Proportion, by Vinyasi In loving thanks to Helen and Charlie Lutes Notations ( PARENTHESIS ) < SQUARE ROOT > [ CONTINUED FRACTION SERIES ] ________ [ B; a:b, a:b, a:b ] ==> B ==> FIRST PARTIAL DENOMINATOR; MAY NOT OCCUR; SEMICOLON WOULD THEN BE DROPPED a ==> SUCCESSIVE PARTIAL NUMERATORS; MAY OR MAY NOT BE ALL ONES b ==> SUCCESSIVE PARTIAL DENOMINATORS; MAY NOT OR MAY BE ALL ONES IF BOTH ARE ALL ONES, THEN QUADRATIC PHI RESULTS: ~1.618... B, a, b ==> ARE ANY KIND OF NUMBERS, BUT IN THIS TEXT, ONLY RATIONAL NUMBERS WILL BE USED: (positive, negative,integers, fractions) ________ ==> REPEAT BAR FOR a AND b; REQUIRED ONLY FOR IRRATIONAL CONTINUED a:b, a:b FRACTIONS Table of Contents I. Introduction 1. Definition..........................................................pg. 2. Historical Background...............................................pg. 3. Investigative Technique.............................................pg. II. Theory 4. Integral Division: The Continued Fraction Versus Euclidean Algorithms A. Real numbers.....................................................pg. B. Algebraic Numbers................................................pg. C. Algorithmic(transcendental) Numbers..............................pg. D. Convergent Series & Ratios.......................................pg. 5. Integer Series and Best Fit Aproximations...........................pg. 6. Plane Geometry: Where Does Proportion Come From?....................pg. 7. Modularity(aesthetics)..............................................pg. 8. Quadratic Roots & Polynomials.......................................pg. 9. Hybridization.......................................................pg. 10. Qualitative Analysis................................................pg. III. Applications 11. Visual A. Design...........................................................pg. B. Optics...........................................................pg. 12. Music(sonic, magnetic)..............................................pg. i. Equal Temperament a. Precise....................................................pg. b. Approximated 1) Euclidean...............................................pg. 2) Best Fit................................................pg. ii. Factored(pure temperament) a. Limited(example; Pythagorean)..............................pg. b. Unlimited 1) Euclidean...............................................pg. 2) Selective Hybrid(example: just intonation)..............pg. iii. Rhythm......................................................pg. IV. Appendix 13. Tables A. A. Phi Series and Ratios i. Odd Polygons..................................................pg. ii. Even Polygons................................................pg. B. Phi Music Scales i. Bi-Factored Euclidean.........................................pg. ii. Selective Euclidean..........................................pg. C. Phi Roots and Polynomials i. Odd Polygons..................................................pg. ii. Even Polygons................................................pg. D. Trigonometric Phi................................................pg. E. Semi-Phi.........................................................pg. F. Infinity versus Zero.............................................pg. 15. Bibliography........................................................pg. 14. Glossary............................................................pg. 16. Index...............................................................pg. Definition Proportion is beauty. All proportion is beautiful. Power is the use which beauty is put to. Finite beauty is represented by rational numbers; infinite beauty, by irrational. Infinite beauty implies infinite potential to manifest beauty to perfection. Beauty lies not in the eyes of the beholder unless the beholder through compassion interprets more beauty than what is manifest. Instead, beauty lies in the coordination between different propor- tionate identities. These identites comprise a set of proportions that are meant to be interpreted and used as a set. Sets are represented by polynomials of integer, or ?rational, coefficients. No one identity derives its beauty by being alone unless its set is composed of just one member. (There is such a number: one.) If an identity appears to be lacking in beauty, it is simply because its membership in a group has not yet been found. Hybrid beauties are produced by the comingling between sets. Imaginary numbers require more than imagination to appreciate their beauty, for they exist in a state akin to limbo. Their beauty is tied up until liberated by squaring. Transcendental numbers transcend class distinctions; their properties are derived from all classes. Thus their identity transcends membership in any one group. Glossary /synonyms apex angle(at the top of an isosceles triangle, the two slopes being equal): identifies the general context(this triangle) of a ratio base angle: the angle at the base of an isosceles triangle which serves as a functional equivalent to a ratio save for chapter ? >>>> ; the isosceles triangle is the context of a ratio if the angle is at the center of a higher curve(i.e., circle, ellipse, etc.), then the slope of the triangle is the distance between the curve and the curve's center; for the purposes of this study, this distance is the unit measure within the curve(i.e., 1), and the ratio is calculated in terms of this: namely, base of triangle divided by slope; if, on the other hand, wavelength: the product of a vibrating substance of different composition 1) cavity, or 2) the temporal equivalent to what ratio does for space beauty/ratio/proportion/(see identity, angle): a pair of numbers related to one another through division; a fraction; if only one number is present, then it is assumed to be the numerator while the number one is the deno- minator; degree polynomial/set/class/group: the number of factors composing a polynomial; number of elements within a set identity/factor/element: a member of a set; a factor of either, 1) a polynomial, or 2) a composite factor of several identities phase relation: the bridge between purely temporal and spacial/temporal phenomena phi/è: the symbol of proportion derived from a Greek letter in honor of Phidias, the architect of the Parthenon in Athens, whose width divided by its height equals ~1.618, the classical standard to human beauty Cosmology Time is. The fact that time remains the same with respect to each observer irres- pective of the observer's speed requires the preeminence of time over space. This preeminence tempts the question: is space defined, or derived, in terms of time? If this be the case, is the "Big Bang Theory" appropriate? It is predi- cated on the notion that time and speed are coorelates dependent on space and vice versa; that if space were to change, so would time frames between obser- vers. What if the isness to time is little different than the isness to the speed of light for all observers? What if the singularity at the start of the "Big Bang" concerns time's lack of "phaseness" as much as it does space's lack of dimensions? What if the "Big Bang" is not necessary to seperate the isness from time? What if the singularity to time coexists presently with its phasic nature? The "Big Bang" was only needed to understand space's apparent expansion, not its intrin- sic relation to time. The Doppler effect certainly doesn't slow down light; it only changes the position of the light source in relation to the observer. But the expansion of space was predicated on the exclusion of the other possibility: that light is being bent prismatically around in space. It was postulated that no force of gravity at some center to space could be so great as to bend light without also maintaining all matter in a singularity forever. If such a massive center does not exist, then where does the blue light go and what causes the curvature in this alternate theory? Light cannot travel in a straight line forever; it must bend back and return from wherever it was sent to maintain balance in nature. It is a question of accountability: nothing is allowed to be lost because nothing is created or destroyed. In other words, no massive center to a spherical universe is required to explain the bending of light in this alternate theory; merely the light source itself is necessary to send it out on a trajectory less than perfectly parallel to the force which sent it. This maintains dynamic equilibrium in all phases of activity. The blue light is taking a longer trajectory. This bending of light, along with its built-in requirement to return to its source, precludes any necessity for an expanding and contracting universe. Even if its source moves, the movement will have been taken into account at the inception of its trajectory, making the two move in conjunction with each other. A further possibility: The isness to time is universal; there are no limitations to this nature. But time is also divided into individual frames of reference. These dif- ferences, or phases to time, make relativity possible. In other words, indivi- duation to reference is preeminent to relativity--not the other way around. In order for differences to have any meaning, the initial standard of unity to time must continue to coexist along with its phases. Now a second problem presents itself: entropy. Since nothing is lost in space permanently(it all coming back eventually), then entropy of matter cannot be a problem; but entropy of time can be. Eventually, the myriad individual time frames loose their sense of timing with each other and themselves. , they loose their ability to relate differences accurately. This happens all the time, but initially it's not a problem. It is only a problem when enough matter gets thrown out of alignment with itself. This happens whenever a trajectory interacts with other forces which will alter its course or even absorb its momentum as part of its own. Such imbalances in bookkeeping will eventually wear down the sense of dif- ferences between references to the point where relativity takes on a whitewash appearance. In this state of relative ambivalence, the ambivalence to the rela- tive undergoes an alteration of alignment. The total misalignment to the system is readjusted with each reference frame in a direction common to all misalign- ments. It is like saying that individual debts amounting to a universal crisis are written off globally in proportion to each other. Such a shift is described as "the night of Brahma", by the easterner, when all of nature sleeps in a virtual state of existence. The virtuality would be equivalent to the blurring of time differences. It may require an equal amount of time in this state to sort out the fuzziness as it did to produce it. Purpose I feel the necessity for giving a taste of what is to come later on in the text, even if it feels to early to hint at. This subject is not all mathema- tics and abstraction. It has consequences to its use. Proportion is an identity. It is a consequence of time. Periodic occur- rences create wave phenomena. A wave is nothing more than a periodic surge of some kind. The proportional difference among interacting waves create angles of incidence, or refraction, through space. A proportion is thus equivalent to some angle, as well as a time difference. There are are two broad headings to the diversity of proportion: namely, beauty and power. Each have their play in the world. Identities of beauty derive there existence by being only those portions of a cycle whose relatedness form a set of simple and thorough integration. Identities of power, on the other hand, are only partially integrated within sets. They can be derived from beauty, power, or some combination. Beauty is of two types: manifest and unmanifest, or existencial and potential. Potential beauty is unity; there is only one of these, but it underlies all of beauty--even manifest. Manifest beauty subdivides into an infinite series of sets of identities of various number of components numbering one to infinity. There is only one rational value to manifest beauty; it is in the first set of only one component: the value of one. All other iden- tities are irrational values within sets of plural elements. Power is also of two types: it is derived from either potential or mani- fest beauty. Power derived from potential beauty is "pure" in the sense that it contains no manifest beauty, only unity. This power must be exercised with con- sistency and equanimity, reflecting the unity of its nature, if it is not to be abused. In other words, the user must be fully integrated within himself--99% won't do. Power derived from manifest beauty, on the other hand, contains beauty within it; it automatically is used in harmony with its environment-- no harm may come from its use. It is necessary to demonstrate and define the mechanics of beauty: "how it came to be, and what there is to see, and all that may be realized"; that is one of the purposes of this book. It is also necessary to define power. Although I run the risk of popularizing and oft abused subject, I am also of the opinion that abuse can arise when power is protected from scru- tiny. It is not power's nature to share of itself since it is a reflective, self-bounded stance; but it would be better to keep the uses of power beautiful. Beauty lies not only in the eyes of the beholder, but also in the inte- gration and appropriate application within a subject. When beauty is lacking, then is the compassion and imagination of the beholder called upon to make up the difference. Beauty, as it is manifest, is both partly restrictive as well as expres- sive. It can be anything provided it chooses some quality that is relative to a quantity of relations. Its beautiful nature is thus restricted to a relation within a group and the choice of its expression. The group may be of one or of many elements; but a group it still remains. Power is gained in reflection at opposition--contrast. It is the pruning away of self-inconsistencies, incoherence, and the amplification of what is left. Thus, the self-practice of power provides the basis for the practice of beauty. Alternation between the two is unavoidable. The state induced by either one makes the practice of its compliment very inviting--make full and proper use of either one while you may. Power lacks the degree of integration that is shared by beauty. Not all power is destructive, but some may be. Spiritual power is so natural, for instance, that it is unassuming: you wouldn't notice it. Most power is not pure, but has equal amounts of both beauty and power within itself. But some power, a small class, has no manifest beauty(unless one is in love with such). Instead, it has potential beauty in the form of unity. It is this power that has the potential for being overly eccentuated. The prospect of success(or imagined future beauty) is not justification for the use of such power. Only unity can handle it. In other wordw, most power is beholden to other identities of power and beauty within an integrated group. This serves as a check against its misuse. Other than this, power devoid of beauty must have recourse to consis- tency if it is to be beholden to anything. Beautiful power is impure power which is beholden to its internal manifest beauty. Spiritual power is unity. Beauty contains no power within it, but instead has integral self- relatedness. All beauty has this. Beauty has no self-fulfilling value. It has little concern for its well being. It shares; it has no time for self improvement. Thus, it too can fall victim to abuse through self-negligence; while with power, it is a matter of self-indulgence. Balance is required between both and variety within each. Both beauty and power have identity. They also have modularity. Identi- ties of power known herein as "proportion" contain miniature copies of itself within itself, while beautiful proportional identities contain every member of its set, including itself, within itself. Identities of power cannot contain the members of its set within itself, but when combined with them it can serve to duplicate each member of the combination. These dual natures of power and beauty exclude simultaneous integration with each other, although they can be found in a mixed state within beautiful power. Experience of either serves to spur alter- nation between them. Evolution of identity as something beyond structural differences, but not excluding them, is the result. Somewhere, sometime, enough of the qualities of both will cultivate the unity of the one who finds itself caught up in a seemingly endless parade of alternation between opposites along with an intolerable seperation from its own unity. Potential unity has to literally become bored, or more or less familiar with duality, before it can aspire towards itself gradually, more often then not, in stages. Historical Background It all started with an Egyptian hieroglyph. A figure facing left and leaning to the right with arms extended upward represent the hypotenuse of a 3:4:5 right triangle from the bottom of his feet to the tips of his fingers. The right side and bottom edge of the hieroglyph represent the other two sides of the triangle. From his crown to his navel divided by from his navel to his heels, represent the golden proportion "Phi", or ~1.618. (Phi is so named in honor of Phidias of ancient Greece, the architect of the Parthenon in Athens. Its width divided by its total height equals Phi.) This inspired a treasure hunt that still continues. I entertained a belief of infinitely many Phi's of pro- gressively larger and larger values. But the computational model provided by Fibonacci of breeding rabbits confined to a room, although romantic(if you're a rabbit), is inadequate to model this belief. Lucas(pronounced: loo-ka) and Taylor series of numbers offer different terms but the same geometric progres- sion as Phi. Tribonacci and similarly constructed series offer novel means of increasing growth ratios approaching the number two, but never equalling it, much less exceed it. Enter a model from optics. It is described by S. Vajda and H. E. Hunt- ley(see references). Light entering sandwiched panes of glass is asked to bounce off the interior faces a non-negative integer of times(i.e., 0,1,2,3,4,5,6,7,...). The number of possible pathways the lightbeam can take are successive terms of an infinite series. If the number of glass panes is two, then Fibonacci's series minus the first term results: 1,2,3,5,8,13,21,34, 55,89,144,... If there is only one glass pane, then a series of ones occur. But infinitely many glass panes may be sandwiched giving an infinite number of Fibonacci-like series with unique geometric progressions. The ratios represent- ing the growth rates of these series are found as the ratios between sides and diagonals, and diagonals with each other, within regular polygons of odd number of sides. (Traditional phi has always been understood as the ratio between either the side of a regular pentagon(all sides being equal) and one of its dia- gonals, or its reciprocal.) If the Euclidean algorithm is used to find the greatest common divisor between two successive terms of the Fibonacci series an infinite number of terms along in the series, then an infinite series of one's known as the partial denominators of a continued fraction result. These ones identify the Fibonacci as a representation of a well-proportioned, or "golden", progression. If the continued fraction terms had been any different, then the series would be less than "ideal". The Euclidean algorithm is normally understood to work on only two numbers at a time. This gives it a quadratic perspective to number processing. (Quadratic refers to a polynomial of second degree and inte- ger coefficients; i.e., x^2 - x - 1 = (x - 1.618) * (x + 0.618) = 0). This type of polynomial is composed of two ratios, the geometric progression and its nega- tive reciprocal. With a few modifications to the algorithm, it is possible to process an infinitely large set of values(transcendental, irrational, or rational). Irrational numbers may be grouped with each other as the powers of simple roots of numbers{i.e., (2^(0/3),2^(1/3),2^(2/3))} or grouped with one or more ones to get a "Euclidean integration". Summation series representing transcendental numbers can be processed to give continued fraction series of varying degrees of accuracy. The ratios defining the proportion between a chromatic pitch in western music of "just-intonation" and the tonic, or first note of a scale, were disco- vered through experience. It can be shown, on the other hand, to proceed from the first five terms of the Fibonacci series; namely, unison: 1/1, octave: 2/1, perfect fifth: 3/2, major sixth: 5/3, and minor sixth: 8/5. The other ratios are derived from these except for the note between the perfect fourth and fifth, the tritone, of 7/5. It is borrowed from the equal-tempered scale as a Euclidean square root approximation of the number two(the octave's ratio). The Euclidean algorithm produces a continued fraction which in turn can be fed back into the algorithm to produce a Euclidean series(Fibonacci is part of such a series; Taylor and Lucas series are derivatives of the Fibo- nacci). A continued fraction may also be used to produce the Euclidean series. From the Euclidean series comes harmonic ratios, both golden(ideal) and dispro- portioned, of which Phi is an ideal example. The terms of a Euclidean series may also serve as initial values for the Euclidean algorithm thus forming a computa- tional loop. I hope you enjoy this subject as much as I have. Technique, Inductive versus Deductive Reasoning Most of what is discovered here, I obtained by inductive methods. Not knowing what to look for, but having only a vague idea of where to look, I would create a sea of data. If it was the right ocean, a pattern would emerge and from that pattern--an algorithm. Very often the ocean would be created sur- real on purpose to evoke the pattern more readily. Rules would be bent, per- pective would be altered; anything to reach the goal. This is essentially in- ductive, or indirect, reasoning: I don't know where I am; I know where I want to go; I don't know how to get there. Once there, the landscape becomes more familiar, more translucent, and the new reality doesn't seem so surreal any more. In fact, what seemed to originally contradict normal behaviour doesn't contradict anything---the lack of familiarity was at fault. Under those con- ditions, bending of the rules was necessary in order just to get by. Once the whole picture becomes familiar, bending isn't necessary; then it is possible to deduce directly from the relevant facts usefull corrolaries or parrellels. If I had known the whole picture from the start, or possessed omniscience, deduction would have been adequate to generate all this infor- mation. But as it is, I am not, and inductive reasoning along with imagination, faith, and creative mistakes held sway. I leave it to others more gifted in the art of deduction to validate or invalidate anything contained herein. Most of what passes for omniscience is only inference anyway. Least Fit Aproximation A rational or irrational value may be represented or approximated using ratios of smallest numerator/denominator pairs of integers. The only stipula- tions 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: 1/1 = 1 is 317% off from 4.17 2/1 = 2 is 217% off from 4.17 3/1 = 3 is 117% off from 4.17 4/1 = 4 is 17% off from 4.17 13/3 = 4.333 is 16.333% off from 4.17 17/4 = 4.25 is 8% off from 4.17 21/5 = 4.2 is 3% off from 4.17 25/6 = 4.166 is 0.333% off from 4.17 121/29 = 4.172 is 0.241% off from 4.17 146/35 = 4.171 is 0.142% off from 4.17 171/41 = 4.1707 is 0.073% off from 4.17 196/47 = 4.1702 is 0.021% off from 4.17 221/53 = 4.1698 is 0.018% off from 4.17 417/100 = 4.17 is 0% off from 4.17 The Euclidean Algorithm I will have to preface this chapter by making a few remarks. I will not seek to prove these remarks, but instead will simply explain and demonstrate by way of example in the appendix. -1 * ZERO = INFINITY -1 * INFINITY = ZERO 1 / ZERO = INFINITY 1 / INFINITY = ZERO A set of numbers are processed by the Euclidean algorithm to produce a continued fraction. The process may be coined as "integral division", since it integrates the varied set into one continued fraction series through the pro- cess of group division among the members of the set. If the initial set of numbers are all rational, then a greatest common divisor occurs as a member of the continued fration series, and the series terminates. If all the initial values are irrational, then the continued fraction has infinitely many cycles of succession of terms, with no greatest common divisor among them. ? If the initial set is composed of a transcendental summation series, then a pair of partial denomi- nator/numerator terms will result. (a,b,c,...) is a set of initial values. The order, or degree polynomial of the set, is equal to the number of elements within the set. is its integration through division as summation terms within the tiered denominator of a continued fraction series; all of the numerators are assumed to be ones, except for the last(if rational), a zero. is a continued fraction series of various pairs of denominator:numerator terms; the last numerator before first terminus is a zero; it indicates a rational terminating c.f.s.; there are no numerators afterward; an irrational c.f.s. would end with three periods instead of a zero indicating nontermination. (a,b,c,...)= xx=g.c.d. of (a,b,c,...) if a,b,c,... are rational. The procedure(for single terms): 1) Count the elements of the set = d(egree) 2) Sort the elements of the set from least to greatest 3) Integer divide each element into its next larger, or equivalent, neighbor; d-1 number of integer divisions will occur 4) From each original element, subtract the product of its next smaller, or equivalent, neighbor and the smallest quotient resulting from step 3); this quotient is the first term of a continued fraction series 5) Replace all of the original elements(except for the smallest) with the differences resulting from step 4) and sort; these are the new elements for the next process cycle 6) Repeat this process from step 3) until a single zero remainder re- sults; this is the first terminus; if the degree is less than three, then it is the only terminus; if the original elements are rational, then the penultimate term of the continued fraction series is the set's g.c.d. 7) Continue the process among the remaining non-zero elements, gradually reducing the number of elements, until a single non-zero element remains---this is the final terminus; mark the continued fraction term associated with the first terminus For example: \ will be the symbol used for integer division (98321,354627,7654) degree=3, (degrees will be used in following chapters) (7654,98321,354627) 98321\7654=12; 354627\98321=3; thus, <3,> and 98321-(7654*3)=75359; 354627-(98321*3)=59664; thus, (7654,59664,75359) 59664\7654=7; 75359\59664=1; thus, <3,1,> and 59664-(7654*1)=52010; 75359-(59664*1)=15695; thus, (7654,15695,52010) 15695\7654=2; 52010\15695=3; thus, <3,1,2,> and 15695-(7654*2)=387; 52010-(15695*2)=20620; thus, (387,7654,20620) 7654\387=19; 20620\7654=2; thus, <3,1,2,2,> and 7654-(387*2)=6880; 20620-(7654*2)=5312; thus, (387,5312,6880) 5312\387=13; 6880\5312=1; thus, <3,1,2,2,1,> and 5312-(387*1)=4925; 6880-(5312*1)=1568; thus, (387,1568,4925) 1568\387=4; 4925\1568=3; thus, <3,1,2,2,1,3,> and 1568-(387*3)=407; 4925-(1568*3)=221; thus, (221,387,407) 387\221=1; 407\387=1; thus, <3,1,2,2,1,3,1,> and 387-(221*1)=166; 407-(387*1)=20; thus, (20,166,221) 166\20=8; 221\166=1; thus, <3,1,2,2,1,3,1,1,> and 166-(20*1)=146; 221-(166*1)=55; thus, (20,55,146) 55\20=2; 146\55=2; thus, <3,1,2,2,1,3,1,1,2,> and 55-(20*2)=15; 146-(55*2)=36; thus, (15,20,36) 20\15=1; 36\20=1; thus, <3,1,2,2,1,3,1,1,2,1,> and 20-(15*1)=5; 36-(20*1)=16; thus, (5,15,16) 15\5=3; 16\15=1; thus, <3,1,2,2,1,3,1,1,2,1,1,> and 15-(5*1)=10; 16-(15*1)=1; thus, the g.c.d. of (7654,98321,354627) is 1(all three values are relatively prime, toward each other) and (1,5,10) 5\1=5; 10\5=2; thus, <3,1,2,2,1,3,1,1,2,1,1,2;> (first terminus) and 5-(1*2)=3; 10-(5*2)=0; thus, (0,1,3) 3\1=3; thus, <3,1,2,2,1,3,1,1,2,1,1,2;3> is the c.f.s. with final terminus, or <3:1,1:1,2:1,2:1,1:1,3:1,1:1,1:1,2:1,1:1,1:1,2:0;3> and 3-(1*3)=0; thus, (0,0,1); the c.f.s. numbers following the first terminus will be used in the next chapter The numbers within the brakets < > are short hand for: < :1, :1, :1, :1, :1, :1, :1, :1, :1, :1, :1, :0;> c.f.s. numerator terms 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 3 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 2 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 2 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 3 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ full c.f.s. ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 2 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 1 + ÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 2 + 0 <3: ,1: ,2: ,2: ,1: ,3: ,1: ,1: ,2: ,1: ,1: ,2: ;> c.f.s. denominator terms The procedure for a continued fraction series of paired terms(more an art than a science): For example: PI/4=~0.7853981 Recommended Reading Huntley, H.E., The Divine Proportion: a study in mathematical beauty, Dover Publications(New York), 1970 Kapraff, Jay, Connections, The Geometric Bridge Between Art and Science, McGraw- Hill, Inc., 1991 Klein, Felix, Famous Problems of Elementary Geometry, Dover Publications(New York), 1956 Lendvai, Erno, Bela Bartok: an analysis of his music, Kahn & Averill(London), 1971 Maori, Eli, e, The Story of a Number, Vajda, S., Fibonacci and Lucas Numbers and The Golden Section: Theory and Applications, Ellis Horwood, Ltd.(Chichester), 1989 References Ogilivy, C.Stanley & Anderson, John T., Excursions in Number Theory, Glossary