HyperFlight Golden ratio yields several solutions when golden numbers are in geometric proportion Before we make the Great Pyramid we will need to understand commensurable and incommensurable numbers.		"There is gold in them hills"
Golden ratio. In the bank. What it means to apply the golden ratio. Pythagoras is known for integers, triangular and square numbers, rationals and irrationals -- and so we are going to derive and apply the golden ratio the Pythagorean way Geometric construction of golden numbers and their relations New and easy way to draw unreduced golden numbers that construct different golden shapes. The application of the Pythagorean Theorem is at its heart How to explain and differentiate incommensurable and commensurable numbers. Practical definition of (in)commensurable numbers 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 golden numbers Take this page out of frames 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 applications. The pure [or perhaps sterile] definition of the golden ratio is as follows: Take a line and find the point that cuts the line such that the whole to the longer part is in the same proportion as the longer part is to the shorter part. But what does it mean? For now, think of the parts a and b as "golden numbers." 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, for a/b: (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 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 is no need to reduce a/b into another number. We will use this ratio -- that is, the golden ratio, in one application below but other relationships among a and b could be even more useful than a/b. While this entire page is as practical as possible, if you are a theoretical math guy you might like the fact that the golden numbers satisfy quartic equations (b/a)4 + (b/a)3 + (b/a) = 1 or (a/b)4 – (a/b)3 – (a/b) = 1. Also, {Jan 16 2008} other higher order polynomial equations can be constructed. Some say that quintic equations have no solution. But with the golden numbers they do: (b/a)5 + (b/a)4 + (b/a)2 – (a/b) + 1 = 0 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 figure 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. New book you will thoroughly enjoy QUANTUM PYTHAGOREANS   More .. 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. The bank would 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 which 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 Since 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 a/b To satisfy the customer, the bank would apply 1.618.. annual multiplier (x) -- which represents 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. It is now easy to see that while the golden ratio is the interest multiplier x = a/b, the interest adder y = b/a is the reciprocal of the golden ratio. (This is because x – 1 = y and a/b – 1 = b/a.) You see that the adder is also a ratio of the golden numbers. You don't have to be a Pythagorean to like rationing -- even if you noticed that 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, the golden consensus might be descriptive. 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 'anti-earth' is the result of two -- plus and minus -- solutions from square numbers -- or did they get as far as the i, the SQRT(-1)? Did Pythagoras' school signed up new students as "investors" with a two-year minimum stay? [Philosophy can be very practical.] Since a galaxy is computationally a single body, there ought to be a sister solar system to our own on the other side of the galaxy. 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 golden ratio convergence is not limited to the Fibonacci series. We show that the golden ratio convergence can be generalized. Irrational numbers are nifty because the 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 number 2. Geometrically, the design and construction of any irrational number is easy to execute through the Pythagorean Theorem. 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) triangles, rectangle, pentacles and pentagrams, pyramids, spirals -- even the trapezoid! For the four sided Great Pyramid, the base will be 4 units wide (see below). 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. 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, too. 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. After a short search on the Internet and in several books, this the unreduced geometric construction of the golden numbers appears to be original {Jan, 2006}. 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.) The 'unreduced' aspect simply means that in the construction we start with the unit length 1 as the shortest length and build up from that. When finished, we do not reduce any numbers and thus the unit 1 remains as unit 1. For perfect right and square angle construction: 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 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. Only once, on an Australian web site, did I come across a dashed diagonal of a square but it could have been incidental. 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 Construction of the SQRT(5) is possibly the most powerful and practical application of the Pythagorean Theorem. The right angle triangle 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 takes 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. There are golden proportion applications in the atom but the exact divisibility of a circle (orbital) is the first consideration there. Take a look at a circle division by five. You will then be able to appreciate the exact division of a circle that results in the pentacle or pentagram along with the golden proportion that is also inherent in the five pointed star. The way we show the pentagon pyramid emphasizes the golden trapezoid. Spirals are interesting because irrational numbers could be re-created in different spatial positions through geometry and rationing and rotation. Spirals with golden proportions are even more interesting but you want to understand how force could arise in nature. Yes, here is a path to gravitation. Spirals are usually drawn as smooth curves but don't forget that atoms can only have multiple discrete solutions. Yes again, algebra plays but a supporting role. 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 proportion can be constructed exactly only through geometry and only as the distance (not the length) via the Pythagorean Theorem. Operations with 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 the constant 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 in addition to being a number.) 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. 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. 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, or on your Valentine's card. 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 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 (one) 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? 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. QUANTUM PYTHAGOREANS The book of numbers and stars and operators. Ready to understand the stars, the pyramids and how the octave fits in? How harmony relates to rational numbers and circular geometry? More .. 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 can then start the count using the monad as the unit 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 unit 1, which is now a counting number 1 as well. Number 2 (or half or doubling) is prominent in symmetry about axis and gives unique properties to all atomic even functions. See, 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 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. Squaring of the circle in Tibet? If the squarish dot in the center were to stand for a zero-dimensional point then that would make sense. There may be a bit more to it if you think the squarish dot has its sides in the golden proportion. Squaring the circle is something worth pursuing. 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 thing. See Pythagoreans. 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 straightforward fashion for ease of understanding. The idea is to find numbers and operators as the elements of nature. 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. Nonetheless, you still have the burden of figuring out who si who and what is hwat. Are there circumstances where reduction is useful? By grouping together, no. By differentiation, possibly. [It may be useful to reduce a sauce but you are not getting rid of all of the water. If you reduce all of the water you don't have a sauce anymore.] Usually, the reductionist "simplifies" a particular thing by discarding or ignoring some apparently inconvenient components. Take the concept of 'space.' People wrote books on 'space' and what it is. 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. When it comes to 'space,' the really good way of addressing it is by differentiation. 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) variable 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. In the 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. The origin, definition, meaning, differentiation, and explanation of Incommensurable (transcendental and irrational) numbers, and Commensurable (rational/finite/exact/absolute) numbers. Proportion: Squaring and rationing Incommensurables: Transcendental and irrational numbers Incommensurables exist for geometric reasons -- that is, they issue from geometry. Squaring of a circle assignment is not possible to do [as given] because the zero-dimensional point is not in general commensurable (is incommensurable) with the length of a line. You cannot use a point to measure distance with it, but the circle exists as an infinite quantity of points nonetheless. It may not be difficult to accept the fact that the circle circumference, by virtue of being incommensurable, has an infinite mantissa of nonrepeating numbers in the measure of its circumference. 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, "Does the 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." Moreover, there are also straight distances that are incommensurable. This is much more difficult to appreciate but most straight distances obtained by the square root are incommensurable, which is how the Pythagoreans got so closely associated with incommensurables. Incommensurable numbers that exist on a straight line are called the irrational numbers. Irrational numbers appear when we try to solve a quadratic equation (an equation with square numbers). A way to visualize the irrational distance is that such distance cannot be measured exactly by some 'minimum unit legths,' but can be filled in with points like "ducks in a row" -- that is, in 1D. The 'minimum unit legth' is length that has finite mantissa -- it is a naturally finite number. The thing is, there is an infinite number of points spanning some distance and so some distances will be expressed as finite numbers while other distances will be irrational (infinite mantissa) numbers. 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 :-) We have a bit more than just ideas on what constitutes the minimum real length. Incommensurable numbers that live on a curve are the transcendental numbers. This is an easy and practical definition. 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 an 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, while infinite, is bounded -- has 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 pyramidal constructs in the actualization of transcendentals. Mainstream scientists will try to dummy you down -- and everybody else, too -- 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 the irrational or 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. As a regular guy you are now in 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 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. 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 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? 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?
Irrational numbers (incommensurables that exist on a straight line) 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 basic one-up of geometry over arithmetic. Irrationals are tractable via 2D geometry but via arithmetic they are not.
Commensurables: Rational or naturally finite or exact 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. 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? (of course!) Is a decimal fraction appropriate when dividing unit 1? (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. 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 giving each planet a note from the octave (below) but that is less than half the story. The time of a planet's orbit is a period and period is the inverse 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 tosses in the corresponding geometric stars for good measure.)	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. Homework 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 so far but it is treated in the Quantum Pythagoreans book). 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. (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? 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 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) [If you get emotional about it you will see the decimal fraction as just a silly earth-bound invention of Catholic France. If you get upset about it you might not see the Heavenly Father as being all that heavenly.] 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 unless you work with the exact division of a circle (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. So, by now you should have fun reading books about the golden ratio in part because the golden ratio is not a rational number. [Science writers like to proclaim they are writing for the general reader but they skip the system building fundamentals and muse at the gaps they leave behind.] 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. The mathematical derivative that finds a tangent, however, nicely differentiates a point from a line because a line tangent needs two points with some minimum separation.
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. 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 homework]. 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).
Rationing is another form of proportion. Perhaps the most common application of rationing is a trigonometric function such as the sin. The numerator is the vertical distance from the circle's periphery while the denominator is always the radius of a circle. Trigonometric rationing, then, unambiguously expresses the quadrant's angle using but one (ir)rational number. Dealing with circle's angles in the rationed format simplifies computations of geometric relationships, particularly from the perspective of the symmetry about a point (radial symmetry). Consequently, math becomes radius-normalized. When we rationed Kepler's periods of planetary orbits on our Venus pentagram page the result became mass-normalized and we did not need to know planetary masses to obtain the correct proportion of planetary radii. It can be said that rationing is about normalization. (In the Quantum Pythagoreans book, the normalization with respect to a particular variable enables the format that facilitates transitioning to the right hemisphere of the brain.) As an "advanced" Pythagorean you will recognize the measure of an angle as a rational or irrational number and not a transcendental number. This means that you can construct (express) any and all angles -- with rational and irrational numbers -- using the Pythagorean Theorem. We need transcendental numbers when taking a curved path but, in addition, for each point on the circle's periphery there exists a number which is not transcendental and is the measure of an angle. Such additional perspective is afforded by rationing. The angle, moreover, is something that has 1D equivalence or 1D existence. This can get interesting. A photon may bounce -- or branch at a half-silvered mirror -- at an angle but its trajectory can be always described using the Pythagorean Theorem because the trajectory, though discontinuous, can be fully described as happening in 1D (because no angle has a value of a transcendental number). This is also why the peripheries of squares or rectangles (polygons in general) can be considered linear, or 1D, structures.		Superposition is about additions and subtractions (and not about proportion). When Newton analyzed optical fringes he was able to measure a 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 a three-body problem. 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 1.6 ratio. Results that have linear qualities such as earnings per share, or miles per gallon, can be carried out with the 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(2). Your first application is with the golden ratio, which is expressed as a fraction because the numerator is an irrational number.
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:
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 that is 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 one 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 pinch -- 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. The a/b ratio of the Great Pyramid face angle was not intended to be the exact match of the golden ratio because it would not work in that fashion. However, the a/b ratio of the Great Pyramid is meant to be close to the golden ratio. 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 kink rather than make the whole pyramid a bit smaller overall. 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.		Homework: Show that h2 = a · b 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 famous transcendental number. 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.
Picture of the Great Pyramid, top view. Creases nicely visible 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		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 while that which is infinite becomes sufficient. {May 14, 2006}