Polynomials: a three-hour tour, by Vinyasi Thanks Helen and Charlie Acknowledgements 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 Progressive Demonstration of Hypothesis: Monics Quadratics Cubics Quartics Pentics Generalized Theory of Polynomials in One Unknown Geometry Aesthetics Polynomials of Multiple Unknowns Appendix Tables Glossary 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 / 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) Introduction The purpose of this text is to delve into the relationship between polynomials and the skill of artistic design: what is mathematics' contribution to art? Through the use of a little used interpretive procedure(the Euclidean algorithm), polynomials are made to yield their light on the subject. Subtext: Polynomials 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: 0 = ax + b 0 = (ax + b) * (cx + d) = acx^2 + (ad+bc)x + bc etc..................... Where a, b, c, d, etc. are known and x is unknown. A simple algebraic expression in multiple unknowns can be similar to these expressions: 0 = ax^b + c 0 = ax^b + cy^d + e 0 = ax^b + cy^d + ez^f + g etc.......................... Or as a simple polynomial in multiple unknowns, their multiplications: 0 = (ax^b + c) * (ey^f + g) = aex^by^f + agx^b + cey^f + cg Where a, b, c, d, e, f, g, etc. are known and x, y, z, etc. are not. This text deals freely throughout with algebraic expressions of both types, multiple and singular unknowns, since they are mutually dependent on each other. Subtext: Incremental Aproximations 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. | (4.17 / ratio) - 1 | * 100% = + error % 1/1 = 1 is 317% off from approximating 4.17 |(4.17/1) -1| *100%=317% 2/1 = 2 is 108% off from approximating 4.17 |(4.17/2) -1| *100%=108.5% 3/1 = 3 is 39% off from approximating 4.17 4/1 = 4 is 4% off from approximating 4.17 13/3 = 4.333 is 3.8% off from approximating 4.17 17/4 = 4.25 is 1.9% off from approximating 4.17 21/5 = 4.2 is 0.7% off from approximating 4.17 25/6 = 4.166 is 0.08% off from approximating 4.17 121/29 = 4.172 is 0.06% off from approximating 4.17 146/35 = 4.171 is 0.03% off from approximating 4.17 171/41 = 4.1707 is 0.02% off from approximating 4.17 196/47 = 4.1702 is 0.005% off from approximating 4.17 221/53 = 4.1698 is 0.004% off from approximating 4.17 417/100 = 4.17 is 0% off from approximating 4.17 This algorithm and its ratios serve as the backbone to this text. From these ratios are plucked choice selections by their integrative merit due to the use of algorithms derived from polynomials. The Euclidean algorithm figures largely in this. Chapter One: Monic Polynomials (x = a) - a or (x = -a) + a x - a = 0 or x + a = 0 This is a monic, or first degree/order, polynomial in one unknown. Looks simple? "A" can be equivalent to any positive or negative integer, including 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". 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] 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: a/1 , it is its "golden" ratio. There are only two golden polynomials for every de- gree, with factors that are equal to each other in absolute value, but opposite in sign. These are the monic 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 +- ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = x 2a If we take a look at this, it looks like: -b ÄÄ +- ÄÄÄÄÄÄÄÄÄÄÄ = x 2a 2a ÚÄ Ä¿ -b ³b^2 - 4ac³ ^ 1/2 ÄÄ +- ³ÄÄÄÄÄÄÄÄÄ³ = p +- q^1/2, where 2a ³ 4a^2 ³ ÀÄ ÄÙ -b b^2 - 4ac p = ÄÄ and q = ÄÄÄÄÄÄÄÄÄ 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 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 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, ÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 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. ÄÄ [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). 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 Ú ¿ ³ h ³ -c/a * j h/j = -b/a * ³ÄÄÄ³ + ÄÄÄÄÄÄÄÄ ³ h ³ h À Ù h (-b/a * h) + (-c/a * j) ÄÄÄ = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ j h Since, h : [(-b/a * h) + (-c/a * j)] :: j : h, then h = (-b/a * h) + (-c/a * j) 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 within series 1 or 2. The auxiliary root is a derivative of the primary. After recycling the output of the formula however many times one desires to achieve some degree of accuracy in determining the primary root, the auxiliary root(s)(only one for this second degree polynomial), can be derived from the last computation of the primary root by dropping the first summation term: h? j? (-b/a * h) + (-c/a * j) Primary Root = ÄÄÄÄÄÄ = ÄÄÄÄÄÄ = Root 1 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ h(?-1) j(?-1) h ÚÄÄ ÄÄ¿ (-b/a * h) + (-c/a * j) ³ -b/a * h ³ Root 2 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ - ³ÄÄÄÄÄÄÄÄÄÄ³ h ³ h ³ ÀÄÄ ÄÄÙ -c/a * j Root 2 = ÄÄÄÄÄÄÄÄ * -1 h c/a * j Auxiliary Root = Root 2 = ÄÄÄÄÄÄÄ h Whenever the two terms: -b/a and -c/a are both equivalent to positive one, then calculation of the final approximations can be written thus: h Root 1 = ÄÄÄ j j Root 2 = - ÄÄÄ h 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). 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 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 1 q2 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 1 q3 + ÄÄÄÄÄÄÄÄÄÄÄÄ 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,....] 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 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ q2 1 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ q3 1 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ etc............ is the case; for this is not a common calculation. If we wish the calculation to look like this: q2 x = q1 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ q2 q1 + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ q2 q1 + ÄÄÄÄÄÄÄÄÄÄÄÄ etc......... q1 = -b/a and q2 = -c/a, then we will need to make the previous gcf searching algorithm look more like the derived Euclidean algorithm. Subtext: Searching For a Pair of Numbers' Polynomial If: 0 = ax^2 + bx + c a = 1 and b and c are integers, then 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 ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄ 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 ÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄ 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) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ choose some value C2 - (q1 * C1 + s) for s, such that: C1 = C2 - (q1 * C1 + s) and C2 = C1 / q2, and C2 - (q1 * C1) ÄÄÄÄÄÄÄÄÄÄÄ 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: Proceed with the partial quotient searching algorithm, but with an addi- tional first step: C1 = non-prime; whichever one satisfies C2 = C1 / q2 C1 = C2 and C2 = the other value C1 < C2 Continue with algorithm. If both values are non-prime, then be ready to try either value as the initial C1. The other method to this madness. Method 2: Root 1 = h?/j? Root 2 = q2 * j?/h? Option 1: k = h? m = j? ~q1 = - Root 1 - Root 2 Ú ¿ -k m ³ m ³ ~q1 = ÄÄÄ + q2 * ÄÄÄ * ³ÄÄÄ³ m k ³ m ³ À Ù Ú ¿ -k ³ k ³ q2 * m^2 ~q1 = ÄÄÄ * ³ÄÄÄ³ + ÄÄÄÄÄÄÄÄ m ³ k ³ km À Ù -(k^2) + q2 * m^2 ~q1 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 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. ÚÄÄ ÄÄÄ¿ ³ -(k^2) + q2 * m^2 ³ ³ ~q1 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ * km ³ km ³ ÀÄÄ ÄÄÄÙ ÚÄ Ä¿ ³ ~(km * q1) = -(k^2) + q2 * m^2 ³ + k^2 ÀÄ ÄÙ ÚÄ Ä¿ ³ ~(k^2 + km * q1) = q2 * m^2 ³ / m^2 ÀÄ ÄÙ k^2 + km * q1 (2) ÄÄÄÄÄÄÄÄÄÄÄÄÄ = ~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 Substitute: x = k/m Into: ~0 = x^2 + bx + c 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) and round q2 up. Check by substitution. If neither of these attempts succeed, then try option 2. All of this was under the assumption that a equals 1, and b and c are integers. "A" serves as the least common multiple for the denominators of b and c. If b had been 1 and c had been 1/2, then a would equal 2 in order that b may equal 2 and c may be an integer equalling 1. The two versions are equivalent, because of multiplying the whole equation by a so that all coefficients may be integers. a = 1 b = 1/1 c = 1/2 lcm of 1 and 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 = 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, or It is found by: = (d * f) / (d, f) = 1 / (1/d, 1/f) See the difficulty? "A" could be anything; the denominators of b and c 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 2. What to do? Notice also that the multiplication of a polynomial by negative one does not change the absolute value of 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) is the only golden, or ideal, quadratic polynomial because it satisfies the cri- teria of having all of its partial quotients equal to each other and to either positive or negative one. I forgot to mention: Euclidean search methods work best for polynomials of real number roots. If any roots are imaginary, then a real version may result, or zeros, or alter- nating one's and zeros. Computing roots the Euclidean way can be cumbersome to derive the cor- rect formulas for all of the factors---especially for higher degree polynomials. To double-check our answers, a simpler method exists. Subtext: A Approach to Approximating a Polynomial's Factors Although this technique is incomplete at the quadratic level, it works superb for the cubic polynomial and beyond. (0 = ax^2 + bx + c) - (bx + c) (ax^2 = -bx - c) / a x = <(-bx - c) / a> Like before, choose a value for the input x that is computable and recycle outputs into next inputs. Only the primary root can be found at the quadratic level, and this is it. If the square root of a negative number results in your computer crashing, try removing the negative sign before square rooting and reattach imediately thereafter. Don't forget to label it as imaginary. 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 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 > As before, if an answer comes out zero, or alternating between zero and one, then the factor is imaginary. It will probably require dividing the poly- nomial by the primary real root and then use the quadratic formula to get the others. 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) ÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄ 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) ÄÄÄÄÄÄÄÄÄÄÄ 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. Chapter ?: Aesthetics 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 length. The skill to estimation is the integration of progressive refinement; 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 lurching. So, harmony is the overriding principle, while angle and length are its elements. Harmony also entails comparative judgement, so proportion is born. Harmony also implies group cohesion, so modularity, or the sense of belonging, is the ability of any angular or proportional value 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 2) Golden 3) Semi- 4) Hybrid 5) Imaginary 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 to no factor of other polynomials. 2) Golden: When all the partial quotients of the ideal Euclidean algorithm are equal to each other and either positive or negative one, then the polynomial, its roots and associated angles are golden. These roots and angles are contained within regular, odd-sided polygons. There are four means of approximating polynomials using the Euclidean algorithm: 1) Incomplete 2) Extended 3) Dynamic 4) Derived 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. phase relation is to angle what time is to space. Table: The Fibonacci, or Phi, Series 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 ³ 1/1 ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ> Tonic, Upscale Perfect Unity 1/2 2/3 3/5 5/8 ................................... Octave Perfect Major Minor Fifth Sixth Sixth ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ> Downscale Increasingly accurate equal temperment approximations of the pure: Table: 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 h1 = q1 * h0 + q2 * j0 j1 = h0 Seed: s = 3 = (q1 * h1) + (q2 * j1) + (j1) h2 = 5 = (1 * 1) + (1 * 1) + (3) j2 = 1 = 1 h3 = 6 = (1 * 5) + (1 * 1) j3 = 5 = 5 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? Table: The Maldekan 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 Beskin, N.M.(Nikolai Mikhailovich), Fascinating Fractions, Mir Publishers (Moscow), 1986 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., 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, Olds, C.D., 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, Vajda, S., Fibonacci and Lucas Numbers and The Golden Section: Theory and Applications, Ellis Horwood, Ltd.(Chichester), 1989