II.
Before drawing and making the Great Pyramid
you want to understand commensurable and incommensurable numbers 


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 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 EditorInChief. We should not expect Euclid to say much about applications — why, he drew circles a lot but not once did he mention the merrygoround. 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 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.
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}: 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: 


After one year,
then, we have the following relation 


The deposit
continues to grow with the multiplier x and after the second
year we have the relation 


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 


Because (from the
very first equation) y = x – 1, we combine both relations into 


Solving for x we get a ratio of two numbers x = (SQRT(5) + 1) / 2 = 1.618.. 


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. 


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. 


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 straightline 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. 

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 halfsquare 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, twoatom valence orbitals such as gas if you think of it in the squareacircle 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 endtoend 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. 


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
Another interesting formulation
with the golden numbers is
And the Pi So what's the tiein 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. 





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 subunity 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 subunity 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. 


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 zerodimensional and onedimensional 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' loosygoosy 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 arithmeticwielding math guys  at least as far as the irrationals are concerned.

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 oneup of geometry over arithmetic. Irrationals are tractable via geometry but via arithmetic they are not. (The first oneup is the ability of some Natural numbers to divide a circle exactly — but arithmetically no Natural number can divide a circle exactly.) 



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.
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. 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. 
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) 


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. 



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..° 
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
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. 


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. 







QUANTUM
PYTHAGOREANS 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 spaceborne energies.
You are invited to walk the true, the Pythagorean road to reality. 

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