The Composition of Number

Transcription

1 An introduction to The Composition of Number Original Work in the Field of Theoretical Mathematics Barbara Mae Brunner Revision Copyright , Barbara Mae Brunner, All Rights Reserved

2 COPYRIGHT , ALL RIGHTS RESERVED The concepts presented in this summary, and in the greater body of work entitled The Composition of Number, are a result of my personal exploration into numbers and number patterns. The popular view that the perfection of the Fibonacci sequence is in its tendency to move toward a phi ratio between progressing consecutive numbers had always felt incomplete to me--not quite reaching the truth of the matter. I felt that there must be perfection, somehow, within the entire series of numbers right from the beginning. Exploring this thought catapulted me quickly into the concept of the composition of numbers and number patterns. I found that perfection did exist from the beginning when looking at that which composes the Fibonacci sequence from under its surface-level appearance. Furthermore, I found that the Fibonacci sequence is composed in the same manner as an unlimited number of other sequences just as perfect, and that the phi ratio acts in the same manner as an unlimited number of ratios within the composition of similar number sequences. The common procedures used to calculate phi and the Fibonacci sequence appeared as incomplete versions of more generalized procedures--procedures whose components represented different aspects of a greater mathematical mechanism. I was beginning to see the hidden nature of numbers and number patterns through the study of their composition. However, without an education in mathematics, I faced the challenge of how to convey my discoveries in a language consistent with the subject matter. I was also unaware of how much of what I discovered is already known by other individuals. In the end, I have opted to let my work stand on its own as the result of a personal discovery process, and to express it as such. This approach may make its content more accessible to others who are not formally educated in mathematics but who, nevertheless, are seeking to know perfection through the study of numbers.

3 GENERAL CONCEPT A mathematical structure, which I call the ratio-based fabric, exists under the surface-level appearance of all numbers and number patterns. The operations of multiplication, addition, subtraction, and division applied to the structure initiate a process of composition for all numbers and number patterns. The ratio-based fabric is comprised simply of points in space. A point references a location within space and is represented by a numerical value. Every number represents both an A and B point. A and B points are not duplicated throughout the ratio-based fabric but, instead, stand in relationship to all other points through ratios and resonance. Space itself becomes defined when held between points. Spaces of the same quantity exist throughout the ratio-based fabric and, under specific conditions, become organized into fields. Multiplication, addition, subtraction, and division are seen as natural forces that bring this limitless, underlying mathematical structure to life--enabling and sustaining the composition of numbers and number patterns of many different types. The composition process occurs when points are connected into mathematical formations-- as will be presented in this introduction. Relationships between numbers and number patterns that would otherwise be undetectable or difficult to detect become visible when working with the ratio-based fabric. In essence, the unique nature of a number or a number pattern is revealed when exploring its hidden composition through the study of the ratio-based fabric. 2

4 SETS OF RATIO-PROGRESSED STRINGS An overview of one particular method used to connect paired A and B points to other paired A and B points will quicken the process of conveying the general concepts. This method involves the use of geometric progressions, referred to as ratio-progressed strings within this body of work. Two identical ratio-progressed strings, String A and String B, are laid side by side in opposite directions and are aligned at the square root of a composition number to create a point of resonance (see displays 1 11), or the two ratioprogressed strings are non-aligned around the focal point of the square root of a composition number (see display 12). The formation of points create a framework from which numbers and number patterns are composed through the operations of multiplication, division, addition, and subtraction all operations are performed in the A to B and B to A directions. The main function of a set of ratio-progressed strings is to x-compose (cross-compose) a number, the composition number, by multiplying the points of String A by the points of String B, and vice versa. The B A and A B ratios between ratio-progressed strings create patterns of ratios that are reflective of a sub-structure within the ratio-based fabric which will be explored further. The operations of addition and subtraction between the two ratioprogressed strings compose various number patterns. Four types of number patterns are composed for each direction: the series of sums (A + B, B + A), the series of differences (A B, B A), and two patterns are composed through the alternate linking of the sums then differences, and vice versa, starting from the point of resonance; i.e., the sum of the current paired A and B points are connected to the difference of the next paired A and B points which are then connected to the sum of the next paired A and B points, and so on and vice versa. The later two number patterns link the sums in the B + A column with differences in the B A column, and vice versa, to create two separate yet intertwined number sequences. The same linking between the sums in the A + B and the differences in the A B columns create a reflective version of the intertwined number sequences. For simplification purposes, the intertwined number patterns are referred to as fibonacci-type sequences throughout this introduction. 3

5 Several categories of fibonacci-type sequences have been identified. All categories of fibonacci-type sequences are calculable from a surface-level perspective using a simple mathematical procedure. Different forms of composition from within the ratio-based fabric constitute the different categories of fibonacci-type sequences. For example, Strings A and B are geometric progressions. However, geometric progressions are also calculable using the same mathematical procedure that is used to calculate a fibonacci-type sequence and, therefore, constitute a category of fibonacci-type sequences. The category of fibonacci-type sequences that result from the alternate linking of sums then differences, and vice versa, between a set of ratio-progressed strings are composed when both points and space weave together within the ratio-based fabric of a primary composition number. The intricate make-up and nature of this category of fibonaccitype sequence is revealed upon much further study of the ratio-based fabric. The mathematical procedure used to calculate any fibonacci-type sequence incorporates a multiplier referred to as the M factor (MF) or the XM factor (XMF). The XM factor is the negative of the M factor. The M factor is used when the numbers of the number sequence are ascending while the XM factor is used when the numbers of the number sequence are descending in order. The procedure is quite simple. The current number is multiplied by the appropriate M or XM factor and then the previous number is added to the result to calculate the next number in the sequence. Of special importance, the M factor of 1 is associated with the Fibonacci sequence. The process of multiplying, as described in the above procedure, by this particular M factor may seem unnecessary. However, the modification of the common procedure for calculating the Fibonacci sequence to incorporate a multiplier is relevant to gaining a greater comprehension of the fundamental nature of the number sequence. The number patterns that emerge as the series of sums or differences, independently, between a set of ratio-progressed strings are calculable with a different but similar procedure. Several categories of this type of pattern have been identified, and also include geometric progressions. This procedure requires the use of a component called the R factor. The R factor represents a sub-structure within the ratio-based fabric, and will 4

6 be introduced within this material. The current number is multiplied by the square root of the appropriate R factor and then the previous number is subtracted from that result to calculate the next number in the sequence. A review of the included displays will aid in understanding these procedures. For additional clarity, note that display 9 will be examined later in this material. A resonance link within the ratio-based fabric defines a family of composition numbers and their resultant number patterns. For example, as shown in displays 2 through 4, a resonance link is generated when the point of resonance (the alignment of the square roots of the composition number) is re-aligned to create a different point of resonance within the same set of ratio-progressed strings. The re-alignment causes the set of ratioprogressed strings to x-compose a different yet related composition number. In essence, the same set of ratio-progressed strings exists within the ratio-based fabric of every composition number that the set x-composes when re-aligning the point of resonance through its limitless yet precisely defined possibilities. This process reveals a highly ordered mathematical structuring of a family of composition numbers and their resultant number patterns within the ratio-based fabric. Sets of ratio-progressed strings are embedded or nested within each other. This form of relationship can be seen when examining displays 5, 3 and 6 in that particular order. This condition happens when the progression ratios used to build sets of aligned ratio-progressed strings for the same composition number are progressively squared. The number patterns that emerge through the addition and subtraction of the sets of ratioprogressed strings also reveal an embedded state. With further study, the intertwined fibonacci-type sequences resulting from the alternate linking of sums and differences, and vice versa, for embedded sets of ratio-progressed strings reveal more of the nature of this particular category of fibonacci-type sequences. Another method of connecting paired A and B points for the same composition number is by pairing and linking specific mathematical means found between points. For instance, the harmonic and arithmetic means between paired A and B points join as A and B points to x-compose the same number as shown in displays 13 and 14. The paired 5

7 mathematical means connect vertically to form a mathematical structure that expands within the ratio-based fabric of a composition number when incorporating the sets of ratioprogressed strings that contain each of the paired A and B points. This structure is compounded to include the vertical alignment of mathematical means for each pair of A and B points within the sets of ratio-progressed strings, and so on. When exploring the mathematical means, it is worth noting that all mathematical formations within the ratio-based fabric of a composition number revolve around the geometric mean or, from another perspective, the square root of the composition number. The geometric mean that exists between all paired A and B points used in the process of x-composition will always be the square root of the composition number used in forming a point of resonance within a set of aligned ratio-progressed strings. This condition generates a far-reaching stabilizing effect within the ratio-based fabric. The ratio-based fabric of each composition number is limitless yet definable, and, upon study, reveals the unique nature of the composition number including the number patterns that are associated with the composition number. THE R FACTOR SUB-STRUCTURE Consistencies in the relationship between all paired A and B points and paired points called RFA and RFB revealed a sub-structure within the ratio-based fabric. RFA and RFB points have the condition of both adding and multiplying to the same number called an R factor. The RFA and RFB points themselves comprise what is referred to as the R factor sub-structure. Paired A and B points will exhibit specific conditions that reflect their association with their underlying RFA and RFB points as well as the R factor itself some of which is included in this presentation. Information used in building as well as observing relationships within the ratiobased fabric is associated with the R factor as shown below. 6

8 R factor AP ratio BP ratio M factor RFA RFB 5 reversed A reversed R factor appears in the above display. It is important to understand that the number represented as the R factor is used as a convenience when working with mathematical procedures. The ratio-based fabric is considered a physical mathematical structure that composes numbers and number patterns through the natural forces of multiplication, addition, subtraction and division. Mathematical procedures are simply tools used to witness the composition of numbers and number patterns. As stated, the fact that RFA and RFB both add and multiply to the same number called the R factor is what identifies A and B points as RFA and RFB points. The two numbers that meet these qualifications play the role of both an RFA and an RFB point as they do in their roles as both A and B points. The two points are in relationship with each other in both directions and as both the RFA and RFB points. This creates some conditions that are reflective of each perspective. The information represented by the R factor is calculable through a few simple procedures. R factors may or may not be whole numbers. However, an R factor must be a number of 4 or greater. This requirement reflects the mathematical limitation that four is the lowest number that two numbers will both add and multiply to. 7

9 AP ratio: BP ratio: M factor: RF (RF 4) = Progression ratio for String A 2 RF + (RF 4) = Progression ratio for String B 2 (RF 4) = Multiplying factor for fibonacci-type patterns RFA: AP ratio = R factor A or RF x AP ratio = R factor A RFB: BP ratio = R factor B or RF x BP ratio = R factor B RFA:RFB: AP ratio 2 = A to B ratio RFB:RFA: BP ratio 2 = B to A ratio Sets of ratio-progressed strings calculated using the progression ratios of a particular R factor are considered to be built from the R factor. Number patterns and relationships resulting from the multiplication, division, addition, and subtraction of the set of ratio-progressed strings are calculable through procedures that incorporate this particular R factor, its square root, and/or the associated M factor. Patterns of R factors along with their paired RFA and RFB points underlie patterns of paired A and B points. This condition exists not only for the formation of ratio-progressed strings but for other mathematical structures within the ratio-based fabric such as the previously presented linking through the mathematical means. Different types of underlying patterns of R factors exhibit conditions that are exclusive to the method of mathematical organization of paired A and B points. The same pattern of R factors along with their RFA and RFB points underlie all sets of aligned ratio-progressed strings built from the same R factor see displays 14 and 15. The pattern of A to B and B to A ratios are reflective of the underlying pattern of R factors. The pattern of ratios also act as a set of aligned ratio-progressed strings that x-compose the number one. However, it is important to note that two different R factors are associated with the series of numbers according to whether they are acting as a pattern of A to B and B to A ratios or as a set of ratio-progressed strings. 8

10 The use of two different R factors can be seen in display 8. The numbers found in columns A B and B A represent the pattern of A to B and B to A ratios associated with R factor 8 the first R factor. A second R factor of 36 is associated with the same numbers as they act as points within a set of aligned ratio-progressed strings to x-compose number one. The progression ratios of R factor 36 build the set of ratio-progressed strings. The difference in association of R factors is based on the role of the series of numbers ratios or points. This second R factor also works with the series of RFA and RFB points that underlie sets of ratio-progressed strings built from the first R factor. For example, in display 16, the series of RFA and RFB points underlying sets of ratio-progressed strings built using the first R factor of 8 (displays 7, 8 and 9) are calculable using the procedure previously described as an alternative to calculating the series of sums and differences, independently, between a set of ratio-progressed strings. However, in this case, the procedure incorporates the information associated with second R factor of 36. The main difference in the association of R factors is seen through understanding the role of squaring the progression ratios. With further study, the general mechanics behind this condition can be more fully understood within the concept of squaring as a naturally-induced process within the composition of numbers and number patterns. The number patterns that emerge through the addition, subtraction, and multiplication of RFA and RFB points that underlie sets of ratio-progressed strings are all calculable with the same procedure. This procedure is also used to calculate the series of RFA points and RFB points, independently, and can also be used to calculate a geometric progression as well as the series of sums or differences, independently, between a set of ratioprogressed strings. As a result of the different forms of composition, different categories of this type of number pattern have been identified. The procedure is a bit more complex than the procedures presented previously. The space between the first previous point and the third previous point from the current point is multiplied by the square root of the R factor. That result is then added to the fourth previous point to calculate the next point, or number, in the series. There is an alternative procedure to calculate this type of pattern. The space between the first and second previ- 9

11 ous points from the current point is multiplied by the square root of the R factor plus one. The result is then added to the third prior point to calculate the next point, or number, in the series. It is important to note that fibonacci-type sequences are not created through the alternate linking of the sums and differences between a pattern of RFA and RFB points that underlie sets of ratio-progressed strings. As noted, the series of sums and differences, independently, between a pattern of RFA and RFB points are calculable using the alternative procedure but not the primary procedure that calculates the sums and differences, independently, of a set of ratio-progressed strings. These conditions reflect that patterns of RFA and RFB points are not geometric progressions. The relationship between paired A and B points and their underlying paired RFA and RFB points, as well as the R factor itself, is visible through a variance ratio. The variance ratio is calculable as the A to RFA and the B to RFB ratio, and is seen in relationships throughout the workings of the ratio-based fabric. For example, the R factor multiplied by the variance ratio will equal the sum of the paired A and B points while the R factor multiplied by the variance ratio squared will equal the product. From another perspective, the product to sum ratio is equal to the variance ratio, while the sum to R factor ratio is also equal to the variance ratio. In the following display, paired A and B points are calculated from paired RFA and RFB points using different variance ratios. All of the paired A and B points x-compose the number The relationship between the sum and the product of the paired A and B points through the variance ratio is also displayed. RFA RFB Variance A B A + B RF x Variance Ratio = A + B RF x Variance Ratio 2 = A x B 5 x 72 = x 72 2 = x = x = x 36 = x 36 2 =

12 The relationship between the pattern of paired A and B points within a set of aligned ratio-progressed strings and the pattern of underlying RFA and RFB points is visible through a pattern of variance ratios. Although the impact of this condition are too involved for the intent of this introduction, the study of display can aid in gaining some valuable insights. Finally, from a different perspective of a mathematical sub-structure within the ratio-based fabric, it should be noted that all sets of ratio-progressed strings that x-compose number one can be used as blueprints from which every set of ratio-progressed strings may be built using a simple procedure. The procedure requires the entire set of ratio-progressed strings to be multiplied, uniformly, by the square root of a chosen composition number see displays 21 and 22 for examples of this form of relationship. The associated R factor will be the same for both sets of ratio-progressed strings. Further examination of the mechanics of this material will allow for greater levels of mathematical comprehension, and theoretical exploration. BRIEF EXAMINATION OF DISPLAY 9 In the following material, display 9 will be briefly examined the set of ratio-progressed strings built from R factor 8 that x-compose number 64. The first step is to identify the progression ratios used in calculating Strings A and B. The BP ratio is calculated as [ 8 + (8 4)] / 2 = The AP ratio is calculated as [ 8 (8 4)] / 2 = The set of ratio-progressed strings are then calculated in the same manner as geometric progressions using the BP ratio to calculate String B and the AP ratio to calculate String A. The numbers comprising a set of ratio-progressed strings represent a mathematical formation of points within the ratio-based fabric. The natural forces of multiplication, division, addition, and subtraction activate the points and the space held between the points to compose numbers and number patterns of many dif- 11

13 ferent types. Although Strings A and B are built as geometric progressions, the series of numbers can also be calculated using the procedure to calculate fibonacci-type sequences. This procedure requires the use of the M factor. In this case, the M factor is calculated as (8 4) = 2. The procedure to calculate any fibonacci-type sequence requires that the current number is multiplied by the M factor (or the XM factor) and then added to the previous number to produce the next number in the sequence. Beginning at the point of resonance defined as the alignment of the square roots of the composition number for String B, the series of numbers is calculated as follows: Current # x M factor Previous # = Next # The XM factor, the negative M factor, is used when the numbers in the sequence are in descending order. For example, beginning at the point of resonance for String A, the series of numbers is calculated as follows: Current # x XM factor Previous # = Next #

14 Another category of fibonacci-type sequence is found when alternately linking the sums and differences as described earlier in this presentation. Different categories of fibonacci-type sequences reflect different forms of underlying composition some of which have not been covered in this material. The familiar Fibonacci sequence is included in this category of fibonacci-type sequences as shown in display 3. Four fibonacci-type sequences of this category result from the operations of addition and subtraction, in both directions, of a set of ratio-progressed strings. Specifically, two sets of two intertwined fibonacci-type sequences emerge by alternately linking the sums and then differences, and vice versa as shown previously. In display 9, starting from sum in the B + A column of the A and B points representing the point of resonance, this series of numbers is calculated as follows: Current # x M factor Previous # Next # Starting from sum in the B - A column of the A and B points representing the point of resonance, this series of numbers is calculated as follows: Current # x M factor Previous # Next #

15 The series of numbers in the sums and differences columns, independently, can be calculated from a surface-level perspective through a similar procedure. However, it incorporates the square root of the R factor rather than the M factor, and it subtracts rather than adds the previous number. The series of difference in column B - A are calculated as follows: N2 (current) X R factor N1 (previous) N3 (next) The series of sums in column B + A are calculated as follows: N2 (current) X R factor N1 (previous) N3 (next) The pattern of ratios found in the B A and A B columns will be the same for any set of aligned ratio-progressed strings built using R factor 8 see displays 7, 8 and 9 and are reflective of an underlying pattern of R factors see display 16. The number patterns that emerge through each of the operations of multiplication, addition, and subtraction of the underlying pattern of RFA and RFB points are calculable using the alternative procedure that can be used to calculate the sums or differences of a set of ratio-progressed strings. However, the R factor of 36 is used in the calculation process. The use of different R factors was covered in the section entitled The R Factor Sub-Structure. 14

16 The series of numbers representing the pattern of R factors found in the B x A and A x B columns are calculable as follows: 1 st Previous # rd Previous # Result x RF th Previous # Next # The above procedure is also used to calculate the sums and differences, independently, between the pattern of RFA and RFB points see display 16. THE TRANSFORMATION OF THE COMMON PROCEDURE FOR CALCULATING PHI The discovery of a hidden mathematical structure supporting the composition of numbers and numbers patterns led to the transformation of the commonly recognized procedures used to calculate phi and, as already presented, the Fibonacci sequence. The first step in the process of transforming the common procedure for calculating phi into a generalized procedure applicable to the entire workings of the ratio-based fabric began with the procedure shown above to calculate the BP ratio. Discovering the correlation between this procedure and the existence of a sub-structure within the ratio-based fabric brought a sense of completion to my discovery process. However, upon further consideration, two aspects of this new procedure appeared unsettled. The hard-coded had transformed to 5 + (5 4) to represent RF + (RF 4). Yet the over 2 division remained hard-coded, as did the minus 4. Suspecting that 15

17 these represented something within the physical nature of the ratio-based fabric, the search was on for a procedure that might work to produce the associated BP ratio for all paired A and B points within the ratio-based fabric. It seemed that there was a strong possibility that the 2 in the procedure represented the harmonic mean between RFA and RFB when considering that two is the harmonic mean between all RFA and RFB points. It also seemed possible that the minus 4 represented two times this harmonic mean. These possibilities required a more detailed examination of the physical relationship between RFA and RFB points. The harmonic mean between two numbers divides the space between the numbers by the same ratio as the numbers themselves. The space between the harmonic mean and RFA (space A) along with the space between the harmonic mean and RFB (space B) were found to both add and multiply to four less than the R factor. This was the insight that made the difference. My original thought was that the A and B spaces added to four less than the sum of the RFA and RFB points due to the subtraction of the harmonic mean of two by two times. The A and B spaces multiplying to four less than the product of the RFA and RFB points proved to be a condition applicable only to RFA and RFB points. The inference of these observations is that the (RF 4) in the procedure translated to [RF (HM x 2)]. This modification did work for RFA and RFB points but did not work for other paired A and B points. This fact indicated that things were not sufficiently resolved, and that the search for the generalized procedure must continue. The first step was to break down the formula even further to better understand the different components. RF became (A x B) and over 2 became over the harmonic mean. These were reasonable modifications that lent themselves to the discovery of a more generalized procedure. The only component not fully understood was (RF 4). Experimenting with the A and B spaces defined as the differences between the harmonic mean and each of the A and B points seemed the logical place to start. 16

18 The answer was found there. (RF 4) became [(HM A) x (B HM)]. The thought that the minus 4 represented two times the harmonic mean was correct but not in the manner I previously thought. The procedure had worked for RFA and RFB points because the A and B spaces not only add but also multiply to the R factor minus 4. To prove whether these modifications were accurate, the following procedure was thoroughly tested with RFA and RFB points as well as paired A and B points. (A x B) + [(HM A) x (B HM)] = BP ratio HM This new procedure calculated the same BP ratio for any paired A and B points associated with same R factor. The following are examples using two different pairs of A and B points associated with R factor 5. Both examples produce the BP ratio of phi. ( x ) + [( ) x ( )] 2 = ( x ) + [( ) x ( )] = A minor yet significant modification of the above procedure resulted after further exploration. I separated the procedure into two parts at the point of the operation of addition. The modification proved to better represent physical aspects within the ratio-based fabric. This perfected procedure for calculating the BP ratio is shown below along with its complimentary procedure for calculating the AP ratio. (A x B) + [(HM A) x (B HM)] = BP ratio HM HM (A x B) - [(HM A) x (B HM)] = AP ratio HM HM 17

19 The following examples represent the paired A and B points taken from the two previous examples when using the modified procedure for calculating the BP ratio. In simple terms, both examples reduce to = The procedure for calculating the AP ratio using the same paired A and B points would reduce to = ( x ) + [( ) x ( )] 2 2 = ( x ) + [( ) x ( )] = The procedure is used in the following two examples to calculate the BP ratio of which associated with R factor 8. The paired RFA and RFB points for R factor 8 are used in the first example and a pair of A and B points from display 9 are used in the second example. The calculations reduce to = Therefore, the procedure to calculate the AP ratio using the same paired A and B points would reduce to = ( x ) + [( ) x ( )] = ( x ) + [( ) x ( )] = The discovery that this new formula would reduce to the exact expression when calculating the AP and BP ratios for any paired A and B points associated with the R factor lent tremendous weight to accepting this formula as that which is truly behind the commonly recognized over 2 formula. Separating the procedure in this manner uncovered relationships between the procedure and the physical workings of the ratiobased fabric. For example, the first half of the procedure was found to represent the arith- 18

20 metic mean that exists between the AP and BP ratios while the second half represented the space held between this arithmetic mean and either the AP or BP ratio itself when these ratios are acting as paired A and B points. The (A x B) and the [(HM A) x (B HM)] portions of the formula represent the square root of the resultant composition numbers for the paired A and B points and the paired A and B spaces created when the space between the paired A and B points is divided by the harmonic mean. These square roots are also the geometric mean between all paired A and B points or spaces that x-compose the two composition numbers. The new procedures for calculating the BP and AP ratios were revealing deeper levels of relationship with the physical nature of the ratio-based fabric only some of which have been covered in this material. However, the portion of the formulas that was thought to calculate the associated M factor only calculated the M factor when using paired RFA and RFB points. This inconsistency needed to be addressed. Although all of the procedures presented earlier for calculating the information associated with the R factor are valid, they are considered short-cuts and acceptable as methods only after the attainment of a fuller comprehension of the mathematical workings of the ratio-based fabric. In order to maintain an appropriate level of integrity, a more generalized procedure for calculating the M factor was found. [(HM A) x (B HM)] = M factor variance ratio [(A HM) x (B HM)] = XM factor variance ratio The procedures require an additional step that incorporates dividing by the variance ratio. This ratio is the variance that exists between the R factor sub-structure and the remainder of the ratio-based fabric. A quick method to determine the variance ratio is to 19

21 calculate the product to sum ratio of paired A and B points. In essence, variance ratio could be replaced by (A x B) (A + B) within the above procedures. The variance ratio for all paired RFA and RFB points is 1 because their product to sum ratio is equal to one. This observation explained why the portion of the procedure to calculate the BP and AP ratios worked for only the paired RFA and RFB points in the calculation of the M factor. In summary, the search for procedures that could be applied throughout the mathematical workings of the ratio-based fabric ended successfully. First, a more generalized procedure for calculating a fibonacci-type sequence from a surface-level perspective was discovered. It incorporated a multiplier called the M factor. The M factor was found to be calculable from any paired A and B points through a simple mathematical procedure. However, different categories of fibonacci-type sequences were defined according to the form of underlying composition of the sequence. The formula of transformed first to RF + (RF 4) to better represent a 2 2 method to calculate a ratio called the BP ratio. The phi ratio is one of an unlimited number of BP ratios that acts in the same manner within the greater framework of the ratio-based fabric. However, this formula still contained hard-coded portions and worked only when using the R factor associated with paired A and B points. A more fundamental or generalized procedure that would work for the actual paired A and B points within the ratiobased fabric had to be found. (A x B) + [(HM A) x (B HM)] stands as the final version of the universal HM HM formula behind the hard-coded formula of over 2. The new formula not only worked to calculate the BP ratio for all paired A and B points but portions of this formula were found to represent different aspects or components within the physical nature of a common mathematical mechanism the ratio-based fabric. 20

22 The mysteries behind the common procedures for calculating phi and the Fibonacci sequence were unraveling to reveal a higher order of mathematical structuring. Although a certain level of satisfaction had already been obtained in understanding the mechanics of the mathematical structure, the theoretical aspects were just beginning to be entertained. The simple observation that the first 1 of the Fibonacci sequence was not the same as the second 1 in the sequence was purely fascinating. It seemed that only a whisper of the mystery had been witnessed. Barbara Mae Brunner bmb192@earthlink.net 21

23 Display 1: Set of Aligned Ratio-Progressed Strings for Composition Number 1. String B progression ratio , String A progression ratio , R factor 5, M factor 1. B - A B + A B A B x A A B A x B A B A + B A - B

24 Display 2: Set of Aligned Ratio-Progressed Strings for Composition Number String B progression ratio , String A progression ratio , R factor 5, M factor 1. B - A B + A B A B x A A B A x B A B A + B A - B

25 Display 3: Set of Aligned Ratio-Progressed Strings for Composition Number.2. String B progression ratio , String A progression ratio , R factor 5, M factor 1. B - A B + A B A B x A A B A x B A B A + B A - B

26 Display 4: Set of Aligned Ratio-Progressed Strings for Composition Number String B progression ratio , String A progression ratio , R factor 5, M factor 1. B - A B + A B A B x A A B A x B A B A + B A - B

27 Display 5: Set of Aligned Ratio-Progressed Strings for Composition Number.2, String B progression ratio , String A progression ratio , R factor , M factor B - A B + A B A B x A A B A x B A B A + B A - B

28 Display 6: Set of Aligned Ratio-Progressed Strings for Composition Number.2, String B progression ratio , String A progression ratio , R factor 9, M factor B - A B + A B A B x A A B A x B A B A + B A - B

29 Display 7: Set of Aligned Ratio-Progressed Strings for Composition Number.125. String B progression ratio , String A progression ratio , R factor 8, M factor 2. B - A B + A B A B x A A B A x B A B A + B A - B

30 Display 8: Set of Aligned Ratio-Progressed Strings for Composition Number 1. String B progression ratio , String A progression ratio , R factor 8, M factor 2. B - A B + A B A B x A A B A x B A B A + B A - B

31 Display 9: Set of Aligned Ratio-Progressed Strings for Composition Number 64. String B progression ratio , String A progression ratio , R factor 8, M factor 2. B - A B + A B A B x A A B A x B A B A + B A - B

32 Display 10: Set of Aligned Ratio-Progressed Strings for Composition Number 1. String B progression ratio , String A progression ratio , R factor 13, M factor 3. B - A B + A B A B x A A B A x B A B A + B A - B