the circle because the implications of the process are richer than a philosophical discussion, a belief, trivia, or the plethora of various approximate constructions. To put it with the flavor of the East -- It's not about the destination, it's about the journey. Even though we are told that the answer at the destination is 'no,' we want to make the trip. As it happens, we will come to a fork in the road, a Tau, which you will never find if you jump to conclusions. A moving object has energy (technically called momentum) that is determined by its speed, which also depends on the distance the object is traversing. Going on a straight path the moving energy of an object has some particular value. If the curved path were to have the same distance as the straight path, the moving object's energy would be conserved at all times. But what if the distance of a straight path cannot be the same as the distance of a curving path? The true question then comes up: Because the moving energy of an object is conserved, what happens to the object's energy if the path changes from being straight to curved? One way of seeing the 'straight' vs. 'curving' assignment is by taking a square and then turn the sides into flexible lines such that the circumference of the original square is the same as the circumference of the resulting circle -- or vice versa. This appears a simple thing to do and possibly in the recreational category, too. Applying the benefits of geometry -- and if its exactness brings something special to the table -- then geometry may well help us decide this question. In essence, if we can make the exact match between a circular path and a straight line path, then the moving energy is conserved at all times and we could not use the parameter of energy to tell us the difference on what is straight and what is curved. But the straight line of a square and the curved line of a circle or an ellipse cannot be matched in length. Every time the geometry of a movng object changes between the straight and the curving geometries, energy is released. The energy is released as force in the case of a gyro (realy force times distance) -- or the energy radiates away or into the object. After a short search on the Internet, the Ankh symbol above is nowhere to be found as being associated with the squaring of the circle. But it is. Ancient Egyptians practiced and likely understood magic as it is embedded in geometry and in natural life forms. (A feather, for example, is both.) We are going to give the ancient Egyptians a lot of credit for understanding their environment and for dealing with complex topics such as the eternal life and the soul. The square (straight, 1D) aspect and the curving (2/3D) aspect also deal with the intercept of cosmic energies and the Ankh symbol is about the human's fundamental geometric duality: the circle and the square -- the curving and the straight geometries. The rounded top, moreover, is not really a representative of a head or a body, but it is the round rib cage that expands during breathing. The air moving in and out (through your diagonal-passage nose, no less) is also undergoing a transformation between linear and circular/spherical geometries. It is nice to point out that Buddhists do practice and possibly understand this aspect of the "breath of life." Would you believe there is an alphabet based on the squaring of the circle? Yep, it's that important.		"Making a turn is not straightforward" The making of a circle is also about taking a step from one to two dimensions. There, you will find the friendly transcendental number Pi. The circle can be divided into exact and equal segments that allows you to construct various perfect stars and regular polygons. Because not all numbers can divide a circle exactly, there is a page devoted to perfect stars construction -- yes, it is also about geometry vs. arithmetic. Squaring of the circle is about the addition or subtraction of energy. Because this energy is not mechanical (kinetic), we are dealing with the virtual energy. Free energy page gets into that while the summary as well the roadmap offer additional classification. New book you will thoroughly enjoy QUANTUM PYTHAGOREANS More .. From ancient Egypt comes a story of magic. The guy, or shall we say a god, in charge of magic was then Thoth, the Ibis-headed. He was also into records keeping because he was there when it was time to weigh your soul and that is when the truth and how much you did for truth is important. A point can be made that 'truth' and 'thought' are the two components of 'Thoth.' When Osiris Jr. was fighting to get his kingdom back he fought Seth and both inflicted heavy damage on each other. Osiris Jr. lost an eye. His compatriots were able to come up with pieces of it but could not get all of it. Thoth took what was left of the eye and restored Osiris Jr. eye through magic. No additional details are available. A case can be made that the eye restoration story is about the making of a circle from finite number of components which, moreover, requires some magic. Thoth knows a lot about geometry and you can tell by the way he holds his arms. The beak of the bird Ibis has a nice curve to it, too, and so you will not find a compass sticking out of Thoth's pocket. You can have a field day finding geometric constructs on Thoth. Just a few: Hyperbolic cut on his skirt, diagonals across his chest and skirt, papyrus scroll as a cylinder, angles everywhere. Is he making ellipses with his fingers? Why did the Christian religion "drop the ball" and just kept the straight cross? After all, Coptic Christians in Egypt adopted the Ankh as their own symbol. Tough to say. A circle is quite difficult to understand and might be elusive -- and possibly painful -- even to the most ardent students of the Bible. The Ankh is also considered "the key" to the eternal life and two keys are on the Pope's coat of arms. For the rest of the people (read followers), however, the Beast of Reduction has made his first mark: "Off with their heads. We will tell them what to think!" To get the two for the price of one, the key would be in the hand of a priest who positions himself in the gateway to your salvation. It is quite amazing how actively the Catholic Church wants their followers to stay away from a circle. So much so the circle is to the Catholics as the cross is to the devil. Perhaps the circle is the escapement from the dogma of the Catholic Church.. And so it comes back from antiquity to the basic two questions: Can people govern themselves? and Is an individual responsible for his or her own salvation? If so, is your salvation a matter of learning or belief -- or perhaps learning and belief? Many interpretations of the Ankh center on sex and reproduction, particularly in connection with keywords such as life, the womb, and the phallus. Well, the reproductive mechanics are simple enough and it certainly would not explain why the ancient Egyptian gods run around clutching the Ankh. Reduced to the basic mechanics of sex, the horizontal bar on the Ankh could well stand in the way of doing it, too. The reproduction of species is an exceptionally complex affair. There is quite a functional difference between a womb and a vagina, for example, and all of the components need to be looked at with their geometric shape in mind. The function of the Couplex point is not understood at all but reproductively it is critical. (The Couplex is our English name for Chinese 'dantien' and Japanese 'hara' that is coined, explained, and applied in the Quantum Pythagoreans book.)
When squaring the circle, every pebble on the road is a component of the solution In two dimensions the exact squaring of the circle through geometry or arithmetic is not exact and so the squaring of the circle in 2D is not possible. Even the Pythagorean Theorem does not do it. The equivalence impossibility between 1D (square, straight line) and 2D (circle, curve) shows nicely there is hard separation between straight and curving geometries. In fact, the impossibility of the squaring of the circle proves that the spatial 'straight' vs. 'curved' definition is not by convention. Said another way, the straight and the curved are individually enforceable. But you also know that the separation is not insurmountable, for atoms with their curving orbitals and the straight-line photons manage to interact in some exact way because the conservation of energy holds. There then must exist a transformative agent or agency. A spinning gyro can move in a straight line just fine, but how does the mindless gyro know the path is curving? A gyro has an angular momentum and an introduction of another curving -- that is angular -- momentum by the curving path is readily differentiated from the linear momentum. So it might be fun to work this anyway, for the squaring of the circle could be possible with, say, 3D constructs (pyramid?, angular momentum is always in 3D) or the transformative agent is an additional step we will have to go through. The ancient Greek geometric constructions always call for a solution with finite number of steps. The idea is that solutions that have some useful applications are those that are tractable. The point is that you don't want to deal with intractable methods, for they deliver solutions in some far away time in the future and only approximately at that. You also do not buy the argument that it just might be okay to use intractable methods, for intractable methods would at some point yield an error that is very small. You do not succumb to the "small enough -- good enough" invitation, simply because you are thinking and working with energy. When applying intractable methods the result could be close all right, but because the energy is conserved you will not be able to explain what is happening with the energy's deficit or excess/surplus [and you may want to do better than Thoth, who chalks it up to magic]. When making a wooden round table you will finish in finite time and of course you are the practical guy who gets the job done. But now we are working with energy and that is an important context in its own right. [Yes, it is in your right brain.] You also feel that for the paths of a circle or a sphere the energies might be closely related to the atom because atomic orbitals have a curved and closed topology. Here is where it gets a bit Zen. The Zen part is that now we got ourselves into quite a mess. We think we need to use tractable methods, for we want to be done by the end of the day -- but tractable methods are not available. Intractable methods do not help us with energy -- but working with energy is very useful. The only option open to us is to work with infinities while finishing in finite time! And for that we will need all components to be available to us right at the beginning -- the full infinity of them. And then we will need instant action. [No need to be shy.] The way of seeing the square-the-circle issue is in the context of transforming the linear geometry into curving geometry and vice versa, which is about the separation and coexistence of the Euclidean and Riemann geometries. The separation of the two is in the Fifth Proposition of Euclid's Elements where two parallel lines take the straight-only path. Riemann, however, opens the extra dimension by allowing a curvature. While Euclid's Fifth works fine for 'parallel' as two lines in a plane (think light beams), Riemann sees a sphere with longitudinals that, with identical curvature, converge and meet at the poles at finite distance where they close upon themselves (think atomic orbitals in a closed 3D topology). From the Fifth Proposition separates yet another, a third, possibility and that is of a line winding ("snaking") around a straight line, for both of these lines are equidistant -- just as parallel lines are -- but do not meet each other and do not close on themselves (coaxial or helix topology -- that is, open 3D topology). (All three aspects are incorporated in the Quantum Pythagoreans book from the Pythagorean perspective -- that is, each aspect has its representative in nature that meshes with its unique geometry.) The Fifth Proposition of Euclid, it is said, was difficult to formulate for Euclid himself. This proposition defines parallel lines and their properties. Euclid's difficulties point to his (and others') weak understanding of the Pythagorean concepts, which explicitly differentiate one dot (point, 0D), two dots (line, 1D), three dots (plane, 2D), and four dots (volume, 3D) geometries. The increasing number of independent dimensions provide different contexts within geometry. On top of that, each of the zero, one, two, and three dimensional constructs are individually enforceable. If Euclid were to say that his 5th proposition is valid for parallel lines in 2D -- that is, in any plane, he would have been okay [this is what leads me to suspect that Euclid was a bit removed from the Pythagorean teaching]. Sometimes you can but oftentimes you cannot increase the dimensionality of the interacting geometry while holding your previous proposition valid -- and this is the basis of the group theory that deals with variant and invariant attributes. Variances are quite easy to see going from 1D to 2D, for example, but many people expect the 3D environment to be just a bunch of 2Ds stacked up like pancakes and essentially "the same." But there are many things around us that are naturally flat, including our solar system, and this is because tractable computing is easier in 2D. Carl "The Prince" Gauss knew there were deficiencies with the Fifth and encouraged Riemann to work this area. (It is a worthwhile exercise to see if the 4D environment is in some respects tractable and this is treated in the Quantum Pythagoreans book.) Another way of seeing the dominating nature of individually increasing 0, 1, 2, and 3 dimensions is that numbers rule. In any beginning there is a number! Even today, challenges to the Fifth are summed up as 'non-Euclidean.' These challenges are trying to "unify" geometry by claiming that 'if another [my] geometry holds then Euclid is wrong [and I am right].' Not so. Simply said, the only diff between 'Euclid's parallel' and 'non-Euclid parallel' is that Euclid is valid in 2D while the rest belongs to 3D. It turns out that the addition of the third dimension allows two new constructs for 'parallel:' one for the open 3D topology and one for the closed 3D topology. In other words, the squaring of the circle is also an issue for the cubing of the cylinder and the cubing of the sphere. In 3D we encounter the problems of (1) Straight-and-round topology of a cylinder, which is in open 3D; and (2) Always-curving topology of a sphere, which is in closed 3D.{1 Jan 2007} The spherical, closed 3D topology is successfully addressed by Riemann while the cylindrical, open 3D topology is qualified but on this site (also see Touching in a Point). There is a way to resolve the sqaring of a sphere as well as the squaring of the cylinder -- all in one swoop. It is closely linked to the interplay between the odd and the even symmetries. In the illustration above the approximate construction of Pi is that which is found in the Great Pyramid. (The angle alpha is on the golden proportion page with comments on the insides of the Great Pyramid.) While the Pi is approximate, you will need to figure out why the circle could be closed with it (think SQRT(5)). If you prefer the esoteric road, think "foam of Venus" to collect the infinities you need. Instant action will happen with QM superposition and that's doable, too. When working the Great Pyramid you may begin to appreciate how the Pi and the proportions of golden numbers are tied together. You could start thinking it is one or the other but after a while you just might see their relationship.		Applying Pythagorean methods to square the circle, the goal is to take a one-dimensional straight line and convert it unambiguously -- that is commensurably -- into a multitude of zero-dimensional points. Since a point is dimensionless and has no length, the conversion is not commensurable and not possible. In other words, you cannot take a point and measure distance with it. If you think this proof is simple, then perhaps that is how it should be. This proof is good for any curve such as an ellipse. However, this proof is a general proof. We cannot square a circle in general -- that is, a circle cannot be squared directly as a whole under all (arbitrary) conditions and under all circumstances, but that does not mean that a particular circular segment cannot be linearized. This is more fundamental than you might think. When something can be unambiguously mapped back and forth from one state to another we can use the equal sign when we figure out the relationship. A complete one-for-one mapping between but two states such as 'potential' and 'kinetic' energy however, is not what usually happens in nature. We can force the environment to yield an equation but at the expense of doing it in a closed system. So we impose a lots of environmental constraints to get a two-way (equation-based) relationship to hold but in the universe there are no such impositions taking place. Yet, the universe organizes nonetheless. If you wish to understand how things really happen, you will need to accept that two-way relationships are exceptions. The relationships are normally over a multi-state (triangular or greater) processes, particularly if a transformation between the real and the virtual domains take place, which means that the equal sign of algebra will not do. In our case, when we force to equate (use equal sign) between 'straight' and 'curving,' we will do so at the expense of tractability -- that is, the process requires an infinite number of steps and becomes intractable. If you come up with a question, "Why would the infinite number of steps be intractable and bad while the infinite number of additions (superpositions) would not?," you are doing well. The "steps" happen in the real domain and each step takes finite time -- and therefore the infinite number of steps takes infinite time. Superposition is instantaneous in the virtual domain. Also consider Euclid's definition of a point as 'that which has no parts.' Another way is through questions. (a) "If you remove a point from a line, will its length change?"; and (b) "If you remove a point from the inside of a line, will you cut the line?" [Just a couple of Western Koans from MI. Feb 2007] Note that the tasks of squaring the circle's periphery or the circle's area are equivalent. Proving or disproving one task is good for both topologies, for either topology is commensurable (proportional) to the other through a constant that is the radius. This issue is very similar to the difference between the momentum and energy of a moving object. First formulated by Leibniz, momentum is proportional to v (velocity) while energy is proportional to v2 (a square number). Momentum and the moving energy of an object are commensurable with each other through v. The implication here is that the conservation of energy holds for the conservation of the object's momentum as well as its energy. Just to make it a bit more complicated (but far more interesting), consider that the circle's circumference is, because of the Pi, a transcendental number. The straight distance of the square's sides can be either a rational or an irrational number because no straight distance has the value of a transcendental number. We cannot take the circumference of a circle that would result in a square (or a straight line distance) with a rational side (or a rational length), because the rational number has finite mantissa while the transcendental number has an infinite mantissa. Now the question becomes: Can a transcendental number equal an irrational number? We answer this as 'no,' because irrationals issue from the Pythagorean Theorem (or an algebraic relation from among polynomials of any degree) while the transcendentals do not. Are we there yet? Almost. What happens if we ratio transcendentals and irrationals? This just might be it. That guy Pythagoras and his ratios are at it again! Rationing is not only about rational numbers, you know. [Would you rather be a Mayan, for Mayans had no fractions?!] Pythagoreans established rationing in the framework of musical heavenly spheres and in the harmony of tones. Using lengths of various strings, their ratios (proportion) resulted in particular harmonious and disharmonious sounds. But what if the length (1D) of a string keeps shrinking and becomes a point (0D)? Think infinity. Better still, think nonlocality. Best yet, think electron. It then also appears that rationing is an ongoing operation in nature and this could also point to the Pythagoreans' admiration for rationing. (Scientists in general think that Pythagoreans admired rational numbers but that just comes from their reductionist thinking.) For a refresher on (in)commensurable numbers, take a quick tour. Some people may run into problems and instead of admitting they do not have all the answers, they work hard to prove there exist no answers. Henri Poincaré is the case in point. He did not defer to geometry and claimed that straight and curving is, well, subjective. If we were to take Henri seriously, a gyro would have never been invented and perhaps not even attempted. Mathematicians and scientists make their proofs from assumptions and.. ..garbage in.. .. Here might be a fertile ground for psychology. We could use a term for people who cannot figure things out and the way they compensate for it is by trying to dumb down the source. The science writer who is clueless about the ancient Egyptian fractions then describes ancient Egyptians as simple farmers, for example. Okay, Henri's copout happened a hundred years ago. One year ago, though, a new vaccine had gone through clinical trials and is presently being marketed. This vaccine emulates a known cancer-promoting virus (or a cancer modality-morphing virus if you are closer to Raymond Rife) of the cervix -- yet the virus emulation is based in the geometry of the virus. No weakened virus or chemical similarities are present -- it is strictly the geometric form of the virus that supposedly triggers the human antibody response. If you know something about Raymond Rife's work, you would know that the geometric representation of the virus' benign form keeps the virus from morphing. The virus' geometric likeness is in the form of a virtual pentagon -- that is, the pentagon is hollowed out into a sphere rather than sticking out as a five sided object. This means the virtual pentagon will attract waves (energies) in the golden proportion, for a pentagonal real object cannot form standing waves. [Could you go on a limb and say that brain functioning imbalance is due to poorly brain-formed geometric factors rather than the catch-all "chemical imbalance?" If our thought patterns have a geometric representation in the brain then this would make sense.] Very common among scientists is the presumption that there is but one way of doing things. Actually, Euclidean and Riemann topologies exist side by side and both can be individually enforced. Transcendental numbers are the gatekeepers between the two [and she-dragons would make fine ones at that]. In the Credits section you will find comments on Leonardo's always-famous Vitruvius man in the context of circle-square and the alchemical 3 vs. 4 Archimedes was the first to apply a method in the calculation of the transcendental number Pi, which allows the computation of Pi to result in a large quantity of decimal places. Pi is the circumference of any semicircle (half-circle). Archimedes' method is intractable because we need an infinite number of steps (moves) to reach Pi. Methods or formulas by Leibniz and Newton that calculate Pi are intractable as well, and so is any other method that obtains Pi through the infinite multiplication/division of suceeding components. (Infinite addition can be accomplished in finite time through superposition but you'll need to be in the virtual domain.) But there is always but .. (think golden proportion). The number of accurate decimal digits of Pi is not arbitrary because we will always run out of paper when trying to express Pi. Can we say that Pi is in some aspects inexpressible? Is such inexpressibility limited to real methods? While many books talk about Georg Riemann accomplishments, they are written mostly by mathematicians. (We reviewed three books on Riemann.) Mainstream physicists bring up Riemann's magnum opus only to make a point about "curved space." So far, nobody differentiated curved space from curved pathways. So far, nobody applied Riemann work to atomic closed topology orbitals and to internal atomic tractability. It is for this reason that the 0, 1, 2, and 3D geometries are not firmly anchored as discrete constructs for spatial behavior, which issue from the unique spatial tractability associated with each increasing degree of freedom.
Squaring of the circle is not strictly about the length equivalence of a circle and a square, for it is about reversible transformations between 1D-straight and 2D/3D-curved geometries. So, when you are working this you may also want to think about the exact division of a circle. You see, a section of a periphery of a circle (an arch) and its corresponding straight segment (cord) start and end at identical points. A particular curved section and the corresponding straight distance then also subtend the same angle. Such angle, moreover, could be an integer multiple of the whole circle. Multiplication or division by an integer or by a rational number is an operation that finishes in finite time, and that is the reason why the circle and the square could differ by a rational number while still addressing the squaring of the circle assignment. If the proportion resulted in a transcendental number we could not use multiplication or division and finish in finite time because we would be dealing with an infinite mantissa and infinite computing time. However, if the proportion resulted in an irrational number, we could use the geometric mean to multiply out the irrational number in finite time even though irrationals also have an infinite mantissa [this goes beyond cool]. Bringing the ancient Egyptian fractions into this helps, too, for the components of the harmonics series represent energy components. As a Pythagorean you want to think about the operation of multiplication and what physically arises from multiplication -- that is, how multiplication manifests in nature. Along the way you may encounter some universal constants, too. Now, the curving portion and a straight portion are in some (ir)rational relationship via the whole circle. Still, the paths (segment and cord) are not equal and the conservation of energy question comes up again. It's back to Zen but this time it is not that difficult. (Keep in mind that a photon cannot be cut in two -- a minor complication.) The application here continues to be the atom -- its stability, energy extraction, and possibly gravitation. If you are a math guy, think Dirichlet. Could we say that only some circles can be linearized -- and then in but a particular way? If you want to have more fun, think about placing two circles (read orbitals) that would have their straight segments in a mutual rational relationship. Yeah, you might need circumpositional numbers (that divide a circle exactly). Bring along the ancient Egyptian fractions, too, and you already know it's about the energy. Happy thinking! Happy workings!		About general proofs Proving something that is not possible to do in general might be easy but such proof is not very useful and is usually irrelevant, too. Proving that an angle is not, in general, divisible by three is pathetic because there are many angles that are. Because a whole circle (an angle of 360) is divisible by three exactly, we can now analyze and work the atomic orbitals with thirds, as well as with other circumpositional numbers. Scientists' proof is irrelevant when we talk about all possible standing waves forming integer-wavelength orbitals that can wrap around the nucleus. Similarly, three-body gravitational equations do not have a general solution and that means that three or more bodies in general behave chaotically (intractably). But we know there is a very specific solution the likes of our very own solar system that is a non-chaotic multibody system. All is quiet on the scientific front on this because -- while it is easy to prove that in general three or more mass bodies are not tractable and are thus chaotic -- the proof would also require an explanation on why our solar system does not subscribe to the scientists' proof. On top of that the scientists' explanation is empty because they just don't know -- likely they do not possess the intelligence -- about creating tractable N-body systems. The scientist is clueless about why and how the specific tractable solution prevails over the chaotic state which is the general and provable state. Some mathematicians can prove that quintic equations have no solution (polynomial equations with the power of five). However, the golden numbers a and b do present solutions to pentic equations provided the golden ratio a/b is not reduced into one number and a/b as well as b/a are allowed as solutions. This brings us to the squaring of a circle. The squaring of a circle is not possible to do in general such as when trying to square a whole circle, but .. .. there are environments and conditions where it is possible to square the circle by linearizing particular segments. The squaring of a circle is logically the very same thing as dividing an angle into thirds exactly. A whole circle and even a half circle (that is Pi) cannot be squared but some segments can. The first application deals with atomic stability. New opportunities make all not-possible-in-general proofs into little footnotes. Finally, a personal story. This is a bit technical but the idea is that there exists a much engraved and authoritative proof regarding sorting that claims there exists the absolute minimum of NLogN operations to complete a sort of N elements. This "proof" is not "wrong" but it makes an assumption. In my [by now patented] implementation of a sorting process the advantage is in realizing that a random string is never really random and chances are good that some substrings (groups of elements) are sorted already. Suddenly the iron clad proof is not a proof, for there exists a several percent chance of a sort completing before NLogN while the new lower bound on sort becomes N. As an example, even a perfectly random generated (shuffled) deck of cards will always contain some card runs that are increasing or decreasing. Moreover, everyday sorting deals with updates, which contain previously sorted strings. Topics such as this are way too technical and they are not in the Quantum Pythagoreans book. The idea, though, is that claims of a proof always rest on assumptions and you should feel free to question their relevance or applicability to what you want to do. QUANTUM PYTHAGOREANS Book by Mike Ivsin Pythagoreans use the knowledge of numbers with straight and curving geometries to construct stable -- that is organized -- systems. Atom building is the first real construction and the means of construction are also the means of maintenance and healing. Quantum Pythagoreans book applies the Great Pyramid geometry -- along with the ancient Egyptians fractions -- in the atomic creation context. Can we build the technology that deals with the infinite and apply it in the creation of real things? The Pythagorean way is the only road to reality. More ..
Go or select another topic from the gold post HyperFlight home Portal in new window  © 2007-8 Backbone Consultants Inc. Copyrights Information Last update April 3, 2008