Polynomial Perceptual Theory: a primer on discrimination in the eyes of math, by Vinyasi For Helen and Charlie Lutes Chapter Contents 1.....Introduction Section One: Linear Aesthetics in No Unknowns 2.....Linear Aesthetic Theory 2a....Eratosthenes' Sieve [ES] 2b....Integer Division Method [IDm] 2c....Base Conversion Method [BCm] Section Two: Planar Aesthetics in One Unknown 4.....Planar Aesthetic Theory 4a....Polynomial Theory 4b....Ratio and Angle Sets 4c....Modularity 4d....Natural Occurence and Historic Use 5.....Overview of Planar Methods 6.....Natural Convergence of Incremental Progressions [NC] [IP] 6a....Incremental Method [INCm] 6b....Interactive Methods [INTm] 7.....Polynometric Progression Methods [PPm] 7a....Serial Progression [SPPm] 7b....Parallel Progression [PPPm] 8.....Flat Plane Trigonometric Methods [FPTm] 8a....Golden Trigonometric Table [GTt] 8b....Golden Trigonometric Method [GTm] 8c....Modular Method [Mm] 9.....Polynomial Self-Looping Methods [PSLm]: 9a....Radical Method [RADm] 9b....Continued Fractions [CF] 9c....Continued Fractions of Square Roots [SQRCF] 9d....Continued Fractions of e and pi [eCF] [piCF] 9e....Reversal of Continued Fractions of Transcendental Numbers for Analysis of Their Convergence Efficiency [RCF] 9f....Derivation of the (Full) Euclidean Algorithm [Ea] 9g....Short Cut Coefficient Method [SCCm] 9h....Ratio Reduction Method [RRm], also known as: Incomplete Euclidean Algorithm [IEa] 8f....The Reflecting Golden Light Model and the Golden Method [Gm] 8g....Averaging Methods [Am] 8j....Reversing Methods to Hypothesize Polynomials [REVm] 10....Algebraic Expression Methods [AEm] 11....Coefficient Inherent Structure Methods [CISm], in Order of their Appearence Alongside the Terms Within a Polynomial "coef. * x ^ ?": (? = zero or degree minus one to infinity) Linear Coefficient Associated With the Term "x^0": 11a...Geometric Progression Methods [GPm] Quadratic Coefficient Associated With the Term "x^(d-1)": 11b...Summation of Power Methods [SPm] Cubic Coefficient Associated With the Term "x^(d-2)": 11c...Summation of Powers of Multiplicative Pairs of Roots Quartic Coefficient Associated With the Term "x^(d-3)": 11d...Summation of Powers of Multiplicative Triples of Roots Etc................................................................... 11e...Word Problems to Construct Polynomials [WP] 12....Hypothesizing a Section Two: 3-Dimensional Aesthetics in Two Unknowns 13....Conclusion Glossary......... Bibliography..... Index............ I) Natural Convergence 1) Incremental Progression II) Integrated Convergence 2) Polynometric Progression 3) Geometry (Trigonometry) 4) Polynomial Self-Looping 5) Algebraic Expressions 6) Coefficient Structures 7) Transcendental Functions I) Natural Convergence [NC] A) Incremental Progression [IP] 1) The Incremental Method [INCm] 2) The Interactive Method [INTm] II) Integrated Convergence [IC] B) Polynometric Progression [PP] 3) The Serial Method [SPPm] 4) The Parallel Method [PPPm] C) Flat Plane Geometry (Trigonometry) 5) Golden Trigonometric Table [GTt] 6) Golden Trigonometric Method [GTm] 7) Modular Method [Mm] D) Polynomial Self-Looping [PSL] 8) The Radical Method [RADm] 9) Continued Fractions [CF] 10) Continued Fractions of Square Roots [SQRCF] 11) Continued Fractions of Transcendental e and pi [eCF] & [piCF] 12) Reversal of Continued Fractions of Transcendental Numbers for Analysis of Their Convergence Efficiency [RCF] 13) The Full Euclidean Algorithm [Ea] 14) The Short Cut Coefficient Method [SCCm] 15) The Ratio Reduction Method [RRm], also known as: The Incomplete Euclidean Algorithm [IEa] 16) The Reflecting Golden Light Model [RGLM] 17) The Golden Method [Gm] ?Four Reversing Methods to Hypothesize Polynomials from their Ratio Sets [REVm] 18) The Reverse Euclidean Algorithm [REa] 19) The Greatest Common Divisor Algorithm [GCDa] 20) Ratio Set Reversal [RSR] 21) The Inverse Euclidean Algorithm in Multiple Unknowns [MUEa] ?) Any More? Two Standard Averaging Methods [SAm] 22) The Bisection Method [Bm] 23) Newton's Special Case for Shell Polynomials of the form: x^a + b E) Coefficient Structures [CS] 24) Geometric Progression Methods [GPm] 25) Summation of Power Methods [SPm] 26) Summation of Powers of Multiplicative Pairs of Roots 27) Summation of Powers of Multiplicative Triples of Roots Etc................................................................... 28) Word Problems to Construct Polynomials [WP] F) Algebraic Expressions [AE] 29) Factoring Real Beautiful Quadratic Polynomials 30) Factoring Real Power Polynomials G) Transcendental Aesthetic Functions Do they exist? "à" "á" "â" "ã" "ä" "å" "æ" "ç" "è" "é" "ê" "ë" "ì" "í" "î" "ï" "ð" "ñ" "ò" "ó" "ô" "õ" "ö" "÷" "ø" "ù" "ú" "û" "ü" "ý" "þ" 1: Introduction Space has an infinite potential for aesthetics. Aesthetics developes as the number of dimensions that must work together increase. At all levels of aesthetic's developement, it is represented by simple polynomials of a determi- nate number of unknowns: Polynomials of: Begin to Function Aesthetically at: 0 Unknowns The Linear Dimension 1 Unknown The Planar Dimension 2 Unknowns The Third Dimension Etc............................................................. Polynomials of a particular class of aesthetics begin their develope- ment on a dimension of space and continue to function in higher dimensions. The nomenclature of a major class of aesthetics is based on the first dimension in which it appears. This first dimension figures largely in the makeup of each class of aesthetics, even as it continues to behave within the context of higher dimensions. The technique of geometric analysis of a polynomial's aesthetic quality is found by taking ratios among different line segments traversing the interior and exterior of a polyhedron in any dimension. Its factors result from such ratios. The dimension of a polyhedra's existence may or may not be dis- torted from the normal along the lines of higher curves. Qualities of aesthetics may be unique to each class. Other methods for analysis are to follow. Graphic representation of a polynomial's factors should not be confused with the graphing of a polynomial as a function. Only the former will be used in section two. No proofs will be offered for any of the ideas in this text. 2: Linear Aesthetic Theory Polynomials in no unknowns appear to be of only one type: universal aesthetics. Numbers appear only as real positive integers. Pythagoreans loved these kinds of numbers, because there was nothing left to the imagination. This may be the secret to simple aesthetics. At this level of enjoyment, aesthetics does not subdivide into smaller subsets as it does at higher dimensions. There is nothing complicated here; just the simple appreciation of integers as aesthetic expressions in themselves. A polynomial of no unknowns is very simple---it is simply a number. The factorization of a number tests for its primality and breaks it down into smaller numbers of prime value if necessary. The factorization of small composite numbers is never any problem--- second guessing usually works fine. Large numbers have elicited many theories. Most are based on knowing what special group of numbers the test number sprang from since number sets contain certain qualities that make their factorization unique. A recent famous example was the announcement in the New York Times dated March 22, 1994 of a reward for the correct factoring of a 129-digit number composed of just two prime factors. It was never solved by anyone other than the advertiser(Dr. Carl Pomerance, University of Georgia) with the help of scores of individuals working around the world on their own computers. Their task was made easy because of the use of the Quadratic Sieve, discovered by Pomerance in 1982. It made possible the breakdown of the problem into smaller pieces, the perfor- mance of parallel calculations on each individual's computer, and the reassembly of the whole solution. What normally would have taken centuries of computing on even the fastest of computers, instead took only a few weeks. The Base Conversion Method, that will shortly follow, can produce the same result on a personal computer in ?. 2a: Eratosthenes' Sieve Eratosthenes' Sieves are made by forming tables of integers of any dimensional array and striking out multiples. Anything left is prime. The aid of these tabular arrays is the ease with which the eye can pick out a pattern of strike-outs for any table. The pattern of multiples for each prime is found by going down a certain quantity of rows and then going a certain quantity of columns either left, right, or both depending on availability of columns. Each prime has its own pattern for that table: -2 -5 -4 -8 -7 -6 -11-10 -9 -8 -14-13-12-11-10 -17-16-15-14-13-12 -1 -3 -2 -5 -4 -3 -7 -6 -5 -4 -9 -8 -7 -6 -5 -11-10 -9 -8 -7 -6 0 -1 0 -2 -1 0 -3 -2 -1 0 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 2 3 4 4 5 6 5 6 7 8 6 7 8 9 10 7 8 9 10 11 12 3 5 6 7 8 9 9 10 11 12 11 12 13 14 15 13 14 15 16 17 18 4 7 8 10 11 12 13 14 15 16 16 17 18 19 20 19 20 21 22 23 24 5 9 10 13 14 15 17 18 19 20 21 22 23 24 25 25 26 27 28 29 30 6 11 12 16 17 18 21 22 23 24 26 27 28 29 30 31 32 33 34 35 36 1 1 2 1 2 3 1 2 3 0 1 2 3 0 5 1 2 3 0 5 0 2 3 0 0 5 0 5 0 7 0 0 7 0 0 0 7 0 0 0 11 0 3 5 0 7 0 0 0 0 11 0 11 0 13 0 0 13 0 0 0 17 0 0 7 0 0 11 0 13 0 0 0 0 17 0 19 0 19 0 0 0 23 0 5 0 0 13 0 0 17 0 19 0 0 0 23 0 0 0 0 0 0 29 0 0 11 0 0 17 0 0 0 23 0 0 0 0 29 0 31 0 0 0 0 0 d = down, l = left, r = right, n = 0, 1, 2, 3, 4, 5,... 2:d2 2:d1 2:d1,lr1 2:d1,lr=2*n 2:d1,lr=(2*n)+1 2:d1,lr=2*n 3:d1 3:d3 3:d1,r1; 3:d1 3:d1,l1,r2 3:d1,l2,r1 4:d1,lr2 5:d1,l1,r4 4:d4 d2,l1 4:d1,r1; 4:d1 4:d1,l1,r3 6:d1 7:d1,r1;d2,l5 5:d5 6:d6 By using these patterns, a pattern of patterns emerges: i(n+1) = the next position of the same number j(n) = another number used as a reference(base) for the other j(n+1) = the next position of the same number These tables exhibit the position of a number as part of a cycle. If a car were to have one size of wheels in the front and another in the back, then the position of either set after traveling a certain distance would be relative to each other. i = a number i(n) = the present position of a number b = the other reference(base) number + = right - = left quotient = down remainder = +,- i / b = quot(ient) + rem(ainder) (quot * b + 1) - rem i(n) = quot down, [right rem = left (b - rem)] i(n+1) = i(n) / base(cycle length) = i(n+1) + remainder(cycle position) By using the patterns of the first table, a pattern of patterns emerges: A multiple of an integer equals the integer plus its previous multiple, or: integer * n = integer + m(n-1). Also, Patterns of composites are not needed since they are already struck out by the multiples of a prime. Using these two observations, a conclusion can be drawn: THE FIRST MULTIPLE M(1) OF AN ARITHMETRIC PROGRESSION OF POSITIVE INTEGERS IS A PRIME NUMBER IF THE PROGRESSION IS NOT A SUBSET OF(AND CONSE- QUENTLY DIVISIBLE BY) ANY OTHER ARITHMETRIC PROGRESSION; BUT IT WILL BE A SUBSET OF THE FIRST PROGRESSION OF INTEGERS M(N)(WHEN P=1). Or, m(1)(p=?) is prime, if and only if, m(n)(p=?) is only an element of m(n)(p=1). And, m(n) = p * n = m(n-1) + p And, m(1) = p * 1 = 0 + p = p n = the index of each term of the series, 0,1,2,3,4,5,6,7,... p = a prime constant Examples: Progression of integers(by one's; p=1): m(n) = p * n = m(n-1) + p m(1) = 1 * 1 = 0 + 1 = 1 n = (0),1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,... m(n)(p=1) = 0, 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,... M(1), when p=1, is not in itself prime, but is the standard by which all other primes and composites are judged by. All series of multiples of primes are exclusively subsets of m(n), when p=1, and no other. All series of multiples of composites are subsets of this and at least two others(prime multiples). Progression of integers by two's(p=2): m(n) = p * n = m(n-1) + p m(1) = 2 * 1 = 0 + 2 = 2 n = (0),1,2,3,4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... m(n)(p=2) = 0 ,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,... m(n)(p=2) / m(n)(p=1) = 2 + 0 m(n)(p=2) = 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,... ö m(n)(p=1) = 0,1,2,3,4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... = quotients = 0,2,2,2,2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,... + remainders = 0,0,0,0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... M(n)(p=2) is a subset of m(n)(p=1) and its elements are evenly divided by the parallel terms(same "n" value) of m(n)(p=1) with a p=2 quotient and a zero remaining. Since there is no other it is also by no other. 2 is prime. Progression of integers by threes(p=3): m(n) = p * n = m(n-1) + p m(1) = 3 * 1 = 0 + 3 = 3 n = (0),1,2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... m(n)(p=3) = 0, 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,... m(n)(p=3) / m(n)(p=1) = 3 + 0 m(n)(p=3) = 0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,... ö m(n)(p=1) = 0,1,2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... = quotients = 0,3,3,3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,... + remainders = 0,0,0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... m(n)(p=3) / m(n)(p=2) = 1 + n m(n)(p=3) = 0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,... ö m(n)(p=2) = 0,2,4,6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,... = quotients = 0,1,1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,... + remainders (n) = 0,1,2,3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... M(n)(p=3) is a subset of m(n)(p=1) and is evenly divided by it, but by no other. 3 is prime. Progression of integers by fours(p=4): m(n) = p * n = m(n-1) + p m(1) = 4 * 1 = 0 + 4 = 4 n = (0),1,2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... m(n)(p=4) = 0, 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,... m(n)(p=4) / m(n)(p=1) = 4 + 0 m(n)(p=4) = 0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,... ö m(n)(p=1) = 0,1,2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... = quotients = 0,4,4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,... + remainders = 0,0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... m(n)(p=4) / m(n)(p=2) = 2 + 0 >>>> Because of this, 4 is not prime. m(n)(p=4) = 0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,... ö m(n)(p=2) = 0,2,4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,... = quotients = 0,2,2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,... + remainders = 0,0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,... m(n)(p=4) / m(n)(p=3) = 1 + n m(n)(p=4) = 0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,... m(n)(p=3) = 0,3,6, 9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,... = quotients = 0,1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,... + remainders (n) = 0,1,2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,... M(n)(p=4) is a subset of m(n)(p=1) and is evenly divided by it, but it is also a subset of m(n)(p=2) and is divisible by it. So 4 is not a prime, but a composite of 2 * 2. Etc............................................................................. The only regularity is the progression of values within each series--- be it multiples of a prime or of a composite integer. If there is an algorithmic relationship linking all series of multiples of primes simultaneously with each other, singling them out from the body of composite multiples, it has yet to be discovered. Living without such an exact formula gives the impression that factoring integers into their prime divisors is more of a skill than it is a science. Two algorithms will be presented. The first is methodical or random guessing. The second is unexplainably(by me) lucky guessing. 2b: Integer Division Method [IDm] A composite integer is made from at least two smaller factors. One of these factors can be no greater than the square root of the original composite, while the other must be greater unless both are equal to the square root. Since this square root boundary is less than half the magnitude of the composite integer, it is more effective to: 1) Search below the square root boundary for just one factor, 2) Divide the composite by the result of the search, and 3) Continue searching for more factors. The search for a factor can be methodic, i.e.: 1,2,3,4,5,..., selective: 3,5,7,..., or random: 23,11,47,... The numbers used to test a composite's divisibilty may be integers of unknown qualities or integers selected from a table of known qualitie(s): prime, etc. For instance, one means of selecting test integers methodically would be to use one counter starting at one less than a multiple of six and incrementing by six, along with a sub-counter incrementing by two. Possible primes not factorable by either two or three hang out on either side of multiples of six: 5,6,7,...,11,12,13,...,17,18,19,...,23,24,25,...,29,30,31,...,35,36,37,........ There is no way to make a series of primes by this method, other than by testing each integer as it comes along. 2b: Base Conversion Method [BCm] Apparently this method can produce both a series of tables of primes(at least as far as the number 48) as well as speed the testing of their divisibi- lity. It is composed of two algorithms: 1) A progression of tables of primes and composites up to "48" 2a) A progression of bases 2b) The testing of possible divisors by their derivation from their dividend 1) Tables of primes and composites are constructed from a couple of formulas: A) The base of each table: base = n! (n = 1,2,3) B) The array of each table(its length of columns and rows): array = [columns = base] * [rows = base + 1] C) The initial value in each table: initial value = base + 1 D) The final value in each table: final value = base * (base + 2) = [(base + 1) ^ 2] - 1 One is the initial first term(n=1) of the series and all succeeding terms might be calculated in terms of the immediately previous term: x(n) = ? --------------------------------------------------------------------------- [x(n-1) ^ 2] + x(n-1) n n! Prime! x(n-1) * (x(n-1) + 1) x(n) = [x(n-1) ^ (n-1)] + x(n-1) - --- ------ --------------------- -------------------------------- 1 1 1 1 1 2 2 2 2 2 3 6 6 6 6 4 24 30 42 222 5 120 210 1,806 10,941,270 6 720 2310 3,263,442 ~10 ^ 35 + 41,27 etc............................................................................. Then again, maybe the series is not intended to go beyond the number six! Whatever... Hypothesis: Each number of the series: 1,2,6 has its own special quality. One is the first integer, two is the first prime, six is the first perfect, etc. It would possibly be difficult to imagine correctly what the next special quality is and consequently the next number in the progression. Each time a successive number is discovered to satisfy the uses we will shortly put them to, a new formula may be required to explain the progression of the series as a whole. Thus, a transcendental-like paradigm would be required to explain the entire series as it progresses towards infinity---nothing short of omniscience. This would fulfill my adolescent hypothesis of a prime series formula: an evolutionary one--- something vain to predict. The first hypothesis may be better: What if the series ends with six? What if the series of tables of primes are not infinite? What if it is good enough that the Creator gave us a series of tables of primes and composites up to and including the number 48 and left it to us to use the third algorithm of this package to test larger composites? I won't worry about it. Like the backbone series having only three terms(to date), the series of tables of primes are also three after one element that initiates the series: 1 The number one is a given. It is not part of the progression of tables, but initiates them(the left- hand branch in terms of its elements; both branches in terms of the column-breadth of their arrays). It is irrefutably prime because of its unity. Preferably ---------- 1 or 2 The number one is a given. It has the quality of 2 3 unity as well as primality. All primes are in base 3 one. 1 2 or 3 6 The preferable version is a magic square. In each 3 4 7 4 example, column one(the reduced residue of base two) 5 6 5 8 contains nothing but primes, while column two are 7 8 all composites. These arrays are in base two. Preferably -------------------- C1 C2 C3 C4 C5 C6 These arrays are said R1 1 2 3 4 5 6 C1 C2 C3 C4 C5 C6 to be in a base of six, R2 7 8 9 10 11 12 or R1 7 14 21 28 35 42 because they recycle R3 13 14 15 16 17 18 R2 43 8 15 22 29 36 the use of the first R4 19 20 21 22 23 24 R3 37 44 9 16 23 30 six columns. All primes R5 25 26 27 28 29 30 R4 31 38 45 10 17 24 up to "47" will be R6 31 32 33 34 35 36 R5 25 32 39 46 11 18 found exclusively in R7 37 38 39 40 41 42 R6 19 26 33 40 47 12 columns 1 and 5 with R8 43 44 45 46 47 48 R7 13 20 27 34 41 48 two exceptions. Except: (written in base six) 25: column 1, row 5 and column 1, row 5 35: column 5, row 10(6) or column 5, row 1 (10 in base six equals 6 in base ten) When performing modular division(division which looks for remainders rather then quotients) with zero as the only projected remainder after dividing by a modulo(base) of six, then 1 and 5 are its reduced residues. This means that they are the only numbers which are relatively prime to the base---they each share no common divisor with the number six, except for the number one. This in some way may have something to do with the occurence of one, five, and one-zero (six in base six) being the only numbers used as the column and row numbers for primes and their exceptions. One is a reduced residue of modulo six: 1 / 1 = 1 + 0, 6 / 1 = 6 + 0 2 / 1 = 2 + 0, 6 / 1 = 6 + 0 2 / 2 = 1 + 0, 6 / 2 = 3 + 0 3 / 1 = 1 + 0, 6 / 1 = 6 + 0 3 / 3 = 1 + 0, 6 / 3 = 2 + 0 4 / 1 = 4 + 0, 6 / 1 = 6 + 0 4 / 2 = 2 + 0, 6 / 2 = 3 + 0 Five is a reduced residue of modulo six: 5 / 1 = 5 + 0, 6 / 1 = 6 + 0 4: Planar Aesthetic Theory The algebraic definition of planar aesthetics is composed of: 1a) Polynomials in one unknown of rational(or zero value) coefficients. If a a polynomial is composed of irrational or transcendental coefficients, then it is missing at least one factor. 2a) The factors of such polynomials are progressively approximated by a infinite series of ratios. Each factor is the limit of its progression. A) Natural convergence: This happens when a factor is analized out of context from its polynomial. There is no relationship between successive terms of the progression, other than an increase in the absolute value of its terms. The series terminates at the limit and the rate of convergence varies. Either the incomplete Euclidean algorithm or the incremental method will produce this. B) Integrated convergence: A series will merely approach its limit, even when the limit is rational, whenever it is analized within the context of belonging to a polynomial(1a). There is a direct coorelation between successive terms. Its convergence rate is an approximating progression of its own, approaching its own limit(a steady state). 3) A series may be skewed from its natural condition. This is called a skewed progression. A minimally skewed progression is a natural skew--- there is only one for each natural progression. There are infinitely numerous skews beyond the minimal. Normally a skew is determined initially, allowing the series to attempt to return to its natural state: the ratio approximation of a factor. Because it is skewed it will never return completely to its natural state, but be forever influenced by the amount and placement(s) of its skew(s). Regardless of whether or not it is skewed, all series are initiated by the use of numbers which are known as the seeds of the series. 4) Each ratio is a vocabulary unit. The units may be multiplied together in any combination of themselves and their reciprocals to create composite units. The set of all vocabulary and composite units composes pure temperment theory. There is an acquired skill to implimenting pure temperment theory because of the jigsaw puzzle nature to assembling ratios multiplicatively while still maintaining small numerators and denominators. 5) Pure temperment theory may express itself as: A) Relative linear measurements within a plane, and between planes. B1) Vibrations within any medium. C1) Vibrations may be pulsed rhythmically. Whereas the relations of linear measurements within a plane and the vibrations of a medium are represented by the division of a numerator by its denominator, rhythmic structures are created by multiplying pairs of integers. Amplitude accentuation of pulses is possible among any factors within a composite structure. 6) Equal temperment theory approximates and simplifies the subtle intricacies of the pure by whitewashing it into a logarithmic progression. It reduces potentially complicated harmonic relations and adds a sense of scale. 7) A second layer of pure temperment can be derived from the equal. This second layer has been modified by its passage through the equal. It shares qualities of the original plus the influence of simple or more complex logarithmic scale structures. The geometric definition parallels and adds a variation: 0) Plane geometry need not be flat. It may be curved. 1b) Polynomials are equivalent to polygons in that: 2b) Ratios may be represented by the values of: A) Dividing the length of the slope of an isosceles triangle by its base. Numerous such triangles are contained within a regular polygon with their apex angles all centered within a common vertice. B) Diagonals are the sides of stellated polygons of the same number of sides as their exterior regular polygon. These diagonals cross themselves as they revolve around their common center. 5) The apex angle is the ratio's corollary. It adds to pure temperment theory's B1) a: B2) Vibrations may be duplicated in wavelength while also being offset from one another as a phase relation. If a single cycle of a wavelength's phase is equated to a full circle of angular measurement, then an apex angle is a phase relation between two or more waves. B3) Postulate: The proportion between two wavelengths is equivalent to an apex angle(as computed above) and the difference between their refractive angles. C2) Two or more rhythmic pulses of waves may be offset from one another by a phase relation. Whereas B2) is a microeffect of coherent waves, this(C2) is a macroeffect of incoherent waves. Stylistic Theory is the interpretation of applied aesthetics. It is dependent on either pure, equal, or both temperments for its analysis of dif- ferent elements of aesthetics: melody, chords and their progression, rhythm, dynamics. Harmony is the cooperation of various elements, all of which are related to a common polynomial. Planar aesthetics comes in several classes. Some are: I) Aesthetics A) Real 1) Beauty a) Undistorted(Flat Plane) i) Golden ii) Hybrid investigate >>>> b) Distorted(Curved Plane) " 2) Power " a) Undistorted(Flat Plane) " i) Golden " ii) Hybrid " b) Distorted(Curved Plane) " B) Non-Real " 1) Beauty: Complex " 2) Power: Imaginary II) Non-Aesthetics: Does it exist? More thoroughly: I) Aesthetics A) Real 1) Beauty a) Undistorted(Flat Plane) i) Golden ii) Hybrid b) Distorted(Curved Plane) 2) Power a) Undistorted(Flat Plane) i) Golden ii) Hybrid b) Distorted(Curved Plane) B) Non-Real 1) Beauty: Complex 2) Power: Imaginary II) Non-Aesthetics: Does it exist? 5: Overview of Planar Methods There are infinitly various methods for solving polynomials. Partial solution is the all of approximation theory. Complete solution methods that work under all conditions do not exist, because the irrational, complex, and tran- scendental fields of numbers can only be accurately defined as functions---not as numbers. They come in categories of: I) Natural Convergence 1) Incremental Progression II) Integrated Convergence 2) Polynometric Progression 3) Geometry (Trigonometry) 4) Polynomial Self-Looping 5) Algebraic Expressions 6) Coefficient Structures 7) Transcendental Functions I) Natural Convergence [NC] A) Incremental Progression [IP] 1) The Incremental Method [INCm] 2) The Interactive Method [INTm] II) Integrated Convergence [IC] B) Polynometric Progression [PP] 3) The Serial Method [SPPm] 4) The Parallel Method [PPPm] C) Geometry (Trigonometry) 5) Golden Trigonometric Table [GTt] 6) Golden Trigonometric Method [GTm] 7) Modular Method [Mm] D) Polynomial Self-Looping [PSL] 8) The Radical Method [RADm] 9) Continued Fractions [CF] 10) Continued Fractions of Square Roots [SQRCF] 11) Continued Fractions of Transcendental e and pi [eCF] & [piCF] 12) Reversal of Continued Fractions of Transcendental Numbers for Analysis of Their Convergence Efficiency [RCF] 13) The Full Euclidean Algorithm [Ea] 14) The Short Cut Coefficient Method [SCCm] 15) The Ratio Reduction Method [RRm], also known as: The Incomplete Euclidean Algorithm [IEa] 16) The Reflecting Golden Light Model [RGLM] 17) The Golden Method [Gm] ?Four Reversing Methods to Hypothesize Polynomials from their Ratio Sets [REVm] 18) The Reverse Euclidean Algorithm [REa] 19) The Greatest Common Divisor Algorithm [GCDa] 20) Ratio Set Reversal [RSR] 21) The Inverse Euclidean Algorithm in Multiple Unknowns [MUEa] ?) Any More? Two Standard Averaging Methods [SAm] 22) The Bisection Method [Bm] 23) Newton's Special Case for Shell Polynomials of the form: x^a + b E) Coefficient Structures [CS] 24) Geometric Progression Methods [GPm] 25) Summation of Power Methods [SPm] 26) Summation of Powers of Multiplicative Pairs of Roots 27) Summation of Powers of Multiplicative Triples of Roots Etc................................................................... 28) Word Problems to Construct Polynomials [WP] F) Algebraic Expressions [AE] 29) Factoring Real Beautiful Quadratic Polynomials 30) Factoring Real Power Polynomials G) Transcendental Aesthetic Functions Do they exist? Serial/Parallel/Direct Progression S...............1) Natural Convergence S & P............2) Polynometric Progressions S...............3) Continued Fractions S...............4) Continued Fractions of Square Roots (both at 2nd deg.) & S..5) Complete Euclidean Algorithm " & P............6) Incomplete Euclidean Algorithm, GCD Algorithm P...............7) Golden P...............8) Golden Coefficient S...............9) Golden Trigonometric S...............10) Radical S & D............11) Geometric D...............12) Coefficient S...............13) Averaging D...............14) Solution The reason why there are so many is because most, if not all, of the methods are derived from the very polynomials they are meant to solve, or from the general knowledge of how polynomials are structured internally. The more factors a poly- nomial contains, the more methods there are that can be derived from them. But they seem to come in a limit of categories. The first four, of the above methods, produce output for aesthetic use. The last is used in tandem with some of the other methods or can be used for scientific purposes. Of the above methods: 1) Is an illustration of what might happen if convergence was the only criteria for generating an approximation series 2) Is a solution for general polynomials in one unknown for all degrees 3) 4) 5) 6) Helps with the conversion of monic polynomials(1x^largest degree of polynomial + ...) into non-monic(ax^larg.deg. + ...) when any of the polynomial's coefficients are fractions rather than whole numbers 7) Produces ratio approximations of the factors of golden polynomials, in all degrees, without foreknowledge of said polynomials 8) Produces the coefficients of golden polynomials in all degrees without foreknowledge of their factors 9) Links all golden and semi-golden polynomials with their associate polygonal factor storehouses 10) Approximates the numeric values of all the real and imaginary(but not complex) factors of all general polynomials 11) Polynomial Theory Polynomials in one unknown are programs for planar beauty. The plane need not be flat. If it is, the beauty contained therein is golden or semi- golden. Algebraic expressions in one unknown take the appearence of: 0 = ax^0 + bx^1 or simply: 0 = a + bx where a and b are known and x is not. These are also known as linear polynomials, or polynomials of only one factor. Linear implies that the graph of this expression is a line within a plane, but nowhere in this text will the graphing of polynomials be undertaken. Polynomials in multiple factors are then built up by multiplying these algebraic expressions: 0 = (ax + b) * (cx + d) = acx^2 + (ad + bc)x + bd For simplicity, the above equation will be written as: 0 = (x - 1st Factor) * (x - 2nd Factor) = ax^2 + bx + c 0 = (x - Root 1) * (x - Root 2) = ax^2 + bx + c 0 = (x - R1) * (x - R2) = ax^2 + bx + c 0 = " = x^2 + (b/a)x + c/a simply: 0 = " = x^2 + bx + c or: 0 = " = a + bx + x^2 Because each factor is written as being equal to zero, then x is equal to either factor, but not both at the same time: 0 = x - R1 or 0 = x - R2 R1 = x or R2 = x R1 and R2 may be positive or negative. For further simplicity and to insure that both factors are real and not imaginary or complex, only real polynomials will be dealt with. Some polynomials are hard to tell if they have only real roots. An easy way to insure a polyno- mial's "reality" is to restrict ourselves to a particular form: 0 = a*cx^0 + b*dx^1 + A*ex^2 + B*fx^3 + a*gx^4 + b*hx^5 + A*ix^6 + B*x^7 + + a*x^8 + b*x^9 + ....., + (a, b, A, or B)*?x^(degree of polynomial) where a * A = b * B = -1 and (a, A, b, B) are elements of set {+1, -1}. Arranging all of the roots in absolute value from greatest to least, the first root is the polynomial's primary factor and all of the others are its auxiliary factors. Also, if a = b and A = B then the even roots are negative. But if a = B and A = b then the odd roots are negative. So changing the signs of the even powers of x (a, A) merely affects the appearence of the whole polynomial without affecting the signs of any roots. But changing the signs of the odd powers of x (b, B) does by inverting the signs of every root. It is not so important what signs the roots are; in this situation, it is merely good housekeeping to insure rational coefficients. It is the distinct absolute values of numbers that will have any effect in the harmonic world of proportional relations. Signs of factors(if they are required) are either the same with respect to one another or they be different since any reference point is valid. 7a: Polynometric Progressions The simplist method to solve this, and equations of higher degree(more factors), for the value of x for aesthetic purposes is to use the Polynometric Method. An arithmetic progression is: - infinity, ....., a, b, c, d, e, f, ....., + infinity where the relation between any two adjacent terms in the series involves addition or subtraction: b = a + k a = b - k c = a + 2k where k is some constant. A geometric progression is: 1, a, b, c, d, e, f, ....., (+, -, or alternately +-)infinity or: 0, ....., f, e, d, c, b, a, (+, -, or alternately +-)1 where the relation between any two adjacent terms in the series involves multiplication or division: b = a * k or: b = a / k where k is some constant. An exponential progression looks like a geometric, except it progresses much more rapidly because the series involves exponentiation: b = a ^ k or: b = a ^ (1 / k) A polynometric progression is geometric when the polynomial is of the first degree. For all other degrees it is an arithmetic progression approxima- ting a geometric. The relation between adjacent terms in the series involves the previous terms used as variables within a polynomial. For a polynomial of the first degree: -ax^0 + x^1 = 0 ax^0 = x^1 If |a| > 1, then: 1, p, q, r, s, t, ....., (+, -, or alternately +-)infinity But if 0 < |a| < 1, then: 0, ....., t, s, r, q, p, (+, -, or alternately +-)1 where |a| means the absolute value of "a". Replacing x^0 and x^1 with two adjacent terms of the series: a * 1 = p And: a * p = q Also: a * q = r Etc.............................. This is a geometric progression. Since: ax^0 = x^1 or: a = x then "a" is the exact solution, or root, of this polynomial. The second alternate series will be dropped for now on because it essen- tially does the same thing, but instead of progressing towards infinity it is progressing towards infinity's opposite: zero. It does this because the overall progressive value of the polynomial is a fraction whose absolute value is between zero and one, as opposed to being between one and infinity. In short, the goal is complimentary(reciprocal) to the path. For a polynomial of the second degree, it actually has two series that run in parallel: -ax^0 - bx^1 + x^2 = 0 ax^0 + bx^1 = x^2 Set: -1, 0, 1, 2, 3, 4, 5, 6, ....., infinity Series 1: y, z, b, p, q, r, s, t, ....., (+, -, or alternately +-)infinity Series 2: 1/a, ....., (+, -, or alternately +-)infinity (*) is undefined. "y" and "z" are seeds which initiate the series. It doesn't matter what they are, just so long as they are integers. If y = 0 and z = 1, then "b" from the above polynomial will result immediately afterward. But sometimes it is justifiable to seed with different values, (1, 1) = (y, z) for instance. Any seed pair will do. In fact, there is no need to stick to this series strictly. Previously determined values may be tweeked(changed slightly) or zonked(far out) of what would have been there natural occurence in order to skew the progression. To wit: If (y, z) = (0, 1): natural series = (anything else): unnatural series For the quadratic(second degree) polynomial: 0 = -1 - x + x^2 If (y, z) = (2, 1): Lucas series (loo`kah; a French name) = (4, 1): Ptah series (name of an Egyptian god) Replacing x^0, x^1, x^2 with three adjacent terms of the series(skipping any undefined term) and calculating only one series since both are identical: a * 0 + b * 1 = b Also: a * 1 + b * b = p And: a * B + b * p = q Etc.............................. This is an arithmetic progression approximating a geometric: a polynome- tric. Its roots are found to be approximately equal to ratios among certain elements within each set(column of terms). For any set: {1, 0), {B, 1}, {p, B}, {q, p}, ....., ....., {element of Series 1, element of Series 2} of Set Infinity or simply, ....., {S1, S2}, ....., Root 1 =: S1 / S2 Root 2 =: -a * (S2 / S1) =: means approximately equal to. "-a" is from the above second degree polynomial. For a polynomial of the third degree, it has three series that run in parallel: -ax^0 - bx^1 + x^2 = 0 ax^0 + bx^1 = x^2 1, p, q, r, s, t, ....., (+, -, or alternately +-)infinity or: 0, ....., t, s, r, q, p, (+, -, or alternately +-)1 Replacing x^0, x^1, x^2 with three adjacent terms of the series: a * 1 + b * p = q Also: a * p + b * q = r And: a * q + b * r = s Etc.............................. Bibliography ?, Denmark: A Social Laboratory Experiment, ? 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