Section 3: A More Indepth Study of Beauty Via Golden Beauty's Source, Hybrid Beauty, Derived From Geometry, Polynomials, and a Light Model: Ch.4: Linear Relations of Aesthetics and the Parallel Method For Categorizing Beauty: The Logical Difference Between Polynomials in One Versus Multiple Unknowns. Numbers Are a By-Product of Polynomials in One Unknown(A Base).Polynomials in One Unknown Are a Way of Creating and Defining Irrational Numbers.Polynomials in One Unknown Can Be Approximated Using Sets Of Polynomials in Multiple Unknowns. Their Limitations. Continued Fractions of Polynomials.Their Extensions From Lower to Higher Degrees.Limitations.The Creation of Alternate Forms of Beauty: General: Simple (Linear), Straightforward(Quadratic), Transformational(Cubic), Approximate (Quartic+), Unclassified General, Golden, Hybrid: Static(Converging), Dynamic (Fractal), Cyclic(Partly Fractal). Extention of the Golden Algebraic Method.Adjusted Multiplication of the Roots. Derivation of the Coefficients From Partial Quotients.The Transformation of One Set of Coefficients Into Another.Relations Within A Set.Integration Within The Quadratic Parallel Method.The Quadratic Quality to Any Polynomial With It's Location in Polygons Ch.: Hybrid Beauty Hybrid beauty is a way of associating individual roots from beautiful polynomials of similar degree. Only roots within golden beautiful polynomials are said to be hybrid, or more accurately---golden, with respect to each other. Roots of all other polynomials of like degree must be taken out of context from their respective polynomials and associated in a loose sort of way with roots from other polynomials. There is a standard, but little understood, law of asso- ciation that determines which root from what polynomial will be grouped with another. If again, in rare circumstances, there is a strong coorelation, than the roots are said to be golden and locatable as ratios among certain diagonals and one side of a single polygon. If, on the other hand, they are weakly related, than the roots are hybrid and must be located as relations among diagonals from bet- ween different polygons of associated number and magnitude of sides. Hybrid beauty can be defined in two ways: algebraically and geometrical- ly. Its algebraic method is derived from the golden's, but its geometric ?may differ from the Golden. Like Golden beauty, these methods help define its existence. To algebraically derive a hybrid beauty polynomial, partial quotients are used. They are either used to calculate the coefficients of a hypothetical polyomial, or else they are used to compute their roots by using an ex- tended form of the golden parallel method. The process of multiplying the roots is then adjusted in an unorthodox way. The direct method of computing the coef- ficients from partial quotients is derived from the adjusted parallel method. Each degree polynomial has its own distinct adjustment process. It is not known whether there is any pattern to either adjustment or direct calculation over all degrees. Only a few degrees are shown here. How well this hypothesis holds is speculative. The Extended Parallel Method: Requires the golden algebraic method: (?),(?+1) are subscripts for successive terms in a series, with (?)=starting at=0 and terminating at infinity(0, 1, 2, ..., ?): h(?+1)=h(?)+j(?)+k(?)...+u(?)+v(?)+w(?) j(?+1)=h(?)+j(?)+k(?)...+u(?)+v(?) k(?+1)=h(?)+j(?)+k(?)...+u(?) ........................... u(?+1)=h(?)+j(?)+k(?) v(?+1)=h(?)+j(?) w(?+1)=h(?) with partial quotient multipliers: q(1),q(2),q(3),q(4),...q(?),q(??),q(???),.... h(?+1)=q(1)*h(?)+q(2)*j(?)+q(3)*k(?)...+q(?)*u(?)+q(??)*v(?)+q(???)*w(?) j(?+1)=q(1)*h(?)+q(2)*j(?)+q(3)*k(?)...+q(?)*u(?)+q(??)*v(?) k(?+1)=q(1)*h(?)+q(2)*j(?)+q(3)*k(?)...+q(?)*u(?) ............................................. u(?+1)=q(1)*h(?)+q(2)*j(?)+q(3)*k(?) v(?+1)=q(1)*h(?)+q(2)*j(?) w(?+1)=h(?) to create an infinite series of a set of values, set: {h(?), j(?), k(?), ..., u(?), v(?), w(?)}. The number of elements within the set is equal to the degree of its polynomial. From any individual set is then calculated an approximation of the polynomial's roots. From set(?) of the series: {h(?),j(?),...} is produced the roots: root(1)=h(?)/w(?) root(2)=-j(?)/u(?) root(3)=k(?)/s(?) .................... root(?-2)=-u(?)/r(?) root(?-1)=v(?)/t(?) root(?)=-w(?)/v(?)*[coefx^0] All even numbered roots are negative and the last root is multiplied by the x^0 coefficient. Alternate versions of the polynomial can be had by either changing all signs to their opposite, or by taking the reciprocal of all the roots making four versions in all. The absolute value range of the roots will be similar to the golden in that either: root(1)...root(?)=infinity...1/2, or 2...0 with the first root always being larger than the next and so on. To find the corrected coefficients of the root's polynomial requires an adjustment which bends the rules during multiplication. This is unorthodox, but it works. Since all the coefficients, except for coefficient(x^degree) and coef- ficient(x^0), are irrational, some means must be found to change them into integers. Integer coefficients are required for all hybrid and golden polynomials of either beauty or power? quality. All of the roots of hybrid polynomials higher than the first degree are usually irrational with the exception of one rational root (the value +-1) within degree (n*3)+1, for all n={0,1,2,3,...}. This is because the algebraic derivation of the parallel method is from quadratic polynomials. Its extention into higher degrees has been done without any logic other than: "Uh, maybe it will work". Cubic example of types of roots: quotients: roots: coefx^3 coefx^2 coefx^1 coefx^0 q1 q2 q3 r1 r2 r3 1 -(q1*q2+q3) -q2 q1 upwards etc..................................................................... 9 8 7 6 5 1 1 quadratic quadratic quadratic 4 1 1 quadratic quadratic quadratic 3 1 1 quadratic quadratic quadratic 2 1 1 cubic quadratic quadratic 1 1 1 cubic cubic cubic -1 1 1 cubic quadratic quadratic -2 1 1 cubic quadratic quadratic -3 1 1 cubic quadratic quadratic -4 1 1 cubic quadratic quadratic -5 1 1 cubic quadratic quadratic downwards etc.................................................... Quartic example of types of roots: quotients: roots: q1 q2 q3 q4 r1 r2 r3 r4 ???????????????????????????????????????!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! upwards etc..................................................................... 5 1 1 1 quadratic quadratic quadratic 4 1 1 1 quadratic rational quadratic quadratic 3 1 1 1 quadratic quadratic quadratic 2 1 1 1 quadratic rational quadratic quadratic 1 1 1 1 cubic rational cubic cubic -1 1 1 1 cubic rational quadratic quadratic -2 1 1 1 cubic rational quadratic quadratic -3 1 1 1 cubic rational quadratic quadratic -4 1 1 1 cubic rational quadratic quadratic -5 1 1 1 cubic rational quadratic quadratic downwards etc.................................................... So solving adjustment problems may simply require identifying each root's algebraic expression within a span of partial quotients, say +1 to +4, and then trying to guess what do all the expressions have in common that would make their multiplications all result in rational values. Once a particular degree's adjustment has been worked out, it is then possible to guess what do all the hybrid coefficients have in common with their production directly from their partial quotients. ?Adjustment formulas for the multiplication of roots when their absolute value range is from infinity to one-half; otherwise, the resulting coeffients are ?swapped in pairs: ?coef.x^1 with coef.x^2; ?coef.x^3 with coef.x^4; etc.: Degree: r,r1,r2,r3,... are roots 1 ax + b = 0 a=1, b=-r No adjustment 2 ax^2 + bx + c = 0 a=1, b=-r1-r2, c=-c*r1*r2 Some adjustment 3 ax^3 + bx^2 + cx + d = 0 a=1, b=q2*-r1 + q2^2*-r2 + q1*q3*-r3 c=r1*r2 + q1*r1*r3 + q3*r2*r3, d=-q3*r1*r2*r3 Major adjustment 4 ax^4 + bx^3 + cx^2 + dx + e = 0 a=1, ? The formulas for directly calculating the coefficients: When the polynomials are equal to 0 and the first coefficient, 'a', is a 1: Degree: 1 q ax + b a = 1, b = -q 2 q1, q2 ax^2 + bx + c a = 1, b = +-q1, c = -q2 3 q1, q2, q3 ax^3 + bx^2 + cx + d a = +-1, b = -(q1 * q2 + q3), c = -+q2, d = q3 4 q1, q2, q3, q4 ax^4 + bx^3 + cx^2 + dx + e a = 1, b = -+?, c = -?, d = +-q3, e = q4 The hybridization of polynomials can proceed in successive order until the process terminates by making the next set of partial quotients equal to the previous set of coeffients from coef.x^(degree-1) to coef.x^0. Each set of partial quotients will have their own unique situation of cycling in pairs or progressing without termination. For instance: For quadratics: quotients: coefficients: q1 q2 b c derivation order: 1 1 -1 -1 first -1 -1 1 1 second 1 1 -1 -1 third For cubics: quotients: coefficients: q1 q2 q3 b c d derivation order: example one: 1 1 1 -2 -1 1 first -2 -1 1 -3 1 1 second -3 1 1 -2 -1 1 third example two: 1 1 -1 0 -1 1 first 0 -1 1 0 1 1 second 0 1 1 0 -1 1 third example three: 1 2 3 -5 -2 3 first -5 -2 3 -13 2 3 second -13 2 3 -23 -2 3 third etc.............................................?th In these cubic examples, either q1 or q2 must be different from both q2 and q3, while both q2 and q3 must equal each other in order that the series of hybrids be nonterminating. Looking at the formulas for directly evaluating coefficients from partial quotients a simple analogy appears that is applicable toward the chapter on planetary biological development (Ch.5). The absolute value of quadratic coefficients with respect to their partial quotients appear to be independent of each other's values, while at higher degree polynomials some coefficients are dependent. This makes life on a quadratic planet appear to be based on freedom and simplicity. It is as if nature wants to balance out the responsibility for decision making with the simplicity of choices. * Integration: It is sometimes wondered why partial quotients of the continued frac- tions of square roots are periodic, but not for roots higher than the square. It is because the only derived parallel method that is globally integrated and complete for its whole polynomial is the quadratic. A continued fraction normal- ly suppresses the second partial quotient to a 1, equivalent to its polynomial's negative x^0 coefficient, thereby forcing its first partial quotient, negative coefficient x^1, to vary. Normally, the polynomial for a square root has a zero x^1 coefficient making the first partial quotient also a zero. This would cause the continued fraction to simplify into a simple fraction: the reciprocal of the squared square root, in other words, the reciprocal of the x^0 coefficient. For example: <2>=1.4142135... x=<2> or x=-<2> 0=x-<2> or 0=x+<2> (x-<2>)*(x+<2>)=x^2 + <2>x - <2>x - 2=x^2 + 0x - 2=0 x^2 = 2 x = <2> x^2 + 0x - 2=0 x^2=0x - 2 x=0 - 2/x x=-2/(-2/x)=x >> kind of useless because of the non-converging periodic x=-2/(-2/(-2/x))=-2/x >> result of x followed by -2/x,... x=-2/(-2/(-2/(-2/x)))=x, but, <2>=int<2> + <2> - int<2>, the integer <2> is 1 1.414=1 + 0.414 or <2>=1 + <2> - 1 =1 + 1/2.414 or =1 + 1/(1/(<2>-1))*(1/1)/(<2>+1)/(<2>+1) =1 + 1/((<2>+1)/(2-1)) =1 + 1/(1+1.414) or =1 + 1/(1+<2>) =1 + 1/(2+0.414) or =1 + 1/(2+<2>-1) =1 + 1/(2 + 1/2.414) or =1 + 1/(2+(1/(1/(<2>-1)))) =1 + 1/(2 + 1/(1+<2>)) =1 + 1/(2 + 1/(2+0.414)) or =1 + 1/(2 + 1/(2+<2>-1)) =q1+q2/(q1+q2/(q1+0.414)) or =q1+q2/(q1+q2/(q1+<2>-1)) etc........................................................ So while q1 varies as 1, 2, 2, 2, etc., q2 does not vary as 1, 1, 1, 1, etc., or: ____ [q1,q2;]=[1,1;2,1;]=<2> The derivation of a parallel method from any degree polynomial always homes in on just one factor at a time unless the polynomial is golden beau- tiful. If one factor is being approximated within the context of a quadratic, the other factor is a single unknown and hence it is a deterministic problem: a problem with only one solution. But if one factor is being approximated for a cubic polynomial, then the solution for the two other unknown factors is a problem with undeterminable solutions relative to the first factor---all of the (degree minus 1) factors must somehow be approximated simultaneously. This happens for any quadratic polynomial of real roots. For golden beautiful polynomials of any degree, it is possible to cull the other roots simultaneously from the formula for the first root. Somehow, ?primary golden roots contain all of the other mem- bers of their set. Also, using the quadratic derived parallel method for some use other than a quadratic one(say, finding the continued fraction for a cube root) is extending the method outside its normal scope without any guarantee of validity ---it is generalizing from a limited supposition. This lack of completeness for the cubic and higher degree derived paral- lel methods is the single most conclusive reason for its failure to provide a repeating continued fraction for cube roots and higher.