Phi: an understandimg of beauty, by Vinyasi For Helen and Charlie Lutes Acknowledgements If it were not for the efforts of all those in the field, this volume would be less than it is. Contents Definition of Some Terms and Notations Introduction to the Themes of the Text Applications: Aesthetics Game Theory Natural Occurences Progressive Demonstration of Various Methods Revolving Around, Linear Beauty: Geometric Solution Polynomials in No Unknowns Generalization of Methods Planar Beauty: Geometric Solution Polynomials in One Unknown: Linear Quadratic Cubic Quartic Quintic Generalization of Methods Spacial Beauty: Geometric Solution Polynomials in Two Unknowns: ? Linear ? Quadratic ? Cubic ? Quartic ? Quintic Generalization of Methods Transcendental Polynomial: e Appendix Tables Glossary Bibliography Index Outline Definition of Some Terms and Notations Introduction to the Themes of the Text: Purpose Definition List of Methods Background in: Polynomials Approximation Theory Illustrated by the Incre- mental Method Short Remark Applications: Aesthetics Game Theory Natural Occurences Progressive Demonstration of Hypothesis Revolving Around, Linear Beauty: Geometric Solution Polynomials in No Unknowns Prime Factorization Methods Generalization of Methods Planar Beauty: Geometric Solution Trigonometric Method Polynomials in One Unknown: Linear Quadratic Formula GCD Method Derivation of the Quadratic Euclidean Algorithm(Ea) Golden Method Polynomial Search Methods Word Problems Arithmetic Method Bisection Method Newton's Special Case Radical Method Lucas & Fibonacci Methods Geometric Methods Cubic Golden Method Extended Quadratic Ea Conditional Formula Extended Quadratic with Adjustments Methods Mixed-Up Extended Quadratic Ea Derivation of the Cubic Ea GCD Method Word Problems Radical Method Bisection Method Newton's Special Case Lucas & ? Fibo ? Method(s) Geometric Methods Polynomial Search Methods Quartic Quintic Generalization of Methods Spacial Beauty: Geometric Solution Polynomials in Two Unknowns: ? Linear ? Quadratic ? Cubic ? Quartic ? Quintic Generalization of Methods Transcendental Polynomial: e Appendix Tables Bibliography Index Definition of Some Terms and Notations = EQUIVALENCE a = a + ADDITION - SUBTRACTION a + b = c c - b = a > GREATER THAN < LESS THAN <> UNEQUIVALENCE a + b > a a < a + b a <> b >= GREATER THAN OR EQUAL TO =< EQUAL TO OR LESS THAN if a >= 0, then a is positive if a =< 0, then a is negative | | ABSOLUTE VALUE |a| >= 0, regardless of a * MULTIPLICATION ^ EXPONENTIATION a * b = ab a ^ 2 = a * a ( PARENTHESIS ) [ BRACKETS ] (a + b) * c <> a + b * c [a * (b + c) + d] * e <> a * (b + c) + d * e < SQUARE ROOT > { CUBE ROOT } ^ 2 = a {a} ^ 3 = a << FOURTH ROOT >> <> ^ 4 = a : PROPORTION 3 : 2 :: 6 : 3 ::: 3 : 4 :::: 1 : 1 [ [ (3 : 2) :: (6 : 3) ] ::: (3 : 4) ] :::: (1 : 1) [ [ (3 / 2 = 1.5) / (6 / 3 = 2) = 0.75 ] / (3 / 4 = 0.75) = 1 ] = (1 / 1 = 1) / normal division \ integer division a > b a > c b > r c > r r is the remainder after dividing a by b, or a by c If r = 0, then b and c are factors of a If r <> 0, then b and c are factors of (a - r), and partial factors of a a - r = b * c (a - r = b * c) + r a = b*c + r (a = b*c + r) / b (a = b*c + r) / c a/b = c + r/b = d a/c = b + r/c = e a\b = c a\c = b d and e are quotients of a/b and a/c, respectively; while, c and r/b are both partial quotients of a/b, and b and r/c are both partial quotients of a/c if r = 0, then a/b = a\b if r = 0, then a/c = a\c if r <> 0, then a/b > a\b if r <> 0, then a/c > a\c thus, c =< d and b =< e and b and c are INTEGERS: for example, ...-4,-3,-2,-1,0,1,2,3,4,5..., and d and e are RATIONALS: i.e., ...-3,-15/7,-2,-3/2,-1,0,4/5,1,5/3,2... There can be any number of factors, partial or complete, to a composite number. These factors may have a positive, or negative, exponent: a = (b ^ (+f) * c ^ (-g) *...) + r Integer division is the extraction of all factors (be they partial or complete; r<>0 or r=0) that have negative exponents. Almost all, or none, of the factors with positive exponents will be extracted. This will result in a (composite of) factor(s) equivalent to an integer. Normal division is pretty straight forward, while integer division is a creative process in that any divisor and structure of factors with remainders may be imagined. [ PARTIAL QUOTIENTS OF THE EUCLIDEAN ALGORITHM ] and [ REPEATING CONTINUED FRACTIONS ], or simply [ QUOTIENTS ], appearing either as [q1, q2, q3,...q?] or [q1, q2,...q?;...?] ( GREATEST COMMON FACTOR/DIVISOR ) < LEAST COMMON MULTIPLE > (a, b, c,...) (see: chapter two, subtext: sifting (see: chapter two, subtext: a pair of numbers for their greatest searching for a pair of numbers' common factor/divisor) polynomial--towards the end) Ea the Euclidean algorithm GCF GREATEST COMMON DIVISOR LCM LEAST COMMON MULTIPLE (See end of chapter two) Inf INFINITY ~ APPROXIMATELY ` DEGREE (ANGULAR) =: APPROXIMATELY EQUAL =; ALTERNATING APPROX. rounding to the nearest alternately rounding either up or down to the decimal position nearest decimal position in a series of roundings (decimal position is any position of choice: one's place, ten's place; tenth's place, hundredth's place, etc.) Introduction Ars Longa, Vita Brevis: Art (is) long, life (is) short. (arz longa, veeta brevis) The purpose of this text is twofold: 1) To delve into the relationship between polynomials and the skill of artistic design: what is mathematics' contribution to art? 2) To approximate the solution of simple polynomials using varied methods Polynomials are made to yield their light on these subjects with the use of geometry(and some trigonometry) as the central focus and supported by alge- braic algorithms where possible. Polynomials are general programs for beauty. They are in cryptic form. Cryptic implies that polynomials are both terse as well as mysterious: how can so much information be incapsulated within their simple selves? Polynomials do not contain factors, they imply them. It is their accurate or inaccurate inter- pretation which furnishes scientific or artistic use. Methods of approximation are derived from polynomials. The more factors contained within a polynomial, the more various are the methods of approximation. Methods once derived are then good for all polynomials of the same type: golden, similar degree, same number of unknowns, etc. Different methods all result in either numeric values, or a series of either angles or ratios which converge on approximating either: the geometric forms that act as storehouses for the roots of their respective poly- nomials, or on the factors' numeric values. Numeric values are used in science for their accuracy or as intermediate steps for aesthetic applications. The latter produce a vocabulary of aesthetic building blocks. These building units are the pure temperment of their respective fields. Their progression towards accuracy is arithmetically inclined. Equal temperment is a reasonable fascimile of the pure along the lines of a geometric progression. Scientific accuracy is taken to be equivalent to an answer. Pure temperment approximation is one step removed from being an accurate depiction of reality, while equal temperment is two steps removed. Different methods of approximating a polynomial's roots, along with variations within methods, produce individually variant building blocks---although their net result is the same when compared one to the other. Polynomials classified in terms of either their analogous geometric forms, the quotients of their derived quadratic Euclidean algorithms, or the conditions of their polynomial's coeffi- cients, are equivalent to different styles of beauty. The converging trend of a series of ratios may be conveniently divided into two segments: those from the beginning of the trend on up to a point that is perceived by the mind as being no different than the root it is approximating, and those ratios from that point onward sharing the same characteristic as the turning point. (The mind's method of of determining the cut off point is still in question.) The first segment con- stitutes art; the second, science. In effect, science in its bid to be accurate looses its field of discrimination: the minute a field of ratios are no longer inaccurate, there is no basis for differences of opinion. Art never started any war, but gallantly did what no science could ever do: it kept differences simple. Once differences are perceived as becoming too complicated to appreciate in their construction, their differences vanish. Art never starts a conflict because it assumes no similarities. Accuracy does, for it is only good as an ideal; in actual practice it is a misconception. I may have given the impression that this work is on computation theory. It is, but with a focus on those methods for solving polynomial's factors which take their time in their developement. It is only these which have the highest value in art. Also included are those methods which are speedy. They are the norm among mathematicians and are included for their supportive help as well as for contrast. The Definition and Scope of Beauty Beauty exists at every plane of reality: from the simplest dimension to infinity. Beauty in each dimension is represented by a simple polynomial of a determinate number of unknowns: 0 Unknowns >>>> Linear (Elemental) Beauty 1 Unknown >>>> Planar (Physical) Beauty 2 Unknowns >>>> Spacial (Vital) Beauty 3 Unknowns >>>> Personal (Emotional) Beauty 4 Unknowns >>>> Concrete Mental Beauty 5 Unknowns >>>> Abstract Mental Beauty 6 Unknowns >>>> Intuitive (Insightful) Beauty 7 Unknowns >>>> Guardian Angel (Self) Beauty Indeterminate Unknowns >>>> Transcendental Beauty Specific types of polynomials over each unknown represent different styles of beauty. There are four broad categories: 1) General Beauty 2) Golden (Ideal) Beauty 3) Semi- (-General or -Golden) Beauty 4) Hybrid Beauty Among Related Forms General beauty incompasses and extends beyond the three latter types. All four will be discussed further in the chapter on aesthetics. Since this definition of beauty is primarily geometric, mathematics is used solely to seek out the application of programmed beauty using algorithms that satisfy the geometrical requirements. Approximate Solution Methods There are infinitely many ways of approximating polynomials' factors using repetitive partial solution methods, most of which are self-looping. Each succesive degree polynomial has more methods then the previous due to their in- creased number of coefficients, roots, or number of layers of radical signs in the algebraic solution form for roots. Geometry adds a fourth element of varia- tion by increasing the number of different star figures that can be contained within forms of increased number of outer limits(points, sides, faces, etc.). Many of these various methods are only good under speciic conditions; under others they break down. But a few are good for all conditions. Thus, there are two broad categories: 1) Dependable 2) Conditional The structure of these methods are based on: 1) A Polynomial's Coefficients, Roots, and Values Under the Radical 2) Polynomials of Lessor Degree in Multiple Unknowns 3) Geometry and Trigonometry The two broad categories contain: 1) Dependable: a) Trigonometric Method b) Polynometric Series Method 2) Conditional: a) Everything Else Dependability defines the minimum requirement for calculating elements of aesthetic design: 1) From Partial Solution Theory: Roots of Polynomials Can Be Approximated 1a) From Trigonometry: These Approximations Can Be Represented Using Ratios of Shared Terms 1b) From Polynomials of Multiple Unknowns: These Shared Terms Form A Polynometric Progression ? Semi-dependability results on the condition that: 2a) We restrict ourselves to golden polynomials: The Golden Method 2b) We restrict ourselves to cubic+ polynomials and the primary root of quadratic polynomials: Radical Method 2c) We restrict ourselves to a specific degree: Coefficient Methods 2d) We know the first or all of the roots: Geometric Methods (The Geometric Methods are patterned from the Coefficient Methods) Many of the last type require preknowledge of the roots in numeric form. Approximation methods may be arranged in fourteen sub-categories: For Art: Requiring fore-knowledge of a polynomial: 1a) Polynometric Method Requiring only the degree of an unknown polynomial: 1b) Golden Method For Art and Science: Requiring fore-knowledge of a polynomial: 2a) Complete Solution Methods 2b) Continued Fraction Methods Requiring preknowledge of a polynomial and its roots: 2c) Coefficient Methods (also good for constructing polynomials from word problems) 2d) Geometric Methods Requiring only knowledge of a polynomial's roots: 2e) Incremental Method Requiring preknowledge of a set of ratio terms: 2f) Polynomial Search Methods For Science: Requiring knowledge of a geometric form: 3a) Trigonometric Method Requiring knowledge of a polynomial: 3b) Prime Factorization Methods 3c) Radical Method 3d) Averaging Methods As a Novelty: Requiring knowledge of a polynomial: 4a) Extensions of Continued Fractions 4b) and their Conditional Formulas If I had to highlight just a few techniques for their ease and simpli- city, it would be: 1) Golden 2) Polynometric 3) Radical 4) Trigonometric Each achieves its results through increasing amounts of difficulty: 1) Golden: Requires only the degree of an unknown golden polynomial 2) Polynometric: Requires a general polynomial 3) Radical: Same as 2) plus the ability to take fractional exponents of numbers: x ^ 1/y 4) Trigonometric: Same as 2) and 3) plus access to cosine functions The situation is a simple one of interrelationships: 0) Geometry a) These are not the graphs of polynomials, but store the factors as ratios among boundaries of geometric forms 00) Angles that are equivalent to ratios a) A Progression of angles which converge on approximating geometric forms' inner angles 1) Polynomials 2) Factors of Polynomials 3) Ratios which Approximate Factors a) Ratio Terms are shared and contained within Sets of Parallel Terms Progressing as a Series ? The majority are capable of producing ratios: * The Practical Ones ^ Novel . Numeric (Continued Fraction) Capacity / Ratio Capacity | Semi-Integrated Series for Shared Ratio Terms \ Integrated Series for Shared Ratio Terms : Obvious Integration Using Simple Algebra ; Subtle Integration Using Complicated Algebra > Fast Progression and Approximation < Slow Progression and Approximation (All roots of polynomials are integrated. The integration is fully streamlined if the polynomial's coefficients are all rational. It is the appar- ent sense of mutual relatedness among integers used as the numerator or denomi- nator terms of approximating ratios that is being designated as Obvious or Subtle.) 1) Parallel/Serial Methods: a) .| Derived ===== for Continued Fractions and Ea.s of all Poly.s in Gen. .\* Useful at the quadratic level b) \*< Incomplete === GCD Algorithm (also serves all Golden Polynomials) used to find the gcd of sets of numbers of any number of elements and for solving the special case polynomials: x^a - b = 0 c) \*< Golden ======= for all Golden Polynomials (.quadratic) d) ? ^> Extended (the Derived Quadratic) with Adjustments ==== for Coef.s and and Quotient/Coef. relations of all Poly.s in Gen. e) |> Mixed-Up Extended with Adjust.s ====== for all Gen.(? 3rd+)Degree Poly.s 2) Serial Methods: a) /*<: Intermittent (Multiplicative) Progression, with approx. rounding b) |<: Arithmetic/Polynometric (Mixed, Arithmetic/Geometric Prog.) (With, or without, Geometric in conjunction) ===== hi/lo version only ? \*< Only for general quadratic or golden of any degree c) .*<; Radical ===== for all Polynomials in General d) ./*>: Newton's Special Case (Radical Averaging) === for Positive Roots of: ax^? - b = 0 e) ? |\./< Bisection ===== for Gen. Poly.s (presumes there is no sequence to the roots and picks one at random for solving) f) /^>: Incremental ======= 3) Direct Methods: a) ? ./*<; Trigonometric ====== b) ./*>: Immediate (Exponential) Progression, in conjunction with lo/hi rounding c) .*>; Lucas (expon. sum. of roots) ===== for general polynomials d) .*>; Fibonacci (modified Lucas) ======= for general quadratics e) .*>; Quadratic Formula ======= for general quadratics f) .*>; Conditional Formula ======= for general cubics(+) g) ?Euler's? Method ======== a simple relation between the square root of a number and its incomplete partial quotients 4) Experimental Methods(Prime Factorization): a) .*<: Integer Selection ======== b) .*<: Random Integer Selection ====== c) .*>; Base Conversion ======== ??????????????? d) .*>; Sum of Powers of Bases ======= 5) Polynomial Search Methods: Single Solution for Monic Polynomials: a) Complete (GCF) ======= b) Singular Approximation ======= S.S. for Non-Monic Polynomials: c) Incomplete GCF plus Analysis ======== Multiple Solutions: d) Inverse Extended ? Derived Quadratic Ea====== (continued fractions) The incomplete is the gcf searching method; it is good for all degrees. Both the extended and the dynamic are integrated. The extended is for golden polynomials. The dynamic is for generally aesthetic polynomials from the cubic on up. The derived is related to the degree of the polynomial. The quadratic is derived, but taken to higher degrees it is ideal. Derivation at higher degrees is not ideal because simultaneous computation of a polynomial's factors is not in a simple and predictable form. Subtext: Polynomials The Pythagoreans had a yen for simple beauty. Simple integers are easy enough to appreciate. They made the mistake though of failing to appreciate the more complex beauty of irrationals. We get around irrational's inaccessibility by approximating them with fractions composed of integer numerators and denomi- nators. Integer beauty is linear. A simple line segment depicts beauty of no unknowns, in as much as we can designate that segment as a unit of some sort of measurement. So long as we confine ourselves to the first dimension, measurement and the comparison of different measurements is rational. Polynomials of no un- knowns are straightforward in the mechanics of their factoring, since we are merely factoring numbers, not unknown quantities. A polynomial of no unknowns is nothing more than an integer or statement of integers(constants): 5 (4 + 6) / 2 = 5 a + b * c = d, where a, b, c, d are known. Polynomials are pretty neat. If we wanted to factor a composite number as integer factors, we would simply have a * b = c. But if we set a and b equal to zero as, x = a and y = b then x * y = c x - a = 0 and y - b = 0 then (x - a) * (y - b) = 0 This last polynomial is nice, but it >>>> xy - bx - ay + ab = 0 doesn't give us a form that is easy to solve. Solutions are much easier if the polynomial is in one unknown. So let's change the setup. The above example takes two different polynomial systems, x and y, and combines them. X and y are dif- ferent variable names from the start because they were not intended to be com- bined into one system. So let's start them off on a better footing. Let's assume that they are one system making two alternate attempts at solving for either a or b separately, x = a or x = b either, x * a = c or a * x = d x - a = 0 or x - b = 0 then (x - a) * (x - b) = 0 x^2 - ax - bx + (-a * -b) = 0 This last polynomial is very easy to >>>> x^2 + (-a - b)x + ab = 0 solve for either a or b separately, but not both at the same time. A polynomial is a composite expression composed of factors known as algebraic expressions. A polynomial can also be a factor of larger polynomials. An algebraic expression may or may not be a polynomial; that is, it may not be factorable. A simple algebraic expression in one unknown has the following appear- ence: 0 = ax + b Where a and b are known and x is unknown. A simple polynomial in one unknown is built up from this basic unit by multiplication: I 0 = ax + b II 0 = (ax + b) * (cx + d) = acx^2 + (ad+bc)x + bd III 0 = (ax + b) * (cx + d) * (ex + f) = = acex^3 + (acf+ade+bce)x^2 + (adf+bcf+bde)x + bdf etc................................................. Where a, b, c, d, e, f, etc. are known and x is unknown. A simple algebraic expression in multiple unknowns is an extension of the singular: 0 = aw + b 0 = aw + bx + c 0 = aw + bx + cy + d 0 = aw + bx + cy + dz + e etc...................... Where a, b, c, d, e, etc. are known and w, x, y, z, etc. are not. A simple polynomial in multiple unknowns, is built up by multiplying any combination of the above potential factors. The various factors may or may not share similar unknowns: IV 0 = (ax + b) * (cy + d) = acxy + adx + bcy + bd V 0 = (ax + by + c) * (dz + e) = adxz + bdyz + aex + cdz + bey + ce VI 0 = (ax + by + c) * (dx + e) = adx^2 + bdyx + (ae+cd)x + bey + ce etc.............................................................. Where a, b, c, d, e, etc. are known and x, y, z, etc. are not. Roman numeral I above is not only a potential factor of a polynomial in one unknown, but it is also a linear, or first degree, polynomial in its own right. In roman numerals II and III above, notice how the coeficient factors are collected under shared "x" terms. These three examples are said to be fully coordinated with respect to the relationship among their factors. This is because of the singularity of unknowns. Roman numeral IV and V are the least coordinated---no unknowns are shared among the various factors. Roman numeral VI is partially coordinated. The algorithm used to interpret polynomials as aesthetic systems of internal harmony is the Euclidean. But unlike the common understanding of limiting the Ea to its quadratic polynomial derivation, it is actually derivable from any degree polynomial. It is used to approximate a solution(partially solve) the polynomial from which it is derived. Its activity is based on two simple premises: algebraic expressions in multiple unknowns and self-looping(the recycling of outputs as next inputs) to create a plausible series of sets of solutions. The solutions are linked arithmetically and geometrically to each other and the initial starting values creating a diversity of possible solu- tions. This linkage is the way self-looping algorithms integrate(non-calculus) sets of solutions with each other in a harmonious way. The coordinated propor- tional relations among a set of parallel series terms approximates the factor's values within a polynomial. So long as we remain loyal to any one particular series sets of solutions, no exteme discordant harmonics is experienced. ? save the mild discord of proximity among coexisting values(close enough to differ without approximating: the result is the perception of phase differences among values assumed to be similar--- incoherence). The injection of an exponential multiple, of what a series set is attempting to approximate, into the series as a part of its vocabulary produces predictable effects that are considered acts of destructive, as opposed to constructive, power. Harmony is power, but of a constructive type. Unpremedi- tated mixing is cacophony. I know this is a little much to take in at this time, but bear with me. ? Since polynomials of singular unknown are the most definitive to solve and the most harmonious in their internal coordination, I will restrict my dis- cussion to their partial solution in terms of multiple unknown expressions of lessor degree. The use of the Ea for multiple unknown polynomials is not unlike the singular, only more complicated. If/then relationships would be the rule, making a tree of many branches for equally plausible solutions---not unlike interactive fiction. For brevity, coefficients of any "x" term may be labeled according to the exponent on the "x": from zero to the degree of the polynomial. For example: 0 = ax^n + bx^(n-1) + cx^(n-2) +....+ px^2 + qx^1 + rx^0 The "n" coefficient is "a" The "2" coefficient is "p" The "0" coefficient is "r" Subtext: Incremental Approximations Through Trial and Error A rational or irrational value may be approximated using ratios of numerator/denominator pairs of integers. The only stipulations are: 1) The first ratio is 1/1. 2) Each succeeding numerator, or its denominator, is minimally larger than its predecessor, such that 3) Each succeeding ratio is more accurate than the previous one. Example: 4.17 The error is figured as the absolute value of the difference between unity(one) and the proportion between 4.17 and its approximating ratio. | 1 - (4.17 / ratio) | * 100% = + error % 1/1 = 1 is 317% off 2/1 = 2 is 108.5% off 3/1 = 3 is 39% off 4/1 = 4 is 4.25% off 13/3 = 4.3333 is 3.7692% off 17/4 = 4.25 is 1.8824% off 21/5 = 4.2 is 0.7143% off 25/6 = 4.1667 is 0.0800% off 121/29 = 4.1724 is 0.0578% off 146/35 = 4.1714 is 0.0342% off 171/41 = 4.1707 is 0.0175% off 196/47 = 4.1702 is 0.0051% off 221/53 = 4.1698 is 0.0045% off 417/100 = 4.17 is 0% off ? Two themes: the definition of partial division and simple polynomials form the basis to this text, while trial and error approximation, is the back- drop. From the above ratios are plucked choice selections representing beauty, integrated linkage, and the polynomials from which they are derived. This text uses algebra and trigonometry to explain a hypothetical defi- nition of aesthetics. To prepare a little for what is to follow, let's take a look at a few geometric figures along with some predefinitions. The circle is ideally beautiful. It exhibits balanced proportion. Regu- lar polygons of odd number of sides inscribed within it offer a golden view of beauty. Such polygons can not be folded any number of times along any diameter without distorting the polygon. They have only radial symmetry. If the polygons were even sided, their beauty would be radially incomplete. If a polygon were to have its corners clipped off creating an inner polygon of some multiple number of sides to the original, then the beauty born out of their relationship would be a hybrid. If the circle which surrounds them were to become squashed as an ellipse, their beauty would no longer be ideal but merely general. Certain algorithms are used to elicit from simple polynomials the voca- bulary of stylistic aesthetics. These algorithms are based on the definition of division(both partial and complete)---the linear polynomial in any number of unknowns---plus a few twists. The infinite scope of polynomials yields an infi- nite variety to beauty, while the imperfection of approximation yields an infi- nite vocabulary for expressing beauty. It is no longer necessary to withhold beauty within the context of the beholder. Beauty can be defined mathematically as the integration to the art of approximating perfection. Perfection is a potential with which to play and we are given infinity with which to play it. Beauty automatically incorporates power within itself, but such power is totally subservient to the strict discipline of unity of purpose. Hence, beauty is spiritual power. When an ego(that's the expression) is a reflection of the system of beauty to which it is born(that's the potential), then the purpose of life is fulfilled by an incremental degree. Systems of beauty are singularly planetary in their scope. Hopefully, this allegory will come across with a minimum of conjecture. A Note to Educators Educational theory presupposes there is a time for everything and every- thing in its time. There are exceptions: sometimes there are individuals who will not want to learn a thing until later than usual or possibly never, while others may want a taste for curiosities' sake only to put off more thourough study until lator. But generally there are seven periods of a person's develope- mental focus when certain concerns and activities appropriate to that period are endeavored for: Seven years Developement of the: 1st " Physical Body 2nd " Health and Vitality 3rd " Emotions, Personality, Drives, and Relations 4th " Concrete Mind 5th " Abstract Mind 6th " Intuition 7th " Self Purpose, or colloquially: Mid-Life Crisis 8th set of 49 years: A new cycle that Transcends the previous It is hoped that this knowledge be used to benefit individuals at all ages in a manner appropriate to their appetite: Seven Years Spent Studying: 1st " Play at: Music, Movement, Arts & Crafts 2nd " Personal Hygiene with: Body Care(including Sex Education), Diet, Exercise, Recreation & Leisure(Reading, Writing, the value of Natural Resources, etc.), Self Culture 3rd " Interactive Psychology, Sociology, Anthropological Study of Cul- tures Past & Present, World Literature, History, Geography & Government, Dance, Theatre, etc. 4th " Natural Sciences, Crafts & Job Skills(Business Admin., etc.) 5th " Theoretical Sciences 6th " Graduate Work 7th " Self Application of all of the above towards a self-guided pur- pose I would classify this as an abstract study of aesthetic theory appro- priate to the 5th seven years, or as a job skill in the use of computer software already developed for this purpose and as an introduction to life and natural science appreciation at the 4th, or as listening and exercise therapy for good health during the 2nd seven years, or as musical instruments and singing already prepared by adults for young people to use in the 1st. It is up to educators to be prepared to apply this knowledge tactfully where appropriate. It is better to render rights and privileges based on academic comple- tion, rather than on age requirements alone. That may or may not be the case with licencing to drive, but should also be for all others: voting, etc. ? A scaled down version of this subject may be taught as an introductory subject on aesthetic theory. It is a good answer to an often posed criticism: what am I studying this math for? It is not necessary to get into number theory. Geometries' role in determining interrelatedness among ratio terms is a key in- gredient. Its ability to catalog beauty into a hierarchy of developement and de- fine it by type is simple. Polynomial inclusion is supplemental. Algorithms for producing ratios may be confined to two: 1) Incremental 2) Golden 7) Incomplete (for accurately/approximately reducing fractions) with the inclusion of eight more for hungry students(especially if there is a calculator or computer available for the last five: 3) Intermittent >>> (may be confined to the 4) Immediate >>> quadratic for simplicity) 5) Derived (Quadratic) (with and without a seed) 6) Arithmetic (Quadratic) ? 8) Prime Factorization Methods (four) 8) Radical 9) Special Case 10) Trigonometric 11) Lucas 12) Fibonacci The real touch will be to inspire their confidence in the purposeful role math and geometry plays in determing our sense of beauty and its expression in nature. Basic music theory can be presented concerning how intervals are determined in both pure and equal temperment systems and their effect on sound quality. The role of pure temperments' intervals in creating a sense of melodic themes may be touched upon. Rhythm's analog to interval construction should not be left out. The use of proportion and angle, whether it is in design or dance theory, is a fertile field to incourage their pursuit. Game theory, emphasizing classroom experience, will bring a welcome relief. And finally, the as yet unexplained ocurrence in nature of both the Fibonacci and Lucas series, along with man's as well as nature's approximate use of the golden mean, is a hot topic for speculation and discussion. Chapter One: Linear Polynomials (x = -b/a) * a (ax = -b) + b ax + b = 0 This is a linear, or first degree/order, polynomial in one unknown. Looks simple? "A" and "b" are positive or negative integers, while "b" can be zero. Dividing: (x = a) / a x/a = 1, with no remainder, hence "A" completely factors x. In this illustration, restricted as we are to a very simple case, "a" is the only factor of x---barring one. x = a + 0 Dividend divided by divisor Equals Quotient Plus Remainder Over divisor ( D / d = Q + R / d ) * d D = d * Q + R But we are barring one as a factor of x, and admitting no factors at this point to "a". So we will need to modify this formula to read: D = dQ + R , making dQ a single entity. "dQ" is both the factor, as well as the divisor, of "D". Notice how division is little different from a linear polynomial. Substituting: x = a + 0, for D = dQ + R "A" becomes the full quotient and divisor of x. ( Quotient = q ) = a Being the only quotient/divisor of x, it is written thus: [q] or [q1] If: q = + 1 or q = - 1 Then: x + a = 0 and x - a = 0 , are "golden" polynomials for the first degree, or polynomials of "ideal" propor- tion. "A" is its ideal root. Written as: (+ or -) a/1 it is its "golden" ratio. If the first quotient of a polynometric series of quo- tients is either positive or negative, then two polynomials with roots equal in absolute value but opposite in sign result. If the first quotient is either positive or negative one and all the other These are the linear ideal polynomials. Later on in the chapter on Aesthetics, these terms and others will be discussed in more detail. Chapter Two: Quadratics Subtext: Solution We saw in the introduction how a polynomial can be produced by multi- plying two factors. Now let's see how it can be factored. 4a * (0 = ax^2 + bx + c) -4ac + (0 = 4a^2x^2 + 4abx + 4ac) b^2 + (-4ac = 4a^2x^2 + 4abx + 4ac - 4ac) (b^2 - 4ac = 4a^2x^2 + 4abx + b^2) ^ 1/2 -b + (+- = 2ax + b) (-b +- = 2ax) / 2a -b +- �� = x 2a If we take a look at this, it looks like: -b �� +- �� = x 2a 2a �� Ŀ -b �b^2 - 4ac� ^ 1/2 �� +- ��ĳ = p +- q^1/2, where 2a � 4a^2 � �� -b b^2 - 4ac p = �� and q = �� 2a 4a^2 Now let's create a polynomial from: x = p +- q^1/2, by multiplying: x = p + or x = p - (x = p + ) - (p + ) or (x = p - ) - (p - ) 0 = x - (p + ) or 0 = x - (p - ) 0 = (x - (p + )) * (x - (p - )) 0 = x^2 - (p - )x - (p + )x + (p +- )^2 0 = x^2 - 2px + (p^2 + p - p - q) 0 = x^2 - 2px + (p^2 - q) Remembering, (0 = ax^2 + bx + c) / a 0 = x^2 + (b/a)x + c/a, thus b/a = -2p and c/a = p^2 - q, thus (b/a = -2p) / -2 b/-2a = p, substituting: c/a = (b/-2a)^2 - q q - c/a + (c/a = (b/-2a)^2 - q) q = (b/-2a)^2 - c/a Since: x = p +- , then x = b/-2a +- <(b/-2a)^2 - (c/a * 4a/4a)> x = b/-2a +- x = b/-2a +- /2a, thus -b +- x = �� 2a The advantage is in the ease of remembering that all quadratic solutions are of the form: x = p +- q^1/2 If it is too difficult to remember the quadratic formula, or how to complete the square of a second degree polynomial, you can always create the quadratic formula by deriving it from this one. Subtext: Sifting a Pair of Composites for Their Greatest Common Factor/Divisor If, a * b * c = d and b * c * e = f, then the gcf of (d, f) = b * c To find the gcf of d and f: sort as, Column 1 = d and Column 2 = f, or Column 1 = f and Column 2 = d, such that C1 is less than C2 integer divide: C2\C1 = q1, -0 -(q1 * C1) and subtract, �� C1 C2 - (q1 * C1) swap and equate, C2 - (q1 * C1) as the new C1 and C1 = new C2, so that new C1 < new C2 Go back and repeat integer division, subtraction of quantities, swap, and equate steps until Column 2 zeros out. Column 1 will then be the gcf of (d, f). The q1, q2, q3, etc. terms generated by this process are called partial quotients of d and f. They are written thus: [q1, q2, q3,....] If any part of the sequence repeats itself, a bar is placed above the repeating terms. �� [q1, q2, q2, q2, q2,......] = [q1, q2] Writing: (d, f) indicates that the gcf of d and f is being sought. If their gcf is 1, then they are relatively prime(to each other), in that they have no factors in common with each other, except the factor of 1. Example: 4.17 >> (417, 100) q = C2 \ C1 C1 < C2 -0 - (q * C1) �� C1 C2 - (q * C1) 4 = 417 \ 100 100 < 417 -0 - (4 * 100) �� 100 17 5 = 100 \ 17 17 < 100 -0 - (5 * 17) �� 17 15 1 = 17 \ 15 15 < 17 -0 - (1 * 15) �� 15 2 7 = 15 \ 2 2 < 15 -0 - (7 * 2) �� 2 1 2 = 2 \ 1 1 < 2 -0 - (2 * 1) �� 1 0 The GCF of 4.17 and 1 is: (4.17, 1) = 1 While the partial quotients are: [4, 5, 1, 7, 2] This is accepted as the Euclidean algorithm, essentially. But we will call it the gcf algorithm. To do more then merely look up common divisors, let's first derive the fuller version from its corresponding polynomial. Subtext: The Quadratic Euclidean Algorithm ax^2 + bx + c = 0 (ax^2 + bx + c = 0) / a (x^2 + (b/a)x + c/a = 0) - ((b/a)x + c/a) (x^2 = -(b/a)x - c/a) / x -c/a x = -b/a + ----- x The x on the right hand side of the equation is known as the input variable. It can be set equal to anything, provided it isn't something that is lacking a definition for computation---zero in this case. But division by zero shouldn't be a hindrance: see Appendix. The x on the left hand side is the output variable. This is known as a continued, or partial, fraction of x. It is one of several possible partial solution methods for what will be called the primary, or first, root/factor/divisor of a quadratic. It is partial because it is incom- plete: there is no definite answer as to what x is. But it doesn't matter what value for x is chosen at any time; the structure of the polynomial, as well as its slant(we'll get to that in higher orders), molds our guess into an infi- nitely approximate answer. All we have to do is keep feeding the output back in as the next input. Since division of x^2 by x was used to derive this method, it would be good to substitute a ratio for x; we'll call this ratio h/j. With every successive pass of h/j, it will begin to approximate the value of the primary root with greater and greater accuracy as a ratio, rather than as a number. The beauty of this substitution is in the creation, or translation, of what would otherwise be an example of just one more dry and humanly inconsequential mathe- matical formula into a definition of what constitutes beauty, aesthetics, and the power of destruction(better known for its purifying value when used in mode- ration). In number theory, these series of ratios are termed: approximates, or convergents, of either a numeric value or of a ratio, in that they approximately converge on their equivalence without ever reaching it. The answer is the limit of this series. The process is regulated by its overall structure in general, as well as by the values of -a/b and -c/a, in particular. Let us continue with the derivation: Substituting: h/j = x -c/a j h/j = -b/a + ---- * --- h/j j � � � h � -c/a * j h/j = -b/a * ��ĳ + �� h � h � � h (-b/a * h) + (-c/a * j) �� = �� j h Since, h : [(-b/a * h) + (-c/a * j)] :: j : h Then: h = (-b/a * h) + (-c/a * j) And: j = h With a slight modification: Series 1: h = (-b/a * h) + (-c/a * j) [+ (s)] Series 2: j = h What we have here are two infinite series that run in parallel with one another: a series of h's and j's. These two series constitute a quadratic, or second degree, Euclidean series that is partly both arithmetic as well as geo- metric in its structure, while approximating a geometric series in its output if we divide at any point along the way: output h divided by output j. The two series will also be a carbon copy of each other accept for one addi- tional term at the beginning of series 2 which will offset the two series in relation to each other. I am referring here to the comparison of parallell terms. The standard approach to the building of a series table is to initialize input h in both series with the value of one and input j in series 1 with the value of zero. An addi- tional seeding agent(symbolized here as s) is sometimes used, but only during the second computation. This seed will offset both series from what would be their norm, since they are computed in parallel sharing the same h and j, but the continual process of recycling the previous output as the new input will approximately take them forward to the same h/j ratio. This will create a two-dimensional table extending to infinity in one direction. The table can further be extended infinitely long in a third dimension by multiplying this table by a progres- sion of integers. The ancient Egyptians may have applied this by using the polynomial: x^2 - x - 1, seeding the second calculation with the number 3, and creating positive integer multiples at least as far as the 48th. In doing this, they may have managed to compute a pragmatically accurate trigonometry in that 48 divisions of a circle's circumference in radians, rounded to four decimals, appears as integer values, along with: 45 degree incre- ment angles of a 360 degree circle, the powers of phi--the golden ratio--from negative seven upwards to infinity in three decimals, and Egyptian measuring units (Refer to Tables). They were savy architects, using both pi and phi for the building of pyramid Khufu at Gizeh. (Refer to P. Tompkins in the bibliography). If this polynomial's Euclidean series is initialized without a seed, or the use of multiples, then the Fibonacci series results. If it is seeded with the number one, then the Lucas series(pronounced Lu-cah`, from the French) occurs. ? Seeded with -4, it becomes the Taylor series. The features of a polynomial are mainly evidenced in what is called its primary root. The primary root is equivalent to the first root. It can be approximated as the ratio between successive terms of series 1 or 2. The auxiliary root is a derivative of the primary. When all the partial quotients are positive, the roots arrange themselves in sequential order from greatest to least in absolute value. After recycling the output of the formula however many times one desires, the roots may then be represented in their approximate value in a number of different ways: q1 = - b/a q2 = - c/a h and j are two unknowns forming a ratio set for the first root sets of h and j are subscripted(indexed) with "?" Series 1: h(?+1) = (q1 * h?) + (q2 * j?) [+ s] Series 2: j(?+1) = h? h0 = j1 = 1 j0 = h(-1) = 0 (*) = undefined ?(infinity implying anything, not implying relative or absolute bigness or smallness) Set: -1, 0, 1, 2, 3, ......, ?-1, ?, ?+1, ?+2, ......, Series 1: h(-1), h0, h1, h2, h3, ......, h(?-1), h?, h(?+1), h(?+2), ......, Series 2: (*), j0, j1, j2, j3, ......, j(?-1), j?, j(?+1), j(?+2), ......, Because it is good to analyze this set of parallel progressions both horizontally and vertically, it is better to overlook set negative one: (0, *). h? j? Primary Root = �� = �� = Root 1 h(?-1) j(?-1) q2 * j? Auxiliary Root = -1 * �� = Root 2 h? If: q1 and q2 are both positive, then: | Primary Root | > | Auxiliary Root | | Root 1 | > | Root 2 | Whenever: q1 and q2 are both equivalent to positive one, then approxima- tions of the roots may be written thus: h Root 1 = �� j j Root 2 = - �� h Dropping the subscripts assumes both unknowns are within the same solu- tion set. Later on for geometry we will be using auxiliary, as well as primary, roots, none of which are negative; but for accounting purposes, to fulfill the requirements of polynomial building of rational coefficients, it is necessary to negate every even numbered root in the sequence of roots. What sequence? Good question. (See Subtext: Approximation Methods, particularly under chapters three and four. More pointedly, see chapter on Geometry). The terms: -b/a and -c/a are the partial quotients of this quadratically derived Euclidean algorithm. There are two of them, because this is the fuller version. Common knowledge often times acknowledges only the first of these, because it is much easier to solve. These two terms would be written thus: [q1, q2;] The brackets identify these values as partial quotients; the commas seperate terms from one another; the semi-colon indicates the termination of one pass, or cycle, of computation while also inferring the potential endlessness of recycling outputs into inputs. Although it is harder to solve for two unknowns(two quotients), then it is for one, the repetition of these two quotients as a cycle of terms lends the appearence of a standing wave pattern. If the quo- tients had progressed seemingly at random as the gcf searching algorithm would have us believe, then no pattern would emerge. A polyno- mial is usually seen as a group effect of a number of factors just somehow for- tuitously working together to create a polynomial of rational coefficients. But the Euclidean algorithm is the image of a single, cohesive, slightly intricate engine, creating both polynomial and factors simultaneously as if one were no more important than the other. The engine's movement is imperfect, but goal oriented. If allowed to rework its self-image, any degree of perfection is possible. This is not unique to computation. In fact, all calculation of irra- tional values must rely on repetitive techniques, while some like this one add self-looping---the modification of feedback. Referring to the previous subtext on finding the gcf of two numbers(and their partial quotients at the same time, for that matter--remember q1?), the gcf algorithm would have paralleled the creation of a continued fraction looking like this: 1 x = q1 + �� 1 q2 + �� 1 q3 + �� etc......... Thus the quotients would be better represented as: [q1, 1; q2, 1; q3, 1;] since the number one is part of a quadratic process. But out of brevity, [q1, q2, q3,....] ? (.......) optional will do just fine, and is the standard, whenever we are merely looking up a gcf. Just be sure and write it like this: [1, q1; 1, q2; 1, q3;] rather than simplifying, if this: q1 x = 1 + �� q2 1 + �� q3 1 + �� etc............ is the case; for this is not a common calculation. If we wish the calculation to look like this: q2 x = q1 + �� q2 q1 + �� q2 q1 + �� etc......... q1 = -b/a and q2 = -c/a we will need to make the previous gcf searching algorithm look more like the derived Euclidean algorithm. Euclidean series construction works best for polynomials of real number roots. When roots are rationally complex, it will work most of the time. But when roots are irrationally complex, the Ea never works. Subtext: The Golden Ea Subtext: Searching For a Pair of Numbers' Polynomials--- ? Four Three Methods If: 0 = ax^2 + bx + c a = 1 and b and c are integers q1 = -b and q2 = -c h0 = 1 and j0 = 0 s = ? h1 = q1 * h0 + q2 * j0 j1 = h0 h2 = q1 * h1 + q2 * j1 + s j2 = h1 h3 = q1 * h2 + q2 * j2 j3 = h2 etc........................... h? = q1 * h(?-1) + q2 * j(?-1) j? = h(?-1) etc........................... Given k and m representing h? and j?, we will be looking for q1 and q2. We won't know of which values, either h? or j?, k and m stand for, so both options will need to be tested. Either: Option 1 is true Option 2 is true k = h? and m = j? or k = (-)j? * q2 and m = h? k/m = h?/j? = Root 1 k/m = [(-)j? * q2]/h? = Root 2 (the negative in front of j? may or may not be present) There are two methods. Method 1: Assume Option 1: Substitute Column 1 and Column 2 for k and m and sort, so that: Column 1 < Column 2 integer - 0 - (q1 * C1) :divide �� q1 = C2\C1 C1 C2 - (q1 * C1) factor C1 and choose a factor (q2) to divide C1 by, such that: (C1 / q2) > [C2 - (q1 * C1)] and q1 = (C1 / q2) \ [C2 - (q1 * C1)] We are done. Now, if we wish to check our answer, three approaches may be used. Continuing with the algorithm: swap, so that: the next C1 = C2 - (q1 * C1) and the next C2 = C1 / q2 C1 < C2 integer / q2 -(q1 * C1) :divide �� q1 = C2\C1 C1 / q2 C2 - (q1 * C1) (C1 / q2) \ [C2 - (q1 * C1)] = q1 ! check ! C1 = C2 - (q1 * C1) and C2 = C1 / q2 continuing with the algorithm until, C2 - (q1 * C1) = 0 or q1 <> C2 \ C1, in which case: s <> 0 and C2 - (q1 * C1 + s) �� choose some value C2 - (q1 * C1 + s) for s, such that: C1 = C2 - (q1 * C1 + s) and C2 = C1 / q2, and C2 - (q1 * C1) �� 0 ! double check ! Second approach: If: x = Root 1 or x = Root 2 x = k/m = h?/j? or x = - (m/k * q2) = - (j?/h? * q2), then substitute either root as x into the original polynomial: a = 1 and b = -q1 and c = -q2 ~0 = ax^2 + bx + c ! check ! (sort of, anyway) The accuracy of obtaining zero will be determined by the size of h? and j? and the magnitude difference between q1 and q2. Third approach: The b coefficient, of the above polynomial, is equal to the negative summation of both roots(refer to Introduction). b is an integer(a = 1) q1 = b ~q1 = - R1 - R2 ~q1 = [-(k/m) + (m/k * q2)] ~q1 = [-(h?/j?) + (j?/h? * q2)] ! check ! If it fails, then try option 2: k <> m Assuming: k = (-)q2 * j? and m = h? Proceed with method 1, but with both steps in reverse order: C1 = non-prime; whichever one satisfies Factor C1; choose a factor to be q2 and divide C1 by it so that both steps produce repeating q's(each q for its respective step): C2 = C1 / q2 C1 = the other value <> C2 -0 - (q1 * C1) q1 = C2 \ C1 �� C1 C2 - (q1 * C1) Continue with algorithm to check for repetition. If both values are non-prime, then be ready to try either value as the initial C1. When k/m is the first root, k > m. When k/m is the second root, k <> m. The other method to this madness. Method 2: Root 1 = h?/j? Root 2 = q2 * j?/h? Option 1: k = h? m = j? Coefficient 1 = -b/a = ~q1 = - Root 1 - Root 2 � � -k m � m � ~q1 = �� + q2 * �� * ��ĳ m k � m � � � � � -k � k � q2 * m^2 ~q1 = �� * ��ĳ + �� m � k � km � � -(k^2) + q2 * m^2 ~q1 = �� km Initially set q2 equal to 0 and solve for q1. Then set q2 equal to the previous value for q1 and solve for the next version of q1. Repeat until q1 is equivalent to q2 to some degree of accuracy. Assuming q1 >= q2, then: Round q1 up to the nearest integer. (1) Replace q1 in the equation with this best guess and solve for q2. �� Ŀ � -(k^2) + q2 * m^2 � � ~q1 = �� * km � km � �� Ŀ � ~(km * q1) = -(k^2) + q2 * m^2 � + k^2 �� Ŀ � ~(k^2 + km * q1) = q2 * m^2 � / m^2 �� k^2 + km * q1 �� = ~q2 m^2 (2) Round q2 down to the nearest integer. There is only one check method we can use here; using our newly found: -q1 = b and -q2 = c 0 = x^2 + bx + c Substitute: x = k/m Into: ~0 = x^2 + q1x + q2 If this fails, maybe q1 < q2; in which case: Round q1 down to the nearest integer; go back to step (1) and proceed through to step (2), but round q2 up. Check by substitution. If neither of these attempts succeed, then try option 2. All of this was under the assumption that -q1 and -q2 are integers(a monic polynomial for its degree). In order that the final form of the polynomial might be written with strictly integer, as opposed to rational, coefficients, the whole polynomial is multiplied by the least common multiple of the denomina- tors of -q1 and -q2. For example, if -q1 is 1 and -q2 is 1/2, then coefficient "a" would equal 2 in order that b might equal 2 and c be an integer equalling 1. a = 1 b = 1/1 c = 1/2 lcm of 1 and 2 = <1, 2> = 2 (0 = x^2 + x + 1/2) * 2 0 = 2x^2 + 2x + 1 If: a = 1 b = 1/2 c = 1/3 lcm of 2 and 3 = <2, 3> = 6 (0 = x^2 + 1/2x + 1/3) * 6 0 = 6x^2 + 3x + 2 The lcm is dependent on the gcf: If, a * b * c = d and b * c * e = f, then the gcf of (d, f) = b * c, and the lcm of = a * b * c * e It is found by: = (d * f) / (d, f), or = 1 / (1/d, 1/f) See the difficulty? "A" could be anything; the denominators of -q1 and -q2 could be anything. We won't be able to use integers to round q1 and q2 up or down to in method 2. Nor will we be able to integer divide two numbers to get q1 in method 1. What to do? ??????????????????????????????????????????????????????????????????????????????? Notice also that the multiplication of a polynomial by negative one does not change the roots in any way. It just gives us an alternate sign way of depicting a polynomial's roots: 0 = x^2 - x - 1 x = +1.618... or x = -0.618... (0 = x^2 - x - 1) * -1 0 = -x^2 + x + 1 x = +1.618... or x = -0.618... According to the definition in chapter one, ? 0 = x^2 +- x - 1 are the two golden, or ideal, quadratic polynomials because they satisfy the criteria of having the first partial quotient equal to either positive or nega- tive one and all of the absolute values of the remaining partial quotients equal to positive one(remembering that partial quotients are opposite in sign to coef- ficients at the quadratic level). Subtext: Word Problems ? Now the ancient Greeks had a different way of saying the same thing. They said, "What the former measurement is to the latter, the latter is to the former"(I'm loosely quoting Euclid?/Plato?). They were referring to the recipro- city of values: R1 = 1 / R2 (1) R1 / 1 = 1 / R2 Or, conversely: 1 / R1 = R2 (2) 1 / R1 = R2 / 1 Substituting "1/R1" for "R2" in equation (1): (1a) R1 / 1 = 1 / (1/R1) R1 / 1 = (1 * R1) / (1/R1 * R1) R1 / 1 = R1 / (R1/R1) R1 / 1 = R1 / 1 R1 = R1 The same can be done to equation (2). (1a) rephrased in proportional language, is: R : 1 :: 1 : (1 : R) In plain english: "R" is to a reference measurement what a reference is to its reciprocal. The Greeks being fascinated with geometry, would state this with a picture: A B B C ---------------- is to ---------- what ---------- is to ------ R 1 1 1/R Or, R 1 A B --- = ----- = --- = --- 1 1/R B C This was their first requirement. Their seconnd requirement was: R = 1 + 1/R Or: A = B + C This was the heart of it. Both together would have made up a pair of simulta- neous relationships. "R" would have to be made up of two segments: an integer (one in this case) and a fraction that is equivalent to its reciprocal. There is only one such example when the integer in question is "1": ~1.618 = 1 + ~0.618 Let us see if we can combine the relationships into one. Since: A = B + C and A/B = B/C (A) Then: (A = B + C) / B A B C 1/C --- = --- + --- * ----- B B B 1/C A B 1 --- = --- + ----- B B B/C But since: A/B = B/C A B 1 Then substituting: --- = --- + ----- B B A/B Or, A / B = 1 + 1 / A/B If "x" were to equal "A/B", then: x = 1 + 1 / x (C) (x = 1 + 1 / x) * x x^2 = x + 1 (D) (x^2 = x + 1) - (x + 1) Oila! x^2 - x - 1 = 0 Now let us take the opposite approach: ~1.618 = 1 + ~0.618 and ~1.618 / 1 = 1 / ~0.618 Substituting for simplicity: A = B + C and A / B = B / C and B / A = C / B A - B = C Or, (C = -B + A) / B C B A 1/A --- = - --- + --- * ----- B B B 1/A C B 1 --- = - --- + ----- B B B/A Since: B / A = C / B C B 1 Then: --- = - --- + ----- B B C/B Substituting "x" for "C/B": (x = -1 + 1/x) * x (x^2 = -x + 1) - (-x + 1) Double Oila!! x^2 + x - 1 = 0 If the Greeks had known about polynomials, they would have derived this. (A) are the pair of simultaneous relations, and (B) is on its way to becoming the de- rived quadratic Euclidean algorithm. Looking at (D), ~1.618 ^ 2 = ~1.618 + 1 And, (- ~0.618) ^ 2 = (- ~0.618) + 1 Where did the minus sign for ~0.618 come from? Let's answer that question by first reviewing what we just accomplished, but with numbers. What are its roots? R1 = Deriving an approximation to a primary root using the continued fraction of a polynomial is one way. To double-check our result, another method exists. Subtext: A Approach to Approximating a Polynomial's Factors Root 1 Root 2 (0 = ax^2 + bx + c) - (bx + c) (0 = ax^2 + bx + c) - (bx + c) (ax^2 = -bx - c) / a (ax^2 = -bx - c) / a x = + <(-bx - c) / a> x = - <(-bx - c) / a> Only one of two roots need be calculated by any of the above methods: Since, -Root 1 - Root 2 = b / a Then, Root ? = -Root ?? - b / a Or, since, Root 1 * Root 2 = c / a Then, Root ? = c / (a * Root ??) Like before, it doesn't matter what the initial input value for "x" is. Also, the larger its absolute value, the longer it will take to stabilize the output at the approximate answer(be it right or wrong). Zero doesn't work too well. Positive versus negative initial inputs vary extremely little in their final result. They are almost exactly the same. But what one misses the other may cover. The best policy is to use +1, -1, compute seperately and compare the results. All of the imaginary and none of the complex roots can be calculated this way. All of the real rational and most(~80%) of the real irrational roots will work out. If negative values occur under the radical, remove the negative sign before square rooting and think nothing of it. What matters is what sign contin- ually gets applied to the outside of the radical. Of course, we have the quadratic formula for doing better. So why go to any trouble? To get use to seeing this. Its simplicity will be appreciated when we start discussing higher order polynomials. Chapter Three: Cubics This text deals with only three kinds of polynomials which can be fac- tored: 1) Linear: 0 = ax^0 + bx^1 2) Quadratic: 0 = ax^0 + bx^1 + cx^2 3) Semi-Aesthetic: 0 = ax^0 + bx^2 + cx^4 + dx^8 + ex^16 +....... 4) Hybrid: 0 = ? All other aesthetic polynomials must have their roots approximated, but with these there is a choice. For aesthetic purposes approximation via ratios is a key ingredient. If we don't mind having irrational values for coefficients, then there is certainly what I would call a Conditional Formula for factoring polynomials created with the use of the Extended Quadratic Euclidean Algorithm. But then ratios composed of inte- gers could not be constructed. I am inclined to be practical and deal only briefly with irrational polynomials. While it's still fresh in our minds, let's develope the search method further. Subtext: Cubic Approximation Method For: Root 1 Root 2 Root 3 (0 = ax^3 + bx^2 + cx + d) (0 = ax^3 + bx^2 + cx + d) (0 = ax^3 + bx^2 + cx + d) - (bx^2 + cx + d) - (ax^3 + cx + d) - (ax^3 + cx + d) ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ (ax^3 = -bx^2 - cx - d) (bx^2 = -ax^3 - cx - d) (bx^2 = -ax^3 - cx - d) / a / b / b ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ {x^3 = (-bx^2-cx-d) /a} ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~ x = +{ (-bx^2-cx-d) /a } x = -< (-ax^3-cx-d) /b > x = +< (-ax^3-cx-d) /b > Only two out of three Roots need be calculated by any of the above methods: Since, -Root 1 - Root 2 - Root 3 = b / a Then, Root ? = -Root ?? - Root ??? - b / a Or, since, Root 1 * Root 2 * Root 3 = d / a Then, Root ? = d / (a * Root ?? * Root ???) As before, if an answer comes out zero or positive or negative one, or an alternation between the two, then the factor is uncomputable using this method. It will probably require dividing the polynomial by the primary real root and then use the quadratic formula to get the others. If all the quotients are positive, the roots will arrange themselves from largest to smallest in absolute value: If: q1, q2, q3 are all positive, then: | (Primary)Root 1 | > | (Auxiliary)Root 2 | > | (Auxiliary)Root 3 | Subtext: Searching For the GCF of Three Numbers Using the Incomplete Euclidean Algorithm (a, b, c) = ? Sort: a, b, c so that: the 1st < 2nd < 3rd. (1): C1 = 1st < C2 = 2nd < C3 = 3rd C2 \ C1 = ? C3 \ C1 = ?? Choose the smallest from among: ? and ??, and label as q1. C1 C2 C3 - 0 - (q1 * C1) - (q1 * C1) �� C1 C2 - (q1 * C1) C3 - (q1 * C1) Make the new column labels equivalent to the results: C1 = C1 C2 = C2 - (q1 * C1) C3 = C3 - (q1 * C1) C3 \ C2 = q2 C3 - (q2 * C2) �� C3 - (q2 * C2) The new: C3 = C3 - (q2 - C1) Go back to step (1) and repeat until a column turns to zero. Then continue with the remaining two columns until left with one non-zero column. That is the gcf of (a, b, c). Subtext: Using the Golden Euclidean Algorithm to Find the Golden Cubic Polynomial Initialize: g0 = 1 and h0 = j0 = 0 Since: q1 = q2 = q3 = +1 and s = 0 This: g = q1 * g + q2 * h + q3 * j >>> The Extended Cubic h = q1 * g + q2 * h Ea j = g Becomes this: g1 = g0 + h0 + j0 >>> The Golden Cubic Ea h1 = g0 + h0 j1 = g0 Calculate: 1 = 1 + 0 + 0 1 = 1 + 0 1 = 1 3 = 1 + 1 + 1 2 = 1 + 1 1 = 1 6 = 3 + 2 + 1 5 = 3 + 2 3 = 3 etc.......... Subsript: 0 1 2 3 4 5 6 7 8 9 10 11........ Series 1: 1 1 3 6 14 31 70 157 353 793 1782 4004........ Series 2: 0 1 2 5 11 25 56 126 283 636 1429 3211........ Series 3: 0 1 1 3 6 14 31 70 157 353 793 1782........ Primary Root = g?/g(?-1) = h?/h(?-1) = j?/j(?-1) = Root 1 = g/j Auxiliary Root = Root 2 = - h/g Auxiliary Root = Root 3 = j/h Root 1 = Series 1 / Series 3 Root 2 = -1 * Series 2 / Series 1 Root 3 = Series 3 / Series 2 Root 1 = 2.2469796 Root 2 = - 0.8019378 Root 3 = 0.5549582 Using: 0 = ax^3 + bx^2 + cx + d = (x - Root1) * (x - Root2) * (x - Root3) a = 1 b = -R1 - R2 - R3 c = (-R1 * -R2) + (-R1 * -R3) + (R2 * -R3) d = -R1 * -R2 * -R3 Results in: 0 = x^3 - 2x^2 - x + 1 So: +- (0 = x^3 - 2x^2 - x + 1) are two golden cubic polynomials. Notice that not all of the coefficients equal the quotients directly, as they did for quadratics. If we were to infer strictly from the quadratic as an example, then: When: a = 1 Then: b should = -q1 c should = -q2 d should = -q3 But: -2 <> -1 -1 = -1 +1 <> -1 Although the extended Euclidean algorithm cannot yield any correct roots for most polynomials, it can yield coefficients for general cubic polynomials with a little help. Attempting to compute a cubic polynomial of integer coefficients in the general sense from the extended E. algorithm and three partial quotients: g = q1 * g + q2 * h + q3 * j >>> The Extended Cubic Ea h = q1 * g + q2 * h j = g Root1 = g / j Root2 = - h / g Root3 = q3 * j / h Of: 0 = ax^3 + bx^2 + cx + d = (x - R1) * (x - R2) * (x - R3) a = 1 Normally: b = -R1 - R2 - R3 Adjusted: b = (q2 * -R1) + (q2^2 * -R2) + (q1 * q3 * -R3) Normally: c = (-R1 * -R2) + (-R1 * -R3) + (-R2 * -R3) Adjusted: c = (-R1 * -R2) + (q1 * -R1 * -R3) + (q3 * -R2 * -R3) Normally Adj.: d = -R1 * -R2 * -R3 * q3 Adjusted: d = -R1 * -R2 * -R3 * q3 ? Using this method it is possible to yield monic polynomials of integer coef- ficients from integer quotients almost all of the time. From this data it is possible to speculate the correct relationship between coefficients and quo- tients at the cubic level is: a = 1 b = - (q1 * q2 + q3) c = - q2 d = q3 Of: 0 = ax^3 + bx^2 + cx + d Making the golden cubic example: a = 1 b = -2 = - [ (1 * 1) + 1 ] c = -1 d = 1 For: +- (0 = x^3 - 2x^2 - x + 1) The only instances of any amount of correctness or integer quotients are: q1 q2 q3 root1 root2 root3 �� 2 1 1 correct incorrect incorrect 1 1 1 correct correct correct -1 1 1 correct incorrect incorrect -2 1 1 correct incorrect incorrect -3 1 1 correct incorrect incorrect -4 1 1 correct incorrect incorrect -5 1 1 correct incorrect incorrect -6 1 1 correct incorrect incorrect etc.............................................................. ? The incorrect second and third roots are due to the Ea trying to compute the values for complex roots which we know by now is not possible. The reason for the correct first root is still a mystery. Subtext: Geometric Series Method This is the best way to calculate integer series values for the general cubic(three element interactive harmony) and beyond. In fact, it is just about the only way, considering the ease with which other methods break down. This is really two variations on one method depending on whether you desire the result to reflect either the Fibonacci or the Lucas series. In the case of the Fibonac- ci, the Intermittent method dallies with the wildflowers. In the case of the Lucas, the Immediate reflects(best approximates) the goal at every step. ? By studying the behaviour of the Ea when approximating roots of golden polynomials, it is possible to create a pseudo-Ea capable of performing simi- larly as well for all polynomials of any degree. ? A plane in flight on autopilot zigzags its imperfect attempt to straighten out its course. The Ea works similarly. Ratio approximates form pre- dictable patterns of alternating high/low values. Their numerators put them exactly one numerator integer value above or below their goal. (It is as if they were never meant to be exactly accurate.) Or they are shared among themselves along with their denominators. First compute the correct values for the roots using some method of approximation and take their absolute value---the trigonometric and radical work just fine; the bisection method is more troublesome since you won't know what bounds to use for narrowing in on the roots----just about anything might be the case. The derived(continued fraction) or the golden methods may also be used. Next, decide on what effect you want the ratios to have with their use: Do you want to dally? Choose the Intermittent. Do you want accuracy? Choose the Immediate. ~|Root 1| = g? / j? ~|Root 2| = (-) h? / g? ~|Root 3| = (q3 *) j? / h? (Negating root 2 and completing root 3 using q3 comes at the end of searching for g, h, and j.) Assume: g0 = 1 And: h0 = j0 = 0 Skip the zeroth solution set, but carry over one of its values, "g0". Step 1 ~ | Root 1 | = g1 / g0 ~ | Root 2 | = h1 / g1 ~ | Root 3 | = g0 / h1 To find the next values for g1, h1, and j1: Either: Begins at Step 1: g? = [g(?-1) * Root 1] rounded up/down Or: Begins at Step 1: g? = [h? / Root 2] rounded up/down Either: Begins at Step 2: ? h? = [g? * Root 2] rounded up/down Or: Begins at Step 2: ? h? = [g(?-1) / Root 3] rounded up/down Begins at Step 1: j? = g(?-1) Proceeding from left to right: >>>> h1 = 1 g1 = |Root1| * g0 ? h1 = |Root2| * g1 (j1 = g0 = 1) round g1 down to ? round h1 up to the the nearest integer ? nearest integer Step 2 ~ | Root 1 | = g2 / g1 ~ | Root 2 | = h2 / g2 ~ | Root 3 | = g1 / h2 Proceeding from left to right: g2 = |Root1| * g1 h2 = |Root2| * g2 (j2 = g1 = 1) round g2 up to the round h2 up to the nearest integer nearest integer Continuing for: Step 3 ~ | Root 1 | = g3 / g2 ~ | Root 2 | = h3 / g3 ~ | Root 3 | = g2 / h3 g3 = |Root1| * g2 h3 = |Root2| * g3 (j3 = g2) round g3 down to round h3 up to the the nearest integer nearest integer And: Step 4 ~ | Root 1 | = g4 / g3 ~ | Root 2 | = h4 / g4 ~ | Root 3 | = g3 / h4 g4 = |Root1| * g3 h4 = |Root2| * g4 (j4 = g3) round g4 up to the round h4 down to the nearest integer nearest integer Recycle steps 3 and 4 to continue this algorithm indefinitely. Finally: ~Root 1 = g? / g(?-1) ~Root 2 = - h? / g? ~Root 3 = q3 * g(?-1) / h? Or: ~Root 1 = g? / j? ~Root 2 = - h? / g? ~Root 3 = q3 * j? / h? This is the series of sets of unknowns in integers intended to approxi- mate the roots of a cubic. But no algebra was used to build up our answer from a standard starter set(1, 0, 0, 0, etc.). Instead, it was derived from previously known values. This actually makes this a solution in three pseudo-unknowns. The incremental trial and error method discussed in the introduction is another type of algorithm that works with pseudo-unknowns: we know the values, we just don't know to represent the values with ratios to make it look like the Ea pro- duced them from scratch. So this is not a solution, strictly speaking. It is just a converted form of rational ratios for irrational decimals. But the growth of the ratios' numerators and denominators are coordinated with respect to one another in a loose sort of way---just not definitively algebraic. If the computation had proceeded from g(?-1) to g? to h? only, then it is possible to come up with two different theories of computation and two dif- ferent results. Both would accurately converge onto the roots, but only one would exhibit modularity: the ability to derive any root from two other roots either before or after it in a circle of procession(R1, R2, R3, R1,...; or, R3, R2, R1, R3,...). But modularity is insured and only one solution results if the computation is required to be reversible: by proceeding from g(?-1) to g? to h? and from g(?-1) to h? to g?. Notice how each set of approximating solutions are no longer simulta- neous with respect to one another. When general quadratic solutions and the golden cubic were sought using the extended Ea, each set was calculated simul- taneously from the previous set: Golden Quadratic: 0 = x^2 - x - 1 Set: 0 1 2 3 4 5 6 7 8 9 10 11 12 13..... Series 1: 1 1 2 3 5 8 13 21 34 55 89 144 233 377..... Series 2: 0 1 1 2 3 5 8 13 21 34 55 89 144 233..... General Quadratic: 0 = x^2 - 2x - 1 Set: 0 1 2 3 4 5 6 7 8 9 10............. Series 1: 1 2 5 12 29 70 169 408 985 2378 5741............. Series 2: 0 1 2 5 12 29 70 169 408 985 2378............. Golden Cubic: 0 = x^3 - 2x^2 - x + 1 Set: 0 1 2 3 4 5 6 7 8 9 10 11........ Series 1: 1 1 3 6 14 31 70 157 353 793 1782 4004........ Series 2: 0 1 2 5 11 25 56 126 283 636 1429 3211........ Series 3: 0 1 1 3 6 14 31 70 157 353 793 1782........ But here, they are calculated somewhat in tandem and partly simultane- ously. Reversibility gives two equally valid results: General Cubic: 0 = x^3 - 3x^2 - x + 2 Calculating from g(?-1) to g? to h?: Set: 0 1 2 3 4 5 6 7 8 Series 1: 1 1 4 12 38 118 368 1146 3570 Series 2: 0 1 3 11 32 102 316 987 3073 Series 3: 0 1 1 4 12 38 118 368 1146 Or: Set: 0 1 Series 1: 1 3,3 Series 2: 0 2 Series 3: 0 1 Now we have something to analyze and wonder how the Ea behaves alge- braically for cubic+ degree polynomials in general. Subtext: The Derivation of the Cubic Ea So far, it looks pretty good. Beginning with three partial quotients we can set out to find their polynomial using the Extended Ea with adjustments. Once the coefficients are known, the roots can then be calculated using either the cubic method (plus either the quadratic method, or the quadratic formula---if necessary), or other partial solution methods not covered in this text. Finally, a triple infinite approximation series of integrated ratios can be deduced using the pseudo-Ea. All this for the glory of artistic design(see, or wait until, the chapter on Applications: Aesthetics). ? The golden Ea is a set of three simultaneous linear algebraic expres- sions in one, two, and three unknowns. Unknowns ----------- g = g + h + j 3: g, h, j h = g + h 2: g, h j = g 1: g What's more, they are self-looping. There are two derivations of a con- tinued fraction and its consequent Ea form: Based on the "x^3" term: 1) Contains two simultaneous algebraic expressions: one linear in one un- known and one quadratic in three unknowns Based on the "x^2" term: 2) Contains ?: The derivation of x^3: For: (0 = ax^3 + bx^2 + cx + d) - (bx^2 + cx + d) (ax^3 = -bx^2 - cx - d) / a x^3 = -(b/a)x^2 - (c/a)x - d/a Divide by "x" in the form of (h/j)^2: (x^3 = -(b/a)x^2 - (c/a)x - d/a) / (h/j)^2 h --- = j Chapter ?: Aesthetic Applications Inexactitude is taken to a high degree of developement as an art form in aesthetics. Unlike science, which seeks to minimize approximation or the time it takes to get something done, art makes a science out of smelling the flowers. Geometry has shown us that the triangle is the basic unit of art and that the triangle contains two qualities to aesthetic design: angle and ratio. The skill to estimation is the integration of progressive atunement; in other words, the creation of a trend which doesn't limp, jerk, or heave. Every time it moves forward towards exactness, it glides instead of lurches. So, harmony is the overriding principle, while angle and ratio are its elements. Harmony entails comparative judgement, so aesthetics is born. Harmony also implies group cohesion, so modularity, or the sense of belonging, is the ability of any angular or ratio to contain within itself only those other ele- ments of its group to which it belongs, and no other. These other elements would be derivable. Since we live within a relative world, harmony itself can be inexact, so there are different categories to aesthetics: 1) General (Aesthetic) 2) Golden (Ideal) 3) Semi- (-Aesthetic, -Golden) 4) Hybrid 5) Higher-Euclidean 1) General: General aesthetic criterian are satisfied when a simple polynomial in one unknown has all of its x terms intact (by virtue of having none of their coefficients in zero) and all of its coefficients are rational. It will have its roots and their associated angles contained within irregular, odd-sided polygons, encircled by an ellipse. Implication: All of the factors will be generally aesthetic in relation to each other, but not to any factors of other polynomials. 2) Golden: When all the partial quotients of the Euclidean algorithm are equal to each other and positive one, then the polynomial, its roots and associated angles are golden. These roots and angles are contained within regular, odd-sided polygons. 3) Semi- When a polynomial contains non-zero coefficients only on terms whose exponents are the powers of an integer? (or prime?), the polynomial is semi-aesthetic. Even- sided polygons are its associated forms. When these forms are inscribed within a circle, as opposed to an ellipse, they are semi-golden. 4) Hybrid When a polygon has another polygon of some multiple number of sides to the original inscribed within itself, with one set of the smaller polygon's sides centered within the larger, the relationship between them is a hybrid form of beauty. Normally, aesthetics pertains to within a system and only one polygon is associated with it. But the interrelationship among system-polygons, when the polygons themselves are already related in some way, determines a hybrid(cross-breed). 5) Higher- Other styles of beauty not covered by the Euclidean Euclidean algorithm, or its derivatives, probably lying outside of physical appreciation. But higher orders of aesthe- tics are probably structured along similar lines. Chapter ?: Game Theory Chapter ?: Planetary Anthropology Whole planets seem to be governed by an overriding polynomial affecting the harmonic developement of its inhabitants. Case in point: The golden ratio of ideal aesthetics in western culture is popularly 1.618... This ratio, and no other, approximates the average high/low voltage of a stable human neuron. It also approximates the ratio of atomic weights taken between an anion and cation of sodium chloride--the primary solute(liquid soluable compound) in the oceans of the earth and blood of vertebrates. The Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21,..., is its analogue. The ratios of western music intervals, pure temperment theory, are based on this series. Pineapple bracklets(that's the hexagonal eyes on a pineapple's surface), pinecone and sunflower seed arrangements, are in pairs of right- and left-handed swirls of successive Fibonacci numbers. Further speculation: During the first and second century A.D., Roman architecture exhibits the hybrid ratio between a square's side and the side of an inscribed octagon, with four of the octagon's sides centered within the square's. Its value, 2.414..., is the primary root of the polynomial: x^2 - 2x - 1. It is this author's opinion that Roman culture, a parent to our own, is an example of one kind of society having once lived on another planet. My candi- date for other-worldliness is Maldek. In esoteric literature(see A.R.E.: Edgar Cayce, Life Readings), Maldek was the fifth planet in orbit around our sun. All that is left of it is a belt of asteroids between Mars and Jupiter. It was blown to bits by a thermonuclear series of events. The souls of that planet were invited by the Earth's spirit to come and finish their evolution on our own. There is also the information that grey people of the moon, during the Atlantean civilazation, migrated to that culture and quickly became its prominent citizens. It is this author's opinion that Maldek's demise was a combination of unimpeded underground thermonuclear arms testing and a planetary suicidal nu- clear deterent: they ringed their major tectonic plates with an underground nuclear arsonal, hoping that no one would have either the audacity or stupidity of commiting war, lest the offended nation retaliate with an equally stupid response. The seismic upheaval that insued, for such a game plan did in fact occur, cost the planet's stability. This is predicated on a presently unpopular scientific opinion that planets(plus stars and galactic centers for that matter) are hollow. (There is not enough matter in the asteroid belt to compose a solid planet). I make no present claim for having a pseudo-fifth force of physics to equal and negate a gravitational force(although I would love to spe- culate). The deceased souls seeking reincarnation within our own world were sent to the moon as a kind of quarantine. But something was happening to the moon's atmosphere(their doing?). The moon was fast becoming a dead planet. Only then were they sent here---ship load after ship load. "Them" is many of us in the nations of the world. salt compound, the primary root of a polynomial is being sought as an approxima- tion. Appendix The purpose of this appendix is to give some background information to help clarify some confusing discrepancies in procedure. Subtext: Definition of Division by Zero Subtext: Redefinition of Imaginary Numbers: A Loosening of Some Requirements phase relation is to angle what time is to space. Tables Subtext: The Tierran, Fibonacci, or Phi Series (Golden Quadratic) Given the polynomial: 0 = x^2 - x - 1 q1 = 1 and q2 = 1 Initialize: h = 1 and j = 0 Using: h = (q1 * h) + (q2 * j) And: j = h Compute: 1st Cycle: Series 1: 1 = 1 + 0 Series 2: 1 = 1 2nd Cycle: Series 1: 2 = 1 + 1 Series 2: 1 = 1 3rd Cycle: Series 1: 3 = 2 + 1 Series 2: 2 = 2 4th Cycle: Series 1: 5 = 3 + 2 Series 2: 3 = 3 5th Cycle: Series 1: 8 = 5 + 3 Series 2: 5 = 5 etc...... Cycle: 0 1 2 3 4 5 6 7 8 9 10 11 12 13..... (Term) Series 1: 1 1 2 3 5 8 13 21 34 55 89 144 233 377..... Series 2: 0 1 1 2 3 5 8 13 21 34 55 89 144 233..... Basic Vocabulary Intervals of Tierran Music Scales(pure temperment): parallel terms of either series divided by the other: 2/1 3/2 5/3 8/5 13/8................................ Octave Perfect Major Minor Beyond the sensitivity of human Fifth Sixth Sixth hearing to discriminate over much of our range � 1/1 ��> Tonic, Upscale Perfect Unity 1/2 2/3 3/5 5/8 ................................... Octave Perfect Major Minor Fifth Sixth Sixth � ��> Downscale Increasingly accurate equal temperment approximations of the pure: Subtext: The Lucas Series: A Tierran Alternate Subtext: The Theoretical Basis for the Egyptian's Calculation of Phi and Pi Looking at this table it becomes apparent that reason may not be the sole requirement for the advancement of human culture. It probably helps speed it up, as well as add an additional element of self-correction. But trial and error, in addition to intuition plus lots of time, may be sufficient to acquire some semblance of cultural evolution--technology being held as the least diffi- cult, but most prominent, element to develope. (When I say intuition, I am referring in this instance to man's proximity to physical nature; spiritual intuition may require reason to evolve). Archimedes' calculation of pi as an ave- rage between relatedly different polygons is an example of reason working out the derivation of a subject. If the Egyptians used this method for calculating pi as an expression of phi, then this is an example of a fortuitous discovery without the aid of reason. Reason may not be as necessary as inspiration and experience. Although we have no record of their having derived this knowledge from some intellectual understanding of the subject, we at least know that they had a long cultural existence to work out this relation- ship, experientially. Could ancient civilations have existed with an inferior intellectual and moral developement, but with a superior technological develope- ment? It may be more than a mere coincidence that the seed of three, in this technique, equals the summation of h1 and j1 after the second round of computa- tion, had a seed not been included. h = (-b/a * h) + (-c/a * j) [+ (s)] j = h Polynomial: 0 = x^2 - x - 1 q1 = -b/a = 1 and q2 = -c/a = 1 Initialializing: h0 = 1 and j0 = 0 Calculating: h1 = q1 * h0 + q2 * j0 j1 = h0 h1 = 1 = 1 * 1 + 1 * 0 j1 = 1 = 1 Seed: s = (q1 * h1) + (q2 * j1) + (j1) s = 3 = (1 * 1) + (1 * 1) + (1) h2 = q1 * h1 + q2 * j1 + s j2 = h1 h2 = 5 = (1 * 1) + (1 * 1) + (3) j2 = 1 = 1 h3 = q1 * h2 + q2 * j2 j3 = h2 h3 = 6 = (1 * 5) + (1 * 1) j3 = 5 = 5 h4 = q1 * h3 + q2 * j3 j4 = h3 h4 = 11 = (1 * 6) + (1 * 5) j4 = 6 = 6 etc....................... Only Series 1 will be used here. The column numbers at the top are the series' term index: 1st term, 2nd term, etc. The first row is series 1. The values in column 1 are the row numbers in addition to being the integer multipliers of series 1. 1 2 3 4 *5 6 *7 8 9 10 11 *12 13 *14 1 5 6 *11 *17 *28 45 73 118 191 309 500 809 1309 *2 10 12 22 34 56 90 146 236 383 618 1000 1618 2618 3 15 18 33 51 84 135 219 354 573 927 1500 2427 3927 4 20 24 44 68 112 180 292 472 764 1236 2000 3236 *5236 5 25 30 55 85 140 225 365 590 955 1545 2500 4045 6545 6 30 36 66 102 168 270 438 708 1146 1854 3000 4854 7854 7 35 42 77 119 196 315 511 826 1337 2163 3500 5663 9163 8 40 48 88 136 224 360 584 944 1528 2472 4000 6472 10472 -------------------------------------------------------------------------------- 24 120 144 264 408 672 1080 1752 2832 4584 7416 12000 19416 31416 -------------------------------------------------------------------------------- 48 240 288 528 816 1344 2160 3504 5664 9168 14832 24000 38832 62832 These integers may be interpreted in one of two ways, as appropriate: either as integers or as decimals rounded to four places, beginning with column 5. Column 7 depicts degrees of a circle in eigth increments. Column 12 is pretty; so are columns 2 through 4, for that matter. Column 14 displays the 48 arc divisions of a circle in radians. In row 1, columns 4-6, are the three ways the ancients subdivided both the royal Egyptian cubit as well as the Chaldean cubit. In row 2 are the negative and positive integer powers of phi: from phi^(-7) in column 5, through to phi^0 in column 12, followed by phi^1, phi^2, and so on. The fourth row of column 14 contains the royal Egyptian/Chaldean cubit in millimeters: 11 * 17 * 28 = 5236. Since this table is based on phi as the square root of five plus one, the entirety divided by two, the fractional part of the square root of five, and its multiples as well as its divisions, keep popping up all over: columns 9, 11, 13, 14,...?: <5> = 2.236... <5>/2 = 1.118... This table is conjectural; we have no record of the Egyptians, or Chaldeans, ever computing this. But we do know that the Egyptians had phi as 1.618, the square root of one over phi as 0.786 = 1/1.618, and pi as the square of phi times six fifths: 3.1416 = 2.618 * 1.2 = 1.618^2 * 6/5. The square root of the reciprocal of phi is significant in that it approximates pi/4, or arc- tangent(1). This was used in approximately squaring the circle; an impossible feat in reality, but roughly accomplished in the Cheops'(Khufu) pyramid design at Gizeh. This feat also afforded them a spherical projection system for map making(they knew the Earth was round). They also knew the dimensions of the Earth and calculated the meter, on which they based their royal cubit: 1 royal cubit = 0.5236 meters = the circumference of a circle(2Pi) / 12. The Hebrew's cubit was one-half a royal cubit. Moses was no dummy: "Go with what works", must have been his motto. The zodiac played a large role in ancient life; it is no small wonder that the circle was divided into 12 or 24 arc wedges. It is known that the priests withheld information for themselves and their chosen students: royalty, neophytes. With the destruction of their centers of learning, and the subsequent fire of the library at Alexandria, most of the evidence of their technological awareness has been lost. We may never know what traditions they passed down, orally or written, unless some new clue arises from the desert. Pyramid Khufu is radiocarbon dated at 71,000 years of age. But carbon dating figures are usually thrown out altogether when the age reaches beyond 46,000 years, a point well below this amount, because of a tendency to- wards inaccuracy. Are we that far off? Esoteric tradition puts the building of Khufu at 75,000 years ago. Have we underrated the ancients? Subtext: The Maldek Series Given polynomial: 0 = x^2 - 2x - 1 q1 = 2 and q2 = 1 Initialize: h = 1 and j = 0 Calculate: h1 = 2 = 2 * 1 + 1 * 0 j1 = 1 = 1 h2 = 5 = 2 * 2 + 1 * 1 j2 = 2 = 2 h3 = 12 = 2 * 5 + 1 * 2 j3 = 5 = 5 etc.................... Term: 0 1 2 3 4 5 6 7 8 9 10............. (index) Series 1: 1 2 5 12 29 70 169 408 985 2378 5741............. Series 2: 0 1 2 5 12 29 70 169 408 985 2378............. Bibliography ?, Denmark: A Social Laboratory Experiment, ? Barbeau, E.J., Polynomials, Springer-Verlag(New York), 1989 Beskin, N.M.(Nikolai Mikhailovich), Fascinating Fractions, Mir Publishers (Moscow), 1986 Funkt-Heller, Charles, La Bible et la Grande Pyramide d'Egypte, ? Huntley, H.E., The Divine Proportion: a study in mathematical beauty, Dover Publications(New York), 1970 Jones, William B. and Thron, W.(Wolfgang) J., Continued Fractions: Analytic Theory and Applications, Addison-Wesley Pub. Co.(Reading, Mass.), 1980 Kapraff, Jay, Connections, The Geometric Bridge Between Art and Science, McGraw- Hill, Inc.(New York), 1991 ^french architect? Klein, Felix, Famous Problems of Elementary Geometry, Dover Publications (New York), 1956 Larson, Roland and Hostetler, Robert P., Algebra and Trigonometry, D.C. Heath and Co.(Lexington, Mass.), 1993, 3rd ed. Lendvai, Erno, Bela Bartok: an analysis of his music, Kahn & Averill(London), 1971 Lopez, Alvarez, ? Maor, Eli, e: the story of a number, Princeton University Press (Princeton, NJ), 1994 Niven, Ivan Morton, Zuckerman, Herbert S., and Montgomery, Hugh L., An Introduc- tion to the Theory of Numbers, 5th ed., John Wiley and Sons, Inc., 1991 Olds, C.D.(Carl Douglas), Continued Fractions, Random House(New York), 1963 ?Ogilivy, C. Stanley and Anderson, John T., Excursions in Number Theory, ?(?), ? Tompkins, Peter, Secrets of the Great Pyramid, Harper and Row(New York), 1971 Vajda, S., Fibonacci and Lucas Numbers and The Golden Section: Theory and Applications, Ellis Horwood, Ltd.(Chichester), 1989