I. Golden ratio yields several solutions when golden numbers are in geometric proportion II. Before drawing and making the Great Pyramid you want to understand commensurable and incommensurable numbers		"There is gold in them hills"
Golden ratio in the bank: Explaining the golden ratio by means of an application. Pythagoras is known for Natural (whole) numbers, triangular and square numbers, rationals and irrationals — and so we are going to derive and apply the golden ratio the Pythagorean way New and easy way to draw unreduced golden numbers that construct the golden rectangle, triangles, pentagon, trapezoid, diamond (rhombus). From the pentagon arises a brand new five pointed star — the hyperstar. The application of the Pythagorean Theorem is at its heart How to explain incommensurable and commensurable numbers. Practical definition of (in)commensurable numbers, their origin and differences Incommensurables consist of irrational and transcendental numbers, while Commensurables are naturally finite numbers that are also called rational or real or exact or absolute numbers What is meant by proportion and rationing. What is so fundamental about fractions How to draw and construct the four sided Great Pyramid through geometry, and how to begin to work the Great Pyramid with the golden numbers

Intro to the golden proportion The first firm reference concerning the golden ratio is made by Euclid in his Elements. This person is best seen as a scribe with a sharp pen and sharp mind. Euclid was not just a line editor, he was the Editor-In-Chief. We should not expect Euclid to say much about applications — why, he drew circles a lot but not once did he mention the merry-go-round. Dry kind of guy. The way Euclid defined the golden ratio does not say anything about its application or purpose. The pure [or perhaps sterile] definition of the golden ratio is as follows: Take any line and find the point that cuts the line such that the longer part a is to the shorter part b in the same proportion as the whole a+b is to the longer part a. See the solution on right. But what does it mean? For now, think of the parts a and b as the "golden numbers." There is no need to reduce a/b into another number. We will use this ratio — that is, the golden ratio, in one simple and original application below. However, other relationships among a and b could be even more useful than a/b and then the golden proportions or golden relations is a better description of the power and utility of the golden numbers. Growing the golden numbers The golden rectangle on the left has its sides in the a:b (golden) proportion. But you also know that (a+b):a is in the golden proportion as well and so the larger rectangle, though rotated, is also golden. In this example we grow the golden proportion in 2D but the golden spiraling can move in 3D as well. You'll also notice that the growth -- including the rotation -- is quantized. Patterns of leaves rotating about the central stalk is then quantized and it is perfectly alright to see it as the quantum mechanical effects on macro scale. You'll need to appreciate that the waves can be of any size and could easily move outside of the micro scale of the atom.	For a/b to result in the golden ratio, we compute distances a and b to be such that the whole (a+b) is to the longer part a in the same proportion as the longer part a is to the shorter part b. Therefore, (a+b)/a = a/b Now, simplify the left side 1 + b/a = a/b Multiply both sides by a/b and get a/b + 1 = (a/b)2 Rearranging terms results in (a/b)2 – (a/b) – 1 = 0 The unknown here is a/b and the solution for this ratio is obtained easily by applying the quadratic equation formula to x2 - x - 1 = 0 Moreover, we can also solve for b/a: (a+b)/a = a/b Now, simplify the left side 1 + b/a = a/b Multiply both sides by b/a and get b/a + (b/a)2 = 1 Rearranging terms results in (b/a)2 + (b/a) – 1 = 0 The positive solution to the golden ratio a/b equals (1 + SQRT(5)) / 2 You can also see that the numerator a is 1 + SQRT(5) while the denominator b is 2. There are advantages to keeping the values of a and b separate and you don't want to reduce a/b into one number. The SQRT(5) is an irrational number, which also means that its decimal portion goes on and on without repeating. If you operate (±, ÷, •) on an irrational number with rational (finite) numbers the irrational number remains irrational. The operation of rationing (division, fraction, reciprocal) in and of itself does not necessarily result in a rational number. The golden ratio and all golden proportions are irrational (also see below). Moreover, the SQRT(5), though infinite, can be made exactly through geometry, and only through geometry, thanks to the Pythagorean Theorem. While this entire page is as practical as possible, if you are a theoretical math guy you might know that quintic equations have no solution. But with the golden numbers they do {Jan 16 2008}: (b/a)5 + (b/a)4 + (b/a)2 – (a/b) + 1 = 0. The point is that if you reduce a/b you will reduce your opportunities and yourself as well.. You do not want to call a transcendental number an irrational number. Although it is true that a transcendental number (such as Pi or e) cannot be obtained by rationing, a transcendental number cannot be constructed exactly geometrically [with real methods] while an irrational number can be. Irrational numbers such as SQRT(2) or SQRT(5) are constructible exactly via the Pythagorean Theorem as a distance between two points. In other words, it makes no sense to call a transcendental number an irrational number because irrational numbers differ significantly in their properties [the ~Wiki~ crowd tends to be disappointingly superficial]. A transcendental number can be constructed via infinite superposition and to do it in finite time you'll need pyramid geometry and work with waves. Below you will appreciate that irrationals span 1D distances while transcendentals are always curving in 2D -- and they are both incommensurable. There are two pages on this site related to the squaring of a circle. The Circle and Pi talks about the enigmatic properties of a circle while the Proofs page deals with the components needed to square a circle in finite time. Kepler (bio) was fond of the golden ratio and discovered that the ratios of two consecutive terms of the Fibonacci sequence converge toward the golden ratio. This is indeed the case but the golden ratio convergence is not limited to the Fibonacci series. (In the link I show that the golden ratio convergence can be generalized.)
Golden numbers in the bank, the Pythagorean approach
A customer walks into a bank and says: "With your high interest rate and my initial deposit, I want to receive the amount of money equivalent to my initial deposit every year — forever!" Easy enough. As a banker you now find out what it takes: The initial deposit is any amount of money, which is one unit of money that is 1 Deposit 1 and after one year use the rate multiplier x to get what the customer wants: 1 • x = 1 + y where y is the extra money earned. Intrinsically, the (important and deeply significant golden ratio) issue is: What is the multiplication that will give me the same result as addition? In the economics vernacular it is the balance of leverage and informal partnering. Informal partnering can also be seen as public acceptance.	The book you will thoroughly enjoy QUANTUM PYTHAGOREANS   More about it..
After one year, then, we have the following relation x = 1 + y and the amount of money in the account increases to 1 + y	Okay, start with counting
The deposit continues to grow with the multiplier x and after the second year we have the relation (1 + y) • x = (1 + y) + 1	At this point you learn addition and multiplication
The right hand side of the relation is now the total earned at the end of year 2. Looking at the right hand side, the bank could then give the customer the initial deposit 1 and would be left with 1 + y, which forms the base for the next (third) year and is identical to the amount after the first year.
After the second year we have the following relation x = (2 + y) / (1 + y)	Numbers that are up by 1 and 2 create two rows of triangular numbers
Because (from the very first equation) y = x – 1, we combine both relations into x = (1 + x) / x or, x2 – x – 1 = 0	Square number appears via rationing of numbers where the numerator is one larger than the denominator. In the Pythagorean tradition the first four numbers of Tetractys are about dimensions (0D -- 3D). Are we encountering rationing of 1D and 0D? That is, rationing of a line by a point? There could be something better than algebra!! Of course, rationing of a line by a point is irrational because you cannot use a point to measure distance with it. Ditto for rationing of a volume by a plane
Solving for x we get a ratio of two numbers x = (SQRT(5) + 1) / 2 = 1.618..	.. and the golden ratio, the irrational number. The interest rate multiplier x is no other than the ratio a/b of the golden numbers
To satisfy the customer, the bank would apply 1.618.. annual multiplier x — which represents 0.618 .. (61.8 ..%) annual additive interest y — and start paying out the customer at the end of the SECOND year. This golden ratio application from Jan 2006 is original [unless HE had used it] and close to real life, too! Let us know of other practical apps and we'll link to them. In essence, the golden ratio comes up when we delay the first annuity payment by one year.	The golden ratio embodies the notion of 'maturity.' There is a delay, in our case two years, before the earnings can be withdrawn. Also, 'earnings must earn earnings' before we get to the golden ratio ["ask me about my grandchildren"]. If the payment was not delayed, the bank would need to double the initial deposit every year to keep up with the payments. Was Pythagoras a banker? Some say he was -- and with insurance, too. Were Pythagoreans applying computing methods that [partially?] surfaced one thousand years later as algebra? Did Pythagoreans think the invisible 'anti-earth' is the result of two -- plus and minus -- solutions from square numbers -- or did they get as far as the i (SQRT(-1)), the virtual and invisible wavefunction? Did Pythagoras' School signed up new students as "investors" with a two-year minimum stay? [Philosophy can be very practical.] Because a galaxy is computationally a single body, there ought to be a sister solar system to our own on the other side of our galaxy. More on Pythagoreans and their School.
It is now easy to see that the golden ratio is the interest multiplier x = a/b, while the interest adder y = b/a is the reciprocal (inverse) of the golden ratio. (This is because a/b – 1 = b/a.) You see that the adder y is also a ratio of the golden numbers. You don't have to be a Pythagorean to like rationing — even if you noticed that the golden number a is irrational to begin with. You may try to figure out whether a/b or b/a is the "primary" golden ratio. Presently we will take a/b to be the golden ratio but only in deference to the historical usage. Because the ratio a/b is used as a multiplier, a good call on a/b could also be the gold lever or the golden leverage. For the adder b/a, the golden consensus might be descriptive.	Irrational numbers are nifty because rational numbers cannot match them. If you establish that, in a House of Representatives/Parliament, a motion will carry when the ratio of the yes/no votes exceeds b/a (or another irrational number) there is no need to be concerned with the number of lawmakers casting a vote -- the vote will be above or below the irrational number and there will never be a tie.
Golden numbers geometric construction and illustration
The mathematical solution yields two golden numbers: one irrational number SQRT(5) + 1 and one rational (finite) number 2. Geometrically, the design and construction of any irrational number is easy to execute through the Pythagorean Theorem. Geometric mean, for example, yields an exact square root of any real number or any irrational number (something a present day computer cannot do). Moreover, the geometric construction of all golden shapes is exact and includes the Great Pyramid. (All irrationals are straight-line incommensurables — see further on.)   So now we have the distance of SQRT(5) + 1 as well as the length of 2 and use these parameters in designs and constructions such as the golden rectangle, triangles, pentagram, pyramids, diamond, parallelogram, spirals, darts, decagons — even a trapezoid. For the four sided Great Pyramid, the base will be 4 units wide (see further on). You want to figure out why the SQRT(5) is dashed. All scientists make incommensurable distances as solid lines and do not think much of it — that is, they do not know any better.
	After a short search on the Internet and in several books, this the unreduced geometric construction of the golden numbers appears original {Jan, 2006}. It starts with a half-square diagonal (a triangle with sides 1 and 2) and extends the diagonal by the length of the short side. Of all other methods, this one is simple and easy to remember, too. (The book Quantum Pythagoreans presents yet another original and simple construction that is better suited for pyramid analysis.) This golden numbers construction is also reminiscent of the trowel symbol in Masonry. [While the Masonic interpretation of the trowel seems weak to me (it's associated with cementing-binding-joining), I like the symbolism of the 'Ocean foam of Venus' here, which is a component of creation.] The 'unreduced' aspect simply means that in the construction you start with the unit length 1 as the shortest length and build up from that. When finished, do not reduce any numbers and thus the unit 1 remains as unit 1 and, in the case of the golden numbers, b remains at 2. You have been told that a ratio of two integers is always a rational number. All mainstream mathematicians define the rational numbers that way. In geometry, however, a right angle triangle with sides 1 and 2 makes an irrational angle of 26.5650..° from arctan(½). The ratio of ½ in arithmetic differs from 1:2 proportion in geometry. For perfect right and square angle construction, and for dividing any distance exactly in half:
One new construction: Pentagon and the rest
Using the right angle from a half-square above, we easily constructed the distances of the golden numbers and are ready to have some fun drawing a pentagon. While there are many different methods for making a pentagon — and thus a five pointed star, pentagram, and pentacle — this particular construction appears original and I dubbed it the Boston Pentagon {Dec, 2009}, for Bostonians take fancy to a tilted pentacle. Boston Pentagon is not tilted by the same amount as the "Muslim" five pointed star (that has the two adjacent points vertical).	This pentagon construction is nice because it allows you to pick the length of the pentagon's side. The ratio of the unit length 1 to the length of the pentagon side is 1:2. If you wish the pentagon side to be 10 feet, start the golden numbers with the unit length one half of that (5 feet) -- and it is going to be exact. In the Quantum Pythagoreans book another pentagon construction with dimension Length priority results in an upright pentagon. Boston pentagon construction is also nice because it starts with the shortest length as the unit 1 and the pentagon side ends with the length of 2 -- a rational number. All relationships and distances related to the golden proportion are easy to see, construct, and remember. You might have noticed that the Boston Pentagon construction results in one template allowing you to make two 180 degree out of phase pentagons. Some math guys compete in doing constructions in the smallest possible number of moves, but, yes, in this construction you get two for the price of one. This is not as trivial as it seems if you think how you'd build atomic orbitals.
Boston Pentagon is rotated by the same angle as the ascending passageway in the Grand Gallery of the Great Pyramid: tan-1(½). Are we having fun yet? Boston Pentagon sides can be extended to make a pentagram. If you retain the circles (orbits, orbitals, rings) you have a new pentacle. The result appears cosmic but it could also have an application in molecular, two-atom valence orbitals such as gas if you think of it in the square-a-circle context. So, the HyperStar pentacle it is {Dec, 2009}. Oh, do you see the fish inside the pentagon? Are we there yet? Just taking off .. ..
The hyperstar – five pointed or ten pointed – has all of its triangles (small, large, inside, outside) golden.
	You can appreciate the name HyperStar by imagining the interplay of waves in circular geometry while separated by discrete orbital distances governed by the golden proportion. Yes, this is not just a pun on this web site's name. The waves exist in superposition and there are a lot of them: each electron being capable of producing many end-to-end waves. The hyperstar is a construct for any and all waves interacting in a (point symmetry of a) circle. Once you appreciate there are two circles, the molecular organization comes in. So far we've created the hyperstar pentagram from dual circles having their radii in the golden proportion. Two centers of radii are needed and they are at the "hips" of the pentagram. You can also create a pentagram from the golden ratio centered at the "shoulder" of the star and in this link you'll see how the golden ratio grows inside and outside the pentagram.
The root of five In the Boston Pentagon the SQRT(5) distance is dashed. Once you know why the irrational distance is dashed, you will also know why irrationals exist and what is their purpose. If you know what Proclus had said about irrationals you may hesitate to go there but, as with anything, once you get to know the neighborhood you will be comfortable walking there unafraid. Construction of the SQRT(5) is possibly the most powerful and practical application of the Pythagorean Theorem. The right angle triangles you know from school — such as the one with sides 3, 4, & 5 — are great for introduction, for these sides have real – that is rational, lengths, and this property makes them well suited to erect the right angle on paper or in the field. Another triangle — one made from a square — has its hypotenuse (square's diagonal) the multiple of the square root of two (SQRT(2)) and the square's geometry is used as the entry to irrational numbers in general. The Pythagorean Theorem works (provides solution) for both the rational and irrational numbers. The Pythagorean Theorem works with — but does not solve for — the transcendental numbers (Circle & Pi). The Pythagorean Theorem superposes areas, including circular areas such as lunes, but the square root can take you to irrationals and not to transcendentals. The SQRT(5), moreover, has the most astounding applications that are all related to the golden proportion. The (square) root of the number five is prominent in these applications. But of course, irrationals arise from good ol' integers — through geometry no less. Because one of the components of the golden proportion is an irrational number, the golden proportions can be constructed exactly only through geometry and only as the distance (not the length) via the Pythagorean Theorem.	We've got a collection of designs inspired by the five-point star. View select designs — or visit our store at Zazzle (.com/Mike_Geo) to see how you would look in a tee, hoodie, long sleeve shirt, or a polo. You might pick something bold, fancy, really masculine, very feminine, never cute. Every design comes from nature and centers on the golden proportion. The design on left is called Going to 3D, while the next one is called Drawing Three Bows and is great for kids too. Only once, on an Australian web site, did I come across a dashed diagonal of a square but it could have been incidental (my query was not answered). May'06 DSSP topic will treat you to the difference between the solid and the open square in the context of further clarifying irrational numbers. In the search of a proportion between the square's side and its diagonal, the SQRT(2) is the multiplier and the question is, "How can we visualize the SQRT(2)?" Another time I saw a dashed line was not in the strict geometric context but the idea was the same. Ingo Swann and his remote viewing method uses a dashed line in a particular construct. [Are spooks supposed to be spooky?] Finally, if you are comfortable with dashed diagonals and are ready for some magic -- and healing is about magic -- you might also visit Mar 05 DSSP topic

Operations with the golden numbers

We are going to call the spatial distance SQRT(5) + 1 the number a, and the length 2 the number b. The number a is irrational but tractable in geometry. This is important, particularly since the same cannot be said through arithmetic or algebra. Also, we are not reducing the number b to 1 and b then remains at 2. In the Pythagorean tradition, 2 is "not really a counting number." Well, 2 is best thought of as doubling or the octave. Try thinking of 2 as an operational verb, too. Number 2 (or half or doubling) is prominent in symmetry about axis and gives unique properties to all atomic even functions. You could reduce all even functions because they can all be split in half, but all even functions exist as even functions until they are actually reduced. This sounds like a riddle but it should not be if you are familiar with QM. It is a riddle if you think of the number 2 in metaphysical terms[, and you might have a tempest on your hands if you try to reduce her]. It's okay to reduce even functions, though at times it might not be the best thing to do. (There are some hot ways of going about it, especially if you are in the land of the snows. There, you will need to break out another "number 2" — duality in geometry.) [In the ancient Egyptian context think Seth and one of the reducing tools he's using.] We are now ready and define the golden ratio as the unreducible ratio a/b. In the formulation with the SQRT(5) or in its geometric form the golden ratio is exact. At times the golden ratio is also called the divine ratio or divine proportion and once you discover some additional applications you may give it the name the likes of "the fulcrum of the heaven and the earth" — perhaps in the same spirit as the cry of Archimedes. The reason you do not want to reduce a/b into one number [and call it Phe, Phi, Pho, or Phum] is because the exciting relations are not confined to a/b, but also include a+b, for example. A good start is a·b, which is the area of the golden rectangle having close association with Balmer's atomic orbital sequence. In the Great Pyramid the relations between a and b are even more striking. For triangles alone the golden numbers construct three different triangles when a and b are used for sides, base, and height. On these triangles the golden numbers in different spatial positions then produce three different answers corresponding to three different angles while arithmetic produces but one answer (and this is a good example of the disadvantage of reduction). Therefore, keeping the golden numbers unreduced and staying with geometry will allow us to appreciate all proportions of these great numbers. Saying 'golden proportion,' then, is more descriptive than saying golden ratio (or golden mean, ~ section, ~ cut, ~ rectangle — or the mean and extreme ratio). It all starts with the golden numbers. What proportion or what relationship you chose is in the realm of your applications, be it atomic construction, architecture, economics, on your Valentine's card, [or on our Tshirt]. Composing and working the Great Pyramid, we want to keep the denominator of the golden ratio at the unreduced number 2 because the number 2 in that case stands for the octave (doubling). In the beginning we came up with one golden numbers relation a/b = b/a + 1 and this relation is also better looking than doing it with Phi. [Just because the ancient Egyptians did not wear your phunny hat does not mean they knew nothing of hats.] When a/b is used as a multiplier in our banking application we need to subtract 1 to obtain the identical adder. When we subtract 1 from the golden ratio a/b we get b/a, which is the reciprocal of a/b. Could this yield a more succinct definition of the golden numbers' property? Besides, the reciprocal is sooo important, particularly if you ask: "What is the reciprocal of a wavelength?" Another interesting formulation with the golden numbers is a2 – b2 = a·b which relates the squares of golden numbers through the area of the golden rectangle. Big deal? You bet. The left side is the Pythagorean Theorem that takes the area of the hypotenuse and subtracts from it the area of one side of a right angled triangle -- and that produces the area from the last side of the triangle. Take a look at Balmer's equation and spice it up with the geometric mean. (Forget Bohr, for he had corrupted it through reduction.) It is even bigger if you, as a Pythagorean, know the physical representation of multiplication in nature. And the Pi Some people claim the number Pi is built into the Great Pyramid in various very close approximations. On this page we show (below) that the golden proportions are in the Great Pyramid as well. Now, if you use the golden numbers a and b in the expression 3·(a+b)/5 you will get to Pi to within 15 parts per million. If you get involved with energy — either as it relates to your body or as it can be freely harnessed or by analyzing the crop circles — every time you work the golden proportion you will also be working with the SQRT(5). Because the golden numbers are soo close to Pi, you will inevitably think you are working the quadrature of a circle. And quess what: you are. So what's the tie-in between the golden numbers and Pi? The golden number a contains the square root of 5, which is the irrational number while Pi is the transcendental number. The following section is about these two kinds of numbers and .. well, think about the physics entities that subscribe to these numbers. We'll be in the micro (atomic) and it's a good intro to the squaring of a circle. Yes of course, irrationals span 1D while transcendentals curve in 2D.

What? Number 2 not a number? Pythagoras started counting at number 3. This is not because numbers 1 and 2 are not really numbers (they are), but it is because there are so many things that apply to 1 and 2 besides just being a count -- numbers 1 and 2 are best thought of in such prevailing terms [and you don't want to get carried away counting pebbles on the beach]. We need integers 1 and 2 to define the Pi of a circle. In another example, Planck constant is the smallest quantum of energy that is used inside the atom as the smallest increment within the orbital, while Balmer's relation deals with integer energy increments between orbitals. Planck constant and Balmer's orbital numbers are important energy units that should stand outside of the counting number 1. As a Pythagorean you construct the monad as a commensurable entity. (The smallest monad is the atom.) Having a monad as a real thing, you then start the count using the monad as the object 1. As a Pythagorean you should not have difficulty understanding that a monad is composed of other numbers but as a real thing it becomes the object 1, which is now a counting number 1 as well. Natural numbers can be used for counting but they are not just counting numbers. There are odd functions inside the atom as well; odd functions describing symmetry about a single (one) point. There is no symmetry about volume (3D) but then the number 3 starts the count as the monad, which is the first stable (and countable) thing. If you want to be mysterious about it you might say "three is one." See Pythagoreans. Academia loves to discuss precedence. Some even go as far as to say that the Pythagorean Theorem is not original because it is preceded by number triplets from Mesopotamia such as 3, 4, 5. Yet there are two original aspects to the Pythagorean Theorem: 1) The establishment of a (square) relationship between the triangle sides and 2) The result of the relationship yields both the rational and irrational numbers -- that is, the Pythagorean Theorem is valid for any conceivable distances of the right triangle sides. (That is how Pythagoras or Pythagoreans discovered irrational numbers. There is additional discussion about the Theorem on the Pythagorean page.) Any writer could pretty much make any point he or she wishes. They can put together quotes from Aristotle and Plato and Archimedes and Newton .. and make a claim that some concepts are old hat. There is a lot of context shifting and translation gymnastics going on, too. Well, nobody really jumped up on the Pythagoras' concept of starting the count at 3. Nobody was really interested in usurping this very strange idea and it continued to be attributed to Pythagoras. Once you understand the reason behind it you will see the true originality and the depth of Pythagoras. All is number. These reductionists! Well, don't worry about them. They pose greater hazard to each other than to the rest of us. It's okay if they do the mind job on each other and only on each other. So you still have the burden of figuring out who si who and what is hwat, for reductionism is about exclusion. The reductionist "simplifies" a particular thing by discarding or ignoring some apparently inconvenient components. Take the concept of 'space.' Because 'space' is such a loaded (grouped) term, several people can take different positions and, by discarding certain aspects of space, "prove" what they want to prove. You want to differentiate space into distance and length and degrees of independence. Distance is the 1D construct of geometry while the length is the measure of something real. You can then discuss how (or if) you can transform distance and how distance can become a dependent (subordinated) parameter when you transition into the virtual domain. Yes, this seems complex but now you can make some progress and even get there from here. Another example of reductionism is 'the field.' Without defining a field you can become a science writer on it but once you try to define it and differentiate it you will see there is no field without geometry. A field, in and of itself, is great for your vegetable garden. To make a linear field, for example, two flat parallel plates need to be manufactured and placed in position. Speaking of a field without saying how it is created is not even wrong. A nice example of an attempted reductionism is the 'straight' vs 'curving' path (or motion). Mathematician Poincare simply said the straight and the curving is subjective. He did not bother to differentiate the two and likely had no clue on how to do it objectively. Although ancient Greeks thought of the curving lines as godly, Poincare took the easy way out and, well, he was dead by the time the mindless gyroscope was able to tell one from the other. The best example of reductionism is the relative vs. absolute motion. The 'cannot tell the difference' reduces the logic by grouping concepts together but it corrupts reality. But of course you can tell one from the other via the quantum mechanical dual slit experiment. The biggest example of reductionism is the taking out of ether. It's okay if you think it does not exist but make it your own belief and pursue it on your own nickel. If you want to convince me ether doesn't exist I know better and want my money back. In a case of the Newton's term 'corpuscular,' the present day reductionists equate this word with 'material.' This is a wonderful example of trying to reverse the differentiation, for Newton understood matter very well and because he knew that light is not the same thing as mass he differentiated light from mass (materia) by calling it corpuscular. Even today, corpus or corpuscular is not just about something physical -- a 'body of knowledge' aka corpus, is a collection of intangibles in a particular context. Corpuscular is best translated as 'embodiment' -- something cohesive that hangs together logically. In a context of light, embodiment is intangible and based on waves (wavefunctions) that in turn are based on the square root of minus one. A recent case of reductionism comes from Richard Feynman. In his theory he needed to have the subatomic environment a collection of billiard balls and so he reduced electrons and photons into just such little critters. Consequently he could not explain the dual slit experiment and spent a lot of time musing about it, making his musings run into 'not possible to explain,' as if he were a guru rather than a guy who is just clueless. Actually, when he reduced an electron into a local and well defined object he jumped right into intractability of 'all possible path histories.' That is what happens when you ignore or deny the "inconvenient" nonlocality of a wave or, in case of Feynman, do not know what to do with waves. If you are a scientist you'll see right away Feynman is in a self-induced non-polynomial (NP) and intractable attempt hopelessly bogged down in the musings over the traveling salesman problem (and under finite time constraints to boot). A reductionists agenda is to reduce you. One of his or her options is to ignore pertinent stuff and insult the audience. Mario Livio in his Golden Ratio book calculates the theoretical golden ratio and the actual face-to-half-base ratio of the Great Pyramid -- and finds the difference at 0.1%. Because he doesn't know the purpose of the golden ratio, Mario flatly states there is no connection between the Great Pyramid and the golden ratio. Livio first compiles a lot of stuff but when it's time to deliver he slams the reader with crap [it is a form of a betrayal]. Livio must ignore the well known crease running the mid-face of the pyramid and insults the reader with 0.1% arguments. "Glowing reviews" is the hurdle you have to overcome. There are many authors, including myself, who do not seek one-liners for the book's back cover, for these mini reviews are available for a price. Reductionists also parade as skeptics, debunkers, and bad news bears. Reductionists' motivation is simple. Reduce the answers to those few that can make money for but a few. Medical treatment is a good example. Yes, we want health care but if the reductionists were to have their say the alternative and in some cases revolutionary medical treatments would not be covered by insurance. Most of the time the treatment is simply disallowed. You guessed it, reductionists are in the infrastructure and alternative treatments are not making money for them. Another case is in sciences. Totally worthless projects are being funded after the physics has been corrupted and reduced into trivial and irrelevant pursuits: big bang, black holes, dark matter and dark energy, gravity waves. But you are really smart. You trust doctors and hospitals and can afford healthcare premiums -- and big bangs to you have nothing to do with NASA. Yet, maybe, you want to check your retirement portfolio. There may be assets in there the reductionists deemed adequate. These assets do not trade on the open market and they have some value some experts put on it after getting some fee for it. Housing securities got you reduced? Oh, the portfolio managers' cut is long spent. It sure got the whole country reduced. When you finally figure it out [while you have a choice], invest yourself in free energy. It is not as easy as free speech [today, in the US] but you will never be, you don't want to be, and you don't have to be, in the company of the reductionists. Bob Lazar worked to reverse engineer UFOs but his statements regarding the inadequacy of our present knowledge are not reductionist. His willingness to express and explain the alien technology with his MIT and CalTech vocabulary nicely shows his misconceptions of reality in general and physics in particular. Bob and many like him have great difficulty integrating and applying new knowledge. Nevertheless, Bob is not trying to reduce and "fit the new reality" into existing limited-by-reduction mindset -- and that's a good thing. He'd rather look silly than explain it away -- and that's a great thing. Except that now you have to figure out why he did not find any cables in that thing. The irrational distance always exists in 1D because it exists between two points -- such as the often quoted square root of 2. To get to the SQRT(2) we need rational lengths and a 90 degree rotation to construct it -- that is, we cannot get irrationals by staying in 1D. (You cannot get any irrationals through arithmetic -- see below.) So, here is where the Pythagorean Theorem comes in: We construct rational distances (1D) and open up the second dimension by making the right angle. We use areas (squares, lunes -- that is, 2D) to obtain the relationships and then move back to 1D via the square root. Relying on the Pythagorean Theorem, the distance presented across the diagonal contains the square root already. This serves as a nice intro to the computational power of geometry. In general, the square root is constructed through the geometric mean. In the Great Pyramid the geometric mean is its height. As a mainstream math guy you might have thought the above description of moving from 1D to 2D and back is not very exciting. As a Pythagorean, though, you know what the 2D domain is about and so you know what physical (and physics) parameters you will be dealing with when working the irrationals. As a Pythagorean you understand Tetractys as not only representing but actually being the 0, 1, 2, and 3 dimensional constructs that exist within geometry [advanced]. Now, the 0D (a point) is routinely ignored although it is a unique and very useful geometric construct. (It is not ignored in the East, though, for it is well understood as Dantien/Hara. Quantum Pythagoreans book tells you why a point is critical and why some versions of the Tao symbol include dots -- that is, points.) You might think the reductionists were busy erasing the geometric point as a unique dimension in its own right. The reductionists, you might think, deleted the dimension zero and replaced it with time as the fourth dimension. Not so. Reductionists are simply not that smart. The four dimensions issue from Tetractys as the Tetra (Quad) of 0D, 1D, 2D, and 3D. Time is always a dependent variable and cannot become independent as the degree of freedom is. The reductionists are really rather dumb and you just want to stay on track while keeping an eye on possible corruption. A case can be made that reductionism is limited in the creative dimension and, technically, a reductionist will always be corruptive in the long run. So what kind of degree of freedom does a point provide? Rotation, orbits, orbitals, spin -- all the angular stuff. Selftest:–) How much of the real moving energy in the universe exists as rotation and orbits and spin? About 99%, give or take a percent. What is the significance of the golden proportion? What is the deep meaning of it all? Well, how deep do you want to get? The gateway is really about the knowledge of the mathematical representation in Nature. As a Pythagorean you want to know the representation of an area (2D) in the real physical environment. Nobody [as far as I know] understands that the Pythagorean Theorem takes you from 2D to 1D and vice versa via the square root. (Get with it -- our solar system is flat.) Then you want to know how Nature performs addition and subtraction. Finally, what is multiplication and how Nature does multiplication? It is a simple synthesis after that. {April 2006} Why, for example, is the golden proportion concerned with the balance between multiplication and addition? What is the purpose of adhering to these operators while keeping them in balance? Operators arise when numbers begin to move such as when dealing with the trajectory of a circle. The multiplication and addition apply (come from) two separate domains -- real and virtual -- and they need to be in balance. Quantum Pythagoreans book deals with these domains in both the direct and indirect fashion. The idea is to find numbers and operators as the elements of Nature. QUANTUM PYTHAGOREANS The book of numbers and stars and operators Ready to understand the stars, the pyramids and how the octave fits in? What happens to energy when it encounters the pyramid geometry? How harmony and disharmony relates to rational numbers and circular geometry? Continue .. Self test:–) If you think Phi is useful, try expressing the area of the golden rectangle a·b using Phi (Phi is a/b and the usefulness of a·b is described in the left column). If you don't succeed you will be able to see why the ancient Egyptians did not use Phi but used the golden numbers a and b in all kinds of geometric or arithmetic relations. You are now ready to look unreduced and unburdened into the insides of the Great Pyramid in the Quantum Pythagoreans book.

The origin, definition, meaning, classification, differentiation, and explanation of Incommensurable (transcendental and irrational) numbers, and Commensurable (rational/finite/exact/real/absolute) numbers. Proportion: Squaring and rationing
Incommensurables: Transcendental and irrational numbers
Incommensurable numbers issue from geometry and only from geometry. (When sharing a bill in a restaurant you just don't get to use the square root or an infinite series.) It is also not possible to arrive at the exact value of Pi or SQRT(5) via algebra or arithmetic because the sub-unity portion of any incommensurable number goes on and on to infinity without repeating. Irrational numbers, however, can be obtained exactly via geometry. We can construct an irrational distance but not an irrational length (magnitude). An irrational distance is exact because the particular geometric construction arrives at two points in space and the distance between the two points could well be irrational. For example, a right angle triangle with sides 1 and 2 will have the hypotenuse the exact irrational distance of SQRT(5). Similarly, a right angle triangle with sides 1 and 3 will have the hypotenuse the exact irrational distance of SQRT(10). Transcendental numbers such as Pi or e (Euler's number) are separated from all other numbers via the squaring of a circle assignment, which is possible in finite time but only with the virtual numbers. (Virtual numbers issue from SQRT(-1) and are applied in the Quantum Pythagoreans Book.) Our — that is Pythagorean — differentiation of irrationals and transcendentals is easy. Irrationals exist in 1D while transcendentals do it in 2D. The square root operation from the Pythagorean Theorem can take you to irrationals but not to transcendentals. When trying to express a transcendental or an irrational number on paper (as a magnitude), the sub-unity part of a number (mantissa), being infinite, cannot be written on paper (or spoken or stored in a computer) and is, therefore, inexact. What becomes most useful in our discovery that "irrationals exist in 1D while transcendentals in 2D," is the fact that the (atomic) photons are 1D entities and can have irrational values. So now the orbital transfer solutions can include irrational numbers (think geometric mean when squaring a circle). As a Pythagorean you know that numbers become — or come alive. Because a circle is composed of infinitely many points, you may want to figure out what entity could possibly exist as an assembly of points or circles or arches (sweeps). You also want to answer a question, "Could an electron fit the geometric definition of a point?" Makes sense if you think geometry rules.	If you think incommensurable numbers (irrationals and transcendentals) are real numbers then you have some work to do in moving these numbers from your left brain and into the right [no need to have Dedekind kind of cut-cut lobotomy]. This is easier said than done for some people but it can be done. In other words, you will need to figure out that irrational numbers are not real numbers. 'Real numbers' is a compound group composed of rational numbers and integers (see below). Some writers restrict integers to positive integers if these were to qualify as real numbers. Atomic orbitals are, after all, positive integers. Real numbers represent real -- that is tangible -- things. Mainstream mathematicians should stick to the best possible representation to go with their definition and labeling, and you should feel free to question it. For example, should a scientist rely on a numerical sequence reaching infinity then such sequence cannot represent a real thing, for a real thing could be unbounded but is never infinite. The inability to measure some distances exactly is a sort of crisis for the scientist because all of the sudden he or she cannot measure distances exactly in principle. Usually they say, "Ah, chop it off," or "close enough for government work." A really stupid error is equating real and irrational numbers. (A real number is a finite number just like the rational number and can be negative.) This happened via mathematician Dedekind who thought -- and many agreed -- that if one could carry on the generation of an irrational number to as many decimal places as we want then an irrational number is the same as a real number. Never in a million years. The irrational number has an infinite number of nonrepeating decimal digits and to make an irrational number via a computer, for example, takes infinite time. Even on technical grounds, any proof that becomes valid at infinite time in the future is not a proof because the proof never completes and never happens. The acceptance of Dedekind claim of irrational and real (rational) numbers equivalence points to mathematicians taking an easy way out and it is a good example of reductionism. If you stick to geometry when creating irrational numbers you will make irrationals exactly and discover applications the real or rational numbers cannot touch. (For example, the SQRT(2) and SQRT(5) irrationals are specifically and exactly constructed in the Great Pyramid and the book gets into the mechanics and the purpose. A computer cannot store or compute any irrational number exactly.)
Most straight distances obtained by the square root are irrational, and that is how the Pythagoreans became so closely associated with irrationals. A way to visualize the irrational distance is that such distance cannot be measured exactly by some 'minimum unit lengths,' but can be filled in with points like "ducks in a row" — that is, in 1D. The 'minimum unit length' is a length that has a finite mantissa -- it is a naturally finite number and comes from rationing at the limit (yes, the infinitesimals). There is an infinite number of points spanning some and any distances and so some distances will be expressible as finite (rational) numbers while other distances will belong to irrational numbers. The rational and irrational distances are exclusive. Once you appreciate the irrational and rational numbers as coexisting exclusively along 1D, you'll see that you can speak of magnitude for the rational numbers only.	There are such things as a point and a length. Conceptually, a distance can be anything but a length has to be composed of some real thing, for you are measuring the length of something. There is such a thing as a minimum length -- and the minimum absolute length at that -- while there is no such constraint for distance. There is strong differentiation between a zero-dimensional and one-dimensional entity and geometry establishes what is 0D and what is 1D. Yes, geometry rules :-() even if you disagree :-) It is not a coincidence or fancy if the Hanub Ku symbol of Mesoameria is also called the Measurer, among many other names. We have a bit more than just ideas on what constitutes the minimum real length. Random numbers have many applications but before you become infatuated with random number generation, consider  that irrational numbers, random or not, cannot be generated by any computer.
Incommensurable numbers that live on a curve are the transcendental numbers. This is an easy and practical definition (and original, inasmuch it isn't mainstream {2008}). These numbers do not result from a solution to a quadratic equation — in fact, transcendentals do not result from any equation. Transcendentals were first identified mathematically by Leibniz (bio). The construction of transcendentals is by infinite superposition (subtraction and/or addition) of smaller and smaller components. Because the individual components' magnitude decreases rapidly then even the infinite number of them does not make the total distance infinitely large — hence Pi converges to 3.14159.. and does not get larger than, say, 3.1416. Technically speaking the summing series that results in a transcendental number, while infinite, is bounded — has a finite bound. For better or for worse, transcendentals cannot be constructed in a finite number of operations. As a Pythagorean you will differentiate transcendentals by needing the pyramid in their construction. That is, you will need 3D pyramidal constructs in the actualization of transcendentals. Mainstream scientists will try to dummy you down — and everybody else along with you — about the pyramid rather than admit they don't know. Mainstream science writers cut their nose off in spite of their face and treat transcendentals and irrationals as the same class of numbers; these numbers are both incommensurable but issue from different geometries — 2D and 1D, respectively. Taking an irrational or a transcendental number and chopping its mantissa will make it into a real number but the idea is that irrational numbers are infinite — and they not only exist that way as a concept but they are actualized that way in nature [and even gods take notice].	As a mainstream mathematician, you will want to differentiate transcendentals by their inability to come out as a solution from an algebraic equation. This is but another weakness of algebra, which, however, can be used to advantage. The straight geometry of Euclid is solvable through algebra while the curving geometry of Riemann is not. (We have three book reviews on Riemann.) As a regular guy you are now in a position to question math guys' loosy-goosy logic. Many mathematicians label the golden ratio the transcendental number in connection with showing the infinite nested sequence of arithmetic operations such as nested fractions (some say continued fractions). But the infinite arithmetic sequence by itself does not a transcendental number make. It is a good example of arithmetic intractably straining under irrationals that are otherwise easy to work with through geometry. As a regular guy, then, you'll appreciate that imagining geometric structures puts you ahead of arithmetic-wielding math guys -- at least as far as the irrationals are concerned. Infinite series meet certain criteria that make them bounded. Finding the bounding property conditions is an exceptionally important mathematical pursuit, particularly in the virtual (imaginary number) domain. The only and the best start is with the harmonics series while keeping in mind that the ancient Egyptians fractions have a lot to do with it. When working with infinities you could be in good company: Newton, Leibniz, Euler, Gauss, Riemann. There is also the Unabomber, Mr. Kaczynski and his exploding spheres, and so working with infinities holds perils on land, too. Perhaps the best way of saying it is that you would be in a mixed company. Transcendentals also exist in mythology -- or perhaps in everyday life -- as dragons. If so, does it mean that dragons live in mountains (pyramids?) and in twisting rivers (in 2D -- that is, on curving paths of the transcendentals?) If you were a giant and could create a mountain, would you make it such that the spirit might live inside? Could you make it such that an eternal spring would happen? In Tai Chi's Martial arts modality you move in slight curves.

Irrational numbers are obtained arithmetically through the infinite iteration of division and multiplication — such as when doing the square root of 2 in a computer, but all irrationals can be constructed exactly with the Pythagorean Theorem and in finite number of operations (and, therefore, in finite time). This is the second one-up of geometry over arithmetic. Irrationals are tractable via geometry but via arithmetic they are not. (The first one-up is the ability of some Natural numbers to divide a circle exactly — but arithmetically no Natural number can divide a circle exactly.)
Commensurables: Rational or naturally finite or real or exact or absolute numbers
Usually, you can use the finite length of something to measure something else with it and do it exactly. If so, the two lengths are commensurable. Any suit you make or buy is commensurable because it is cut to some size and it makes no difference if the tailor used inches or centimeters. The act of cutting calls for a specific (rational, finite) length using real numbers — and any two rational numbers are commensurable. So, you can get from inches to centimeters exactly because there exists a finite multiplier between the two. Can you generalize and say that all real — that is tangible — things are commensurable? Yes indeed, use the atom as your smallest unit of measure. Can you use the electron as the smallest unit of measure? If not, why not? Can you say that the atom — while it could be divided, i.e., destroyed — is nonetheless the smallest commensurable thing there is? In the Pythagorean vernacular, is the atom the smallest monad? Another way of reaching the same result is by asking a question: When and how does a line unbecome a line and become a point? In other words, if a geometric point is infinitely small, what is the minimum separation between the two points that could be connected and become a legit one dimensional line? Any whole number is commensurable to any other through a unit 1 and that is a trivial result. This unbecomes trivial if you ask "what is the smallest unit 1?" It is said Pythagoreans used pebbles for numbers. Through questions such as these they reached the concept of the atom. This works well for a length (magnitude), which is a measure of a real thing but becomes interesting for distance; magnitude represents something real but for distance you start working with an interesting component of 1D space. So, even though the atom uses 0D, 1D, 2D and 3D geometries the resulting construction is the monad — the smallest unit 1 that is also the smallest commensurable counting number. Does the division (partition) of the unit 1 enter the world of incommensurables? (yes) Are we then dealing with distances rather than lengths? (yes) Is a decimal fraction appropriate when dividing unit 1 in the orbital geometry? (no) What entities live in (occupy) any possible distance? Think Quantum Pythagoreans subtitle: Of Stars, Numbers, Gs and Waves. Because the rationing (division) of any two whole numbers always ends up with a number having a finite or repeating mantissa, any and all fractions issuing from division of whole numbers are commensurable — that is, rational. We take it for granted and it is not intuitive, but any one whole number is commensurable with any other whole number through their ratio (see example on right.) A rational number is always finite in its magnitude/mantissa and the nice part is that it fits in a computer. This becomes important with planetary orbits, which is how the Pythagoreans got so closely associated with the music of the heavenly spheres. In the pleasant sounding ratios of musical strings and in ratios of planetary orbits, the harmony was born. Harmony depends on two tones — that is, two tones may be harmonious and pleasant — or disharmonious and unpleasant. The two tones played will thus produce a ratio of two frequencies. The idea is that if this ratio is rational and, therefore, naturally finite (and it is), then there is no need for a mechanism dealing with infinities and planets' orbits would be stable. You may have seen pictures such as the one below giving each planet a note from the octave — but that is less than half the story. The time of a planet's orbit is a period and period is the inverse (reciprocal) of frequency. Each planet, moreover, makes a different ratio with every other planet. Earth makes one ratio with Venus and another ratio with Mars -- and either one of these ratios can be obtained with several different pairs of musical notes. Different notes can be used to produce the identical ratio -- for it all depends on the ratio -- yet the notes should be harmonious and for planets they are. If two tones are to be found harmonious or disharmonious then both tones need to be played together. A and G notes, as well as A and B notes, are not harmonious and in the illustration above the corresponding objects are out of luck. Actually, the author illustrates the idea of linking celestial harmony to music — a Pythagorean idea — but he cannot sort out the harmonious and disharmonious pairs. Harmony could be debated but it is absolute because we agree on sounds that are, or are not, harmonious. (Quantum Pythagoreans book has the formula and the explanation for notes to be harmonious — and illustrates the corresponding geometric stars.) If you take an irrational number such as SQRT(2) and divide it by a rational number such as 2, you are doing rationing (division) but the result is not a rational number. If you get frustrated by this you just cut the fraction off but the idea is that rationing does not guarantee a rational number unless you work with finite numbers such as integers (or with infinitesimals -- advanced). That is exactly why saying 'commensurable' is better than saying 'rational.' The word 'commensurable' speaks of exact measurable conditions while 'rationing' or 'ratio' speaks of a procedure that is division or cutting. Probably the best label for rational and/or commensurable numbers is, again, naturally finite numbers. A finite number is also an exact or absolute number, for there is no question as to the number's value.	Example. Take numbers 8 and 5. The ratio of 8/5 is 1.6000, which is a rational number that will always have finite magnitude (length) of its mantissa. In this case the repeating number in the mantissa is 0. So, you can take the finite number 1.60 and get the number 8 back exactly by multiplying 5 with 1.6. Numbers 8 and 5 are commensurable through 1.6, which is their ratio. The second rational number from numbers 8 and 5 is 5/8 and, therefore, the reciprocal of any rational number is also a rational number. Another way of saying 'commensurable' is 'exactly proportional.' When you say 'rational,' you are also saying (and you should know) that the ratio deals with two whole numbers. Saying 'commensurable' is then superior to saying 'rational' because 'commensurable' deals with an exact measurement (using a standard) or exact proportion (using two things or whole numbers). 'Rational,' in a way of a definition, refers to the naturally finite result after division. While a naturally finite number comes out of rationing of two integers it does not mean that this is the only way of obtaining it. Any operations with any numbers that results in a finite number can then also be used to define a rational number. Brainwork What two notes of the (Western) octave do you need to play to get 8/5 frequency ratio? Are these notes harmonious? That is, do they sound pleasant together? Of course, you want to figure it out before you play them (this is advanced because you will not find this in any book except in the Quantum Pythagoreans). Aristotle spoke about the "natural propensity" of things. Where it fits nicely is the natural propensity of an electron to spread. Could we say that operations also have a propensity to constantly happen? For example, could we say that the operation of rationing is constantly taking place? Think zero and infinity, like, gee, a/b and b/a. [Aristotle has problems with infinity and he never figured out that numbers are not just for counting.] Questions: 1) Is the result of 3.52/1.8 a rational number? It is. This ratio is also 352/180 -- which is a ratio of two integers (two whole numbers). Any finite number can be made into a whole number if you place it in a ratio. And rationing is found throughout the natural world. 2) Is the ratio of two rational numbers a rational number? That is, because (a/b)/(c/d) = (a·d)/(b·c) then if a product of two rational numbers is a rational number then any and all ratios of rational numbers is also a rational number. A product of two finite (rational, commensurable) numbers is always a finite number and so the result of m·n is a finite number if m and n are both finite numbers. Addition and subtraction of finite numbers results in a finite number as well. Questions above appear to be in the category of crossing the t's and dotting the i's. However, what also becomes apparent is that rational numbers remain inside the rational domain of numbers through these various operations (*, /, +, -), and no transformation takes place. So, if planetary orbits subscribe to certain rational ratios, multiple planets may join in and introduce new ratios while all ratios remain rational -- that is finite (and stable). Yes, this is very similar to -- and a bit more general than -- Euclid's proof that no rationing of numbers that are already rational will "cross the domain" and become irrational. Can you say that multiplication and division are 'real methods' residing in your left brain? This page has rational numbers examples in the micro and macro. Yet, what is the fundamental need for rational numbers? Because the numbers are primary, a better question is the reverse: 'How are rational numbers applied or used in nature?" The rational number is finite and then it is also exact. The exactness comes in handy (required really) when dealing with energy conservation. When a transformation takes place during a collision, say, the conservation of energy holds and all energies have to add up exactly before and after the collision. (Mechanics of electron jumps are different and other numbers are engaged as well.) The new direction after a collision is vector based (vector contains a direction) and then the energy must be conserved along a direction. For direction we need two points -- yes, energy is nonlocal (it's a wave). And the relation between the first and second vector (directions) must be exact (rational) for energy to be conserved exactly. The mathematical derivative that finds a tangent nicely differentiates a point from a line because a line tangent needs two points with some (absolute) minimum separation. Natural it is. In our rush to discovery and attempt at dominance, we tend to think that everything is the way it is because we define it that way. Not so. There are variables that are derived such as the time, and, again, men cannot make time into an independent variable because it is the nature that makes (derives) time from organized systems. In the Buddhist vernacular the derived variables are said to exist "by convention." In the economics vernacular the instruments called derivatives have just the right meaning and are in agreement with our discussion. To a math guy a derivative is also about finding a tangent and that is not what we mean by a derivative.
A rational number is a single number but it originates from two integers — and that is why Pythagoreans love integers. Do irrational numbers originate from integers? Of course they do. Not through rationing (division, cutting) but through geometry they do. Not only that. To get to irrationals you first have come up to the squares and this makes it a triangular process. In the (near?) future we may also have integers [again?] originating and constructing transcendentals through pyramidal geometry (think Grand Gallery — esoterically it is the foam of Venus).
Proportion
Proportion relates two things or two parameters. When multiplying two numbers we are usually moving from 1D to 2D and work with an area. In nature, area is usually associated with .. [your brainwork]. As a Pythagorean you will be working with square numbers. When we are rationing we are relating two parameters in the act of normalization. Rationing takes two parameters — and discovers their relationship — without needing to know their absolute or actual value. Multiplication or squaring is one form of proportion. You will need to discover the physical parameter(s) that exist as a product of two parameters or a square of a single parameter. For example, the tone of a string depends on two parameters: the length and the tension (force). You now must work with the proportions of the two variables to get at the result. You will also find the concept of a continuum here but only if one of the proportioned parameters is energy (explained in the Quantum Pythagoreans book in the quantum mechanical context). But of course, spatial distance and time cannot make a continuum, for neither distance or time contains energy. If your math teacher encouraged you to leave the result as a ratio and did not require you to complete your calculations with a decimal fraction, he or she is very good. This is applicable to those results that have cyclic qualities such as inventory turnover. Leaving the result as a ratio is also applicable to area ("energy") qualities such as the computer display that is, say, in the 16:10 ratio and not in the 1.6 ratio. Results that have linear qualities such as earnings per share or miles per gallon can be carried out with a decimal fraction because these results have a straight dimension (or linear motion) rather than cyclic (rotational, repeating) or area (energy) qualities. Irrationals and transcendentals should never be expressed as a decimal fraction unless you have time on your hands and you, to be true to the value of the number, continue on to infinity. Get comfortable and show your understanding by having your result contain fractions, Pi or SQRT(5). Your first application is with the golden ratio, which is expressed as a fraction because the numerator is an irrational number — and so the denominator stays at 2. So far we talked about proportion in the context of getting one tone using two parameters and also about getting a better visualization when talking about an area. If you have two tones of different frequencies, you can get them in proportion when you play them together. So we say the two tones are in the 3:2 ratio, say. But in a musical proportion the two frequencies A and B have the same result for A:B as well as for B:A. This seems, and is, obvious but algebraically A/B and B/A are not the same. This is the diff between proportion and rationing (and the new book coming out late 2012 also gets into that). You might also reach the conclusion that the decimal fraction does not lend itself to working with the atomic orbitals, particularly from the energy (orbital jumps) perspective that deals with waves having point symmetry (some say radial or odd symmetry). So now you have to figure out that 1.333 is really a rational number originating from 4/3 or that 22/7 goes to six decimal places before its fraction starts to repeat. Do the ancient Egyptian fractions deal with this "problem?" (yes) Did they figure it out or has someone [come down and] told them? Can you see the orbital energy components by looking at the decimal fraction? (no)	Superposition is about additions and subtractions (and not about proportion). When Newton analyzed optical fringes he was able to measure the specific wavelength of light without needing to know lightspeed. He measured the wavelength associated with a yellow ring by using the distance across a gap. Because yellow results from the absence of blue, he measured the wavelength of a blue color as blue color was zeroed out across a known gap distance. (Illustrated and extended in the Quantum Pythagoreans book.) While it is true that multiplication (or division) can be accomplished with a plurality of additions (subtractions), there are conditions and situations where such methods become intractable. However, if you could accomplish addition and subtraction with zero time delay the tractability is shifted up and includes non-polynomial level problems (think QM superposition). Even with instant superposition, though, intractability does not completely disappear and stays with the general solution to the three-body problem.
All commensurable (rational) things are real things and all are finite. You may call the left brain the rational brain as it deals with real things. You guessed it, the right brain deals with virtual concepts and that includes infinities.
Golden Proportions in the Great Pyramid
Construction and Illustration Now that you know how to make the golden numbers, the construction of true to life four sided Great Pyramid is easy. The longer distance a is rotated until the vertical line is intersected, which forms the side as well as the height of the pyramid. Because the base is a square, each side will be the size of 4 units of the measure you started with when constructing the golden numbers:	Rotation of the golden distance a is reminiscent of the Masonic 'Raising with the lion's paw.' Such symbolism could be a stretch unless one accepts the human body to have the golden proportion characteristics. There is a subtle difference between 'raising' and 'rising.' Oftentimes 'the rising' ceremonies are associated with being "second born," but there is much more to that besides the ceremony.
The angle alpha from the pyramid's construction above is prominent [I'd say crucial] in the Great Pyramid but you will need to go inside to find it. With the Pythagorean touch you know that no number and no angle exists alone and every number is a result of some proportioning or rationing of other numbers that stand behind it. Unbecoming a reductionist could be an eye opener. Generations may have looked for 1.618.. throughout the pyramid and easily miss the angle alpha that is so central to the golden proportion. The angle alpha = arctan(½) = 26.5650..° {Sep 2006}.	Many a person measured and documented the angles inside the Great Pyramid. Quantum Pythagoreans book, moreover, identifies geometries that lead to these particular angles and puts forth the needs and purposes for structures such as the Trough and the Great Step. What then is the real purpose and the application of the Great Pyramid? You guessed it, it's in the book.
The Great Pyramid as a computational construct In the illustration below, the right angle triangle is one half of the Great Pyramid shown in a vertical profile through the pyramid's mid face. [By now you should be saying, "Yeah, the diagonal a should be dashed."] The actual Great Pyramid ratio a/b is very close to the golden ratio (some say 0.1%), but because the measurement is not "exact," the scientists writing about the pyramid readily take the easy way out and dissociate the golden ratio (really themselves) from the pyramid. Those who think there is a relationship between the Great Pyramid and the golden ratio do so for a variety of reasons, but the most important consideration is that a pyramid built for aggrandizement or even for star observations would not need to have such unusual and expensive-to-execute angles and geometric constructions -- and that is even before we enter the guts of the pyramid. The mainstream scientist abdicates because he or she would not know where to go if indeed there would be a match. Self test:-) If you think that the Great Pyramid builder's intention was to make the exact match with the golden ratio a/b as measured over and across the stone, you have a gap in the understanding of irrational numbers. Again, you will need to think about why the irrational distance is dashed. The Great Pyramid has a kink — really a crease — in the mid-base that runs up the length of the face. This subtle but well known feature is not discussed for lack of understanding of its purpose. The crease is a needed concession in the realization of the transcendental number Pi, and it's the reason the Great Pyramid's golden ratio a/b is not exact when measuring it inside the crease. Now think about the difference between length and distance, tangibles and space. Self test:-) If you think that the Great Pyramid builder's intention was to make the exact match with a/b — but not as measured across the stone — you are doing really well. So, now you have to figure out why they had to go for a crease rather than make the whole pyramid a bit smaller overall. If you are into Tai Chi, consider that your arms and legs do not usually get fully straightened out, and that means .. .. If the height of the King's Chamber is an irrational number, would you lay the stones horizontally or vertically? This is a good question to ask a Freemason [and he would likely be clueless]. Buddhists are not much ahead. Just ask: "Is the unfinished floor in the Queen's Chamber finished?" Buddhists will appreciate the question and perhaps have an inkling of its meaning but they will think it is not possible to explain it.  Dimensions of the King's Chamber have a common multiple of five — and there are then a few more things to think about.
Applying the Great Pyramid illustration above, in computations involving the pyramid height h we use a and b independently (and reducing a/b to one number unnecessarily complicates the math). The Pythagorean relation nicely dovetails with Balmer's math, too. Note that the pyramid base is now [again] the multiple of 4 of some unit of measure because we keep b unreduced at 2.	Brainwork: Show that h2 = a · b where h is the height of the Great Pyramid Area of the golden rectangle in a square pyramid? (Hint: SQRT(a·b) is the geometric mean.) Are you beginning to see the Balmer's formula? Was the geometric mean taught in your school's main curriculum? If not, brush up on that and smile when the so-called experts try to reduce the importance of the pyramid geometry. The experts know the price of everything and the value of nothing.
Now, take a+b and multiply it by 3/5. You will get very, very close to the most famous transcendental number (Pi). This may come in handy when squaring the circle. The crease of the great pyramid can be seen at acute angles of view or illumination: At mid east side is the "funerary" (or what's left of it) with a path leading to it. All pyramids have (had) structures abutting mid east side. Geometric constructions at mid east side such as arrays of columns could indicate a true — that is, (at the time) a working pyramid. Execution of the cube root is also needed, and just for that a plurality of supporting pyramids is called for.
The Great Pyramid from the Iconos satellite. Creases nicely visible. South is on top	The Great Pyramid at Giza is at times called the Pyramid of Khufu. If you spend a couple of hours researching the Great Pyramid, you will recognize that Khufu had nothing to do with this pyramid. If you like the story of Khufu, you will also enjoy the story of the Piltdown Man and evolution in general. There is not a single mention of the pyramid in the Bible. The Operational Principle of the (Great and all other) Pyramids is as follows: 'The exact will happen if that which is infinite has the spatial infinite series.' This is how to square a circle in a pyramid — that is, actualize the transcendental number Pi. (The mechanism of orbital jumps is a subset of this and "free energy," atomic decomposition, or gravitational influencing is simpler.) {June 24, 2010}
Another picture of the Great Pyramid. This photo has no indication of the North (it's from NASA). Adjacent to the East side is a line of smaller pyramids and so the picture was taken mid-morning
QUANTUM PYTHAGOREANS  Book by Mike Ivsin Geometries bring certain advantages to the table — advantages that may find new sources of 'free energy.' In Quantum Pythagoreans, the construction of the golden proportion is with a unique method called the Golden Eye, which introduces the golden proportion in a spinning context. Incommensurable numbers' applications are also found in the Great Pyramid, and the book reveals the critical component in the actualization of incommensurables' infinite length. Irrationals can be constructed — that is, actualized, with geometry. Throughout the book all numbers, including the ancient Egyptians fractions, are applied in their relevant context such as atomic construction. The book tells you the purpose of the pyramid's edges and why they are more important than the sides. The bottom line is you will understand how different geometries interact with energy and the means of working with space-borne energies. You are invited to walk the true, the Pythagorean road to reality. More ..
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