Structural Quantity Analysis Along the Orders of Dimension

An Exploration Through Conceptual

Mathematics and Combinatorics

by

Brother John

“Structural Quantity Analysis Along the Orders of Dimension - An Exploration Through Conceptual Mathematics and Combinatorics”

 TABLE OF CONTENTS Foreword Introduction I.  Dimensions II.  The Unit III.  Structures IV.  Unitary/Structural Relationship V.  The Structural/Dimensional Relationship VI.  Order and Class VII.  Undefined Quantities VIII.  The Matrix Representation     A.  Expanding Unitary Structures     B.  Higher-Ordered Structures     C.  Numeric Representation of Expansion Sets     D.  Matrix Standardizations and Notations     E.  Formula Generation of Proximate Matrix Values IX.  Numeric Patterns Derived From the Matrix     A.  Doubling Vertices     B.  Base-1     C.  Sequence of Sums     D.  Higher Base-System Sequences     E.  Other Sequencial Phenomena X.  Early Extractions of Sx     A.  Root Extraction from S0     B.  A Derivative Relationship of the Earliest Classes     C.  Denominator Progression XI.  Extractions for All Sx     A.  Numerator Exponent Growth Constant     B.  Chained Factors in the Numerator     C.  Simplifying the Chained Factors XII.  Synopsis of the Matrix Formula   Foreword       When I explain to curious inquisitors that mathematics is a favored pasttime that I have enjoyed for the past (?) years I generally receive an amusing grimace for a response.  I tend to chuckle it off and go on about my business.  However, once in a vast while I will encounter that eyebrow raised in curiosity and I will be asked what sort of mathematics I am partial to.  In the moments following the question I am blissfully immersed in explanation.  It is for that reason that I have written this body of work.  I have found that not only does mathematics contain a mysterious charm and beauty but the conveyance of such is as pleasurable an art as writing a song or painting a picture.     I must admit that my methods are amateurish at best.  My formal education in mathematics reaches as far as high school Calculus.  Informally I have gone on to research a great many fields of conceptual mathematics.  This probably puts a damper on my style of notation and may affect my choices of nomenclature but the solidity of math prevails as consistency makes up for formality.  Every concept in this exploration is notated and described consistently so that the information contained within is sufficiently unified.     It is very unlikely that this body of work contains any ideas that have not been previously documented.  Actually, it is very likely that the basis of this study has been known in the realm of mathematics for over two millennia.  A related form of this research has been dated back to around 300 B.C. compliments of Hindi mathematician Halayudha and later developed by Blaise Pascal.  The point of this exploration is not to break ground with uniquely significant research but to simply present a mathematical concept from the perspective of the ancient ponderer who did not have the luxury of referencing volumes of precedent works to achieve understanding.  Pascal did not open a text book and make his discoveries by simply learning the existing theories of mathematics.  Pascal did a great deal of pondering which rewarded him with original knowledge in many aspects of math.     When I began my own research on this topic I did not want to absorb knowledge about a concept then fill the gaps in superficially.  I approached this field of study like a mystery.  I began with a basic idea and worked at it like a puzzle until the pieces fit together to form the whole picture.  I realize that reading this exploration is most similar to the former method of study, however, I would encourage the reader to adopt the latter form of study where this body of work ends.  The knowledge acquired may not be original to mankind but it is certainly original to the ponderer and rewarding as such.     Over two thousand years of mathematical study have proven that mankind has a long way to go before a full understanding of mathematics is attained.  Perhaps there is an infinity of knowledge that we have barely begun to comprehend as a culture.  Why not explore that ocean of information?  Why not satisfy mathematical curiosities while running the risk of becoming one of those ponderers who does indeed develop an original concept?  Along the way just remember to enjoy the grimaces and raised eyebrows.   Introduction     What exactly does the phrase, “Structural Quantity Analysis,” mean?  Taken literally it means the examination of a “count” of structures which it most certainly is.  The structures are of a very specific type and the “count” of these structures becomes very significant when evaluated against one another within a particular environment.  As can be clearly seen further along in the study the environments that these structures are to be counted against are specific orders of dimension and the structures themselves are perfect unit structures based on allowances and limitations of these orders of dimension.     A more accurate title for this publication might be, “Quantity Analysis of Unitary Perfect Structures Expanding Along Subsequent Orders of Dimension.”  This title is not very eloquent but it is a fairly straightforward description of the nature of this study.  It is a thorough analysis of the numbers that present themselves in a particular fashion based upon individual counts of specific structure types within an order of dimension, even those beyond human perception as inhabitants of three-dimensional space.     When presented in such a manner, as later described, these quantities reveal marvelous predictabilities and patterns of behavior which are linked to many interrelated fields of mathematics.  This field of study spans such areas of mathematics as number theory, combinatorics, the Calculus, topology, and many others that relate quite elegantly to one another.     To best understand the mathematics behind this exploration it is beneficial to have some understanding of the theories and concepts from which they derive.  The first few segments of this body of work are dedicated to the concepts behind the mathematics.  This serves as a comprehensive backdrop on which the mathematics may be suitably sketched.  Further along, the concepts develop into formulae and equations; the meat and potatoes, so to speak.   I. Dimensions As a dweller in the third-dimension it may be very easy to conceptualize and visualize objects that lie completely within the confines of this realm. In reality, it is easier to consider the sphere than it is to consider it's lower-ordered counterpart, the circle. This is a result of the underlying need to visualize a depth of some sort which allows the perception of any object to be possible. Without this "third" dimension of depth an object becomes so thin it is difficult to explain it's very presence. It is not until the lower-ordered objects are represented by some three-dimensional means that it's properties may be considered and evaluated within those realms that remain impossible and invisible in this familiar realm. This analysis will span not only the realm of three dimensions but also those that precede it down to zero-space and those that follow it through infinity. It is quite useful to consider how these orders of dimension interrelate so that it becomes apparent as to how certain quantities are arrived upon in the contained methods of counting, or quantification. For instance, there are certain aspects of the order of two dimensions which exist in the order of three dimensions and which may be expanded upon to explain the behavior of objects in the order of three dimensions. Consider the square, an object containing two dimensions, as it is expanded upon to create depth, a concept familiar at first to the order of three dimensions. If this square has a side length of a it has an area equal to a2. Expanded to it's cubic counterpart the square continues to have an area of a2 and is met by four new squares, each with an area of a2 and a sixth square appears at the end of the expansion, again with an area of a2. The object may now be seen as a solid rather than a plane. This solid not only has the combined surface area of all it's component squares but also has a volume determined by three separate axes of dimension. The volume of this object is equal to a3. The notation alone suggests that there are three axes of dimension by which the object may be described as the exponent is equal to the number of axes that exist in the evaluation of the object. II. The Unit To simplify the study of complex structures such things as exact, specific, or even determined values will be ignored. Objects with a determined dimensional measurement will not need to be applied in this study since quantification of such things is arbitrary. The immediate concern is of the "arbitrary unit" or unspecified unit value. Given the necessity a unit may be assigned a specific value and substituted with such to determine specific qualities, but in the following study it will be sufficient to consider the unit as a single value equal to "1."   III. Structures The unitary measure will be necessary to maintain as expansion is represented along ascending orders of dimension. If structures expand by one unit for each ascending order of dimension they will emerge as specific types of structures known as "perfect" structures, that is, structures which contain equivalent values between similar elements. The square is a perfect structure. Each of the side lengths and angle values at each vertex are equal. Topologists may argue that there are exceptions to this contention but this study assumes that it is permissible to consider perfect objects geometrically stable along the axes of expansion, thus, the vertex, the line segment, the square, the cube and all of the structures that follow along the axes of expansion are perfect structures that concern this study.   IV. Unitary/Structural Relationship Since this study concerns perfect structures the element which shall determine the value of each expansion will be the arbitrary unit. In the case of the square, sides that comprise this structure will be equivalent to one unit. Notation of this evaluation may appear as u2 where u is the notated symbol for the arbitrary unit. Likewise, the cube may be notated as u3 since this represents the cubed value of the arbitrary unit. Each perfect structure may be notated as un where n denotes the exponent that describes the number of axes of dimension that the structure contains. V. The Structural/Dimensional Relationship Each order of dimension will contain exactly one structure of equivalent class. This structure will be the most complex possible structure in proportion and accordance with the perceivable limit of the dimensional order. For example, the cube is the most complex structure that may be evaluated within the order of three dimensional axes. Only one of these structures exists to be quantified in this order of dimension. Likewise, there are numerous sub-structures that may be quantified within this order, all of which relate specifically to the most complex structure. In the example there are six squares, twelve line segments and eight vertices in addition to the cube which acts as the superstructure. All of these quantities will vary depending upon the order of dimension that is to be immediately evaluated.   VI. Order and Class The evaluatory axes along the matrix are of ascending order and class. The order axis describes which degree of dimension is to be evaluated where the class axis describes the type of structure to be quantified. Each order and class denomination begins with a subset of zero. The order series begins with a zero subset because for the set of axes used to construct (and restrict) this order of dimension no axes of measurement exist. This order of dimension, sometimes called zero-space, contains only one single vertex. If any one axis of dimension is added to this order the exponent immediately reflects this and the order becomes one. The class series begins with a subset of zero which notes the complimentary structure type associated with zero-space. This class reference states that for the least complex structure type, the vertex, existence is first encountered at the order with subset zero. For reference throughout this study it is prudent to assign notated values to these orders and classes. Any order of dimension may be notated as dn where n represents the number of axes of dimension which comprise the order. Any class of structure may be notated as sx where x represents the number of axes that precisely describe the structure or the exponent value of the unit structure. These notations will play a substancial role in this study as the subset values will be extensively referenced and substituted into formulae.   VII. Undefined Quantities Because every order of dimension contains only one structure that may be described as the most complex structure pertaining to that order, such a structure may be superceded by more complex structures which only first appear as higher orders of dimension are evaluated. Although the cube is the most complex structure within d3 and no structures more complex than the cube may be defined within that order, a higher-ordered dimension may include structures beyond the complexity of the cube, such as the tesseract which first appears within d4. On the matrix d3 contains quantity values for each structure prior to and including the cube, or s3. The tesseract, or s4, does not appear as a structure represented within d3, therefore that quantity is not defined within or prior to that order of dimension. On the matrix this is represented as an undefined quantity rather than as a quantity of zero. Not only does this follow the progression of logic presented here, but also stands in accordance with the formulae to be derived further on in this study.   VIII. The Matrix Representation Thus far, the conventions of notation and basic conception have been touched upon. At this point it will be of use to put some of these elements into place and derive the matrix upon which all of this theory falls. As the matrix is explored and it's roots analyzed the pieces will begin to take form and some unexpected phenomena will become apparent within the mathematics thereof. Following will be a series of subchapters which will culminate in the complete derivation of the matrix.   A. Expanding Unitary Structures This section begins with the root structure and order of dimension. These are the vertex, s0, and the order of zero dimension, d0. Within zero-space there are no values of measurement such as length, width or depth. This lack of dimensional construct results in a binary characteristic of possible superstructures. The vertex, the only structure which contains no properties which may be evaluated by means of measurement, may be either present or absent within d0. Since this study requires that the maximum quantity of the most complex structures for each order of dimension is one it will be assumed that the vertex structure is present within d0, thus the quantity will be evaluated so that (s0,d0)=1. Now that the "seed" value of the expansion sets has been established progress may be calculated as the orders of dimension are ascended through. Following d0 is d1 which is sometimes referred to as the "first dimension" or "one-space." One axis of measurement has been added to the previous order of dimension. This axis determines the qualitative value of length which is arbitrarily one unit in this study. As the vertex is expanded along this axis it becomes a line segment of length one unit and serves as a superstructure containing several substructures. The superstructure of the line segment contains itself, as a superstructure, and two endpoints, as two substructures or vertices. This may be quantified as follows: the order of dimension d1 contains two structures of class s0 and one structure of class s1 or d1=2s0+s1. The actual sum of all structures within an order of dimension will not be addressed until further on since it does not pertain to the derivation of the matrix. The significance of these quantities will become apparent as the fully derived matrix is analyzed for patterns of behavior. The next expansion follows a similar pattern as an additional axis of dimension is introduced. Within d2 each vertex from d1 expands to create a new vertex with a line segment between each old and new vertex. Also, the new vertices are joined by a line segment between them bringing the total substructure count to four vertices and four line segments. This new configuration of substructures presents a new superstructure, a square, the most complex structure possible within d2. The expansion series of superstructures should divulge a noticable pattern of physical expansion. Starting with a single vertex within d0, expanding to a line segment within d1 and expanding once again to become a square in d2, logic suggests that the next expansion results in a cube within d3 and so on for each consecutive order of dimension. B. Higher-Ordered Structures As expansions are completed beyond d3 it becomes more difficult to represent structures visually due to the perceptual limitations governed by this universe of three dimensions. Representations may only be visually acceptable according to the three axes of dimension that govern three-space thus causing difficulties when attempting to represent higher-ordered structures in the visual manner. Fortunately, mathematics easily compensates for such limitations and allows representations to be made on a non-visual basis upon which higher-ordered structures may be evaluated. The most complex structure within d4 is known as the tesseract or hypercube. Visually this structure is impossible to represent but mathematically and conceptually it is not difficult to evaluate. Like it's lower-ordered predecessors the tesseract is an expansion of a predictable nature based on the cube. If an additional axis of dimension is added to d3 and the elements of the cube are expanded along this axis by u, one unit, the tesseract results. Just like the structure expansions previous to it the tesseract contains twice the number of vertices as the previous superstructure, one new line segment connecting each old vertex to it's new counterpart and a series of structure expansions which result from this. Although it takes quite a bit of consideration to conceive of it's dynamics the tesseract is comprised of the following structural elements: sixteen vertices, thirty-two line segments, twenty-four squares, eight cubes and of course, one tesseract that is the superstructure. This predictable pattern of expansion holds true for each expansion beyond d4 adding structures and superstructures uniformly with each level of expansion. The hyper-hypersolid superstructure which follows in d5 is again followed by a hyper-hyper-hypersolid in d6 and so on. Based on these patterns of expansion the physical nature of any expanded structures may be grasped and quantified within any order of dimension.   C. Numeric Representation of Expansion Sets In order to develop the matrix for further numeric study the quantification of structures and superstructures within each order of dimension must be represented in some organized manner. The matrix is a simple construct which may be generated to any degree based on previous information gathered thus far. At this point in the study a method for generating specific values on the matrix based on coordinate values alone may not be accomplished. As progress is made through evaluation of the matrix this will become possible as a generation method emerges. In the meantime a finite map of the matrix will be generated and expanded upon based on what has been covered thus far. The following matrix, a portion of the entire matrix, is limited to the boundaries of perception encountered in three-space. This will serve to get started. Along the columns of the matrix the structural class is noted. In this early representation of the matrix the structural classes have been represented by their common names. In future iterations of the matrix these names will be replaced by notations that are more mathematically sound which will be explained in more detail further on. The rows are represented by the orders of dimension beginning with zero-space, d0, and ascending by each axis added for each subsequent order of ascention. This notation is a specified subset value for dn where n represents the value of the subset and the actual number of axes within an order of dimension. For instance, d2 represents the order of dimension for dn where n=2=the number of axes of dimension that lie within that specific order of dimension.   D. Matrix Standardizations and Notations Based on the afformentioned principals a more standardized representation of quantities within the matrix may now be generated. The following matrix contains the bases for the latter-derived equations and provides a reasonable comparison of quantities, at a glance, up to an arbitrary order and class which extend past the written boundaries of the matrix ad infinitum. Added to this matrix is a column of ordered sums which will be a focus later in this study. This added column of sums will not render quantities that are comparative to other values on the matrix in the same manner that coordinate values relate to one another and should not be considered a part of the actual matrix itself but as a reference set of values for future computations. As can be seen in this and the previous matrix there are quite a few plots which contain no values. These values, left blank in the latter matrix representation, show where quantities may not be defined. This serves to note the irrationality of the existence of a structure of a higher class value as compared to the dimensional order.   E. Formula Generation of Proximate Matrix Values Rather than having to visualize each scenario that comprises the matrix in order to quantify an individual plot value it is more expedient to generate such values based on a formula that can be extracted from what is already known about the nature of the expansions. Since the bases of these calculations have already been considered this formula need not be proven for values governing strictly the earlier orders of dimension and least complex structural classes. Proof of these values will fall under the jurisdiction of the formula that may be used to describe the entire matrix. Upon inspection of the chart many numerical patterns may emerge. Some of these patterns will be explored later within this study. Pertaining to this section of the study are two particular patterns of numeric behavior. The first pattern lies within the set of numbers that appear in the column headed s0. This set includes all quantities of vertices for dn. As developed earlier, each time a superstructure is expanded along one additional axis of dimension the resulting superstructure contains exactly twice the number of vertices. This pattern of behavior justifies the entire generation of values under s0 for dn. Each value for dn in s0 may be evaluated as twice the value of the previous dn, or dn-1. This allows us to notate a coordinate within that column as follows: (dn,s0)=2(dn-1,s 0). This coordinate notation assumes that plots are to be notated in the fashion of (row,element) which is contrary to the more widely used geometric notation of (x,y) as a cartesian standard. (dn,sx) are not geometric coordinates and serve this arrangement more efficiently further on in the study of this topic. The second numerical pattern serves to complete the triangular gap left on the matrix by those numbers that are not yet classified as either undefined or predictably one, since it has already been established that each order of dimension contains exactly one superstructure of corresponding class value. The gap that is left appears to be full of numbers that are quite dynamic yet discreetly orderly*. Every value that lies within the "gap" has a specific relationship to it's neighbors in that any value is equal to twice the number of the value directly "above" it plus the value of the plot above-left of it. This translates to the notation of : (dn,sx)=2(dn-1,s x)+(dn-1,sx-1). To sum up, the entire matrix may be described by the following basic conditions: *At this point I must admit that I spent a great deal of time staring at a very large representation of the matrix and it seemed to eventually just occur to me that all of these numbers followed a very simple and predictable pattern in relation to one another.   IX. Numeric Patterns Derived From the Matrix Now that the matrix has been generated and the values that lie within it are able to be evaluated resulting patterns of numeric behavior can be found. Orderly sequences of numbers appear in many forms on the matrix and following are but a few of the more obvious examples. Some of these patterns will prove very useful later in this study. Others are worth noting strictly for their comparative intrigue. Later in the study a comparison between this matrix and Pascal's Triangle will be made where even more patterns will be explained. The phenomena that follow are merely the few that made themselves apparent during exploration of the matrix on a more intimate and unguided basis. The latter comparisons to Pascal's Triangle resulted from evaluations made after having performed a basic amount of research into related studies.   A. Doubling Vertices Perhaps the most obvious of the matrix-derived patterns is the sequencial doubling of vertices with each increasing order of dimension. This pattern is very easily explained as with each ascending order of dimension the number of vertices contained within a superstructure doubles as an axis of dimension is added. This holds true for all values of (dn,s0) as explained earlier. This particular series of values exhibits the same properites as the series of values that represents digit values of a binary counting system in base-10. Beginning with 20, which is equal to 1, each increment of expansion yields the next corresponding digit value of base-2: 1, 2, 4, 8, 16... = 20, 21, 22, 23, 24... Each of the exponents in the latter series matches exactly the value of the subset for dn where the corresponding value appears under s0 from the former series. In other words, (dn,s0) = 2n.   B. Base-1 Upon consideration of the emergeance of a base-2 pattern from within the matrix it seems appropriate to also note that the more basic and often unappreciated representation of base-1 coexists in a very convenient and significant location. Base-1, or the "tally" counting system is no more than a series of digit values of exactly one, or more precisely, 10,11,12,1 3,14... This everlasting series of ones occurs along the center diagonal of the matrix wherever n=x for (dn,sx).   C. Sequence of Sums Another significant sequence of numbers occurs when the sum of all structures within each order of dimension is evaluated as a discreet sequence. The sum of all structures within a given order of dimension is notated as (dn,sT). This sum includes each individual count of substructures that comprise each and every structure within a superstructure. For example, (d3,sT) contains one cube, six squares, twelve line segments and eight vertices bringing the total count to twenty-seven structures, thus, (d3,sT) = 27. When evaluated as a sequence of numbers beginning with (d0,sT) the sequence emerges as 1,3,9,27,81... This, of course, may be compared with the sequence of digital values of a base-3 counting system or expressed as a phenomenon on the matrix where (dn,sT) = 3n.   D. Higher Base-System Sequences The matrix very conveniently represents sequences of digital values found in counting systems for base-1, base-2 and base-3. Higher base-system sequences may be extrapolated from the matrix but not so elegantly. For instance, base-4 could make an appearance through the context of base-2. This seems to lose some significance in that base-4 is merely a counting system structured around a non-prime integer that has been found sequenced within the matrix in it's factored form.   E. Other Sequencial Phenomena The matrix contains many values which are significant to other values on the matrix in many mysterious and inspiring ways. Further along in the study some more of those phenomena will be analyzed. The more immediate concern is to simplify the formula that describes the matrix so that any value on the matrix may be determined given only a coordinate value (dn,sx).   X. Early Extractions of Sx A. Root Extraction from S0 By noting patterns within early classes of structure related equations may be determined which serve to describe higher-class equations. A logical place to begin extracting these patterns is in (dn,s0). It has been predetermined that (dn,s0) can be described as 2n. The pattern which follows for (dn,s1) is slightly more complex. The sequence of numbers represented by this column of values begins as follows: undefined, 1, 4, 12, 32... For now, the undefined value will not be considered, rather the series of integer values will be evaluated and the undefined value used as a place-holder representing some value corresponding with (d0,s1). The remainder of the series yields a pattern which may be successfully represented as the notation n2n-1. For example, in the case of (d3,s1) the value may be calculated as (3)23-1 which equals 12. This calculation works for all values of (dn,s1) except where n<1. This inconsistency will be clarified later in the study. In the meantime, the equation that has been extracted for all remaining values of (dn,s1) will be utilized.   B. A Derivative Relationship of the Earliest Classes Veterans of calculus may recognize n2n-1 as the first derivative of 2n. This peculiar relationship aides in early extractions in an even more peculiar way. If this pattern of behavior is applied to (dn,s2) it would be expected that this class could be described by the equation n(n-1)2n-2. As this formula is tested against actual values on the matrix, however, it becomes obvious that this is not the case. It is noticable that the results of such tests yield values that are predictably similar to those expected in that these values are exactly twice the expected value. In this respect, the equation for all cases of (dn,s2) where n>2 should be notated as [n(n-1)2n-2]/2. This may be elaborated upon to represent defined values in the next class column and those that follow as well. If the derivative of the numerator of the previous equation is used as the numerator for the current extraction it is evident that values generated are six times the expected value, thus, placing six in the denominator allows the equation to work. The equation which describes all defined values for : (dn,s3) = [n(n-1)(n-2)2n-3]/6 Likewise, if the same logic is used to determine the equation which describes all defined values for (dn,s4) the equation becomes [n(n-1)(n-2)(n-3)2n-4]/24. The derivative operation in the numerator always seems to hold fast as long as the denominator is allowed to compensate for predictable overestimates. As these extractions ascend through each class column it becomes evident that yet another pattern emerges from the sequence of numbers suggested by the denominator values in each equation.   C. Denominator Progression The sequence suggested by the denominators of class-column equations begins as follows: 1, 1, 2, 6, 24... The first two iterations of this sequence have a value of one which is equivalent to having no denominator at all as is the case with (dn,s0) and (dn,s1) having descriptive equations for defined values which are respectively 2n and n2n-1. This sequence exhibits a growth rate between elements that follows the form 1, 1*1, 1*1*2, 1*1*2*3, 1*1*2*3*4... or may be notated as the factorial sequence: 0!, 1!, 2!, 3!, 4!... This sequence works in determining any denominator value for any class column extraction. It is also quite convenient that the number that appears in the factorial is the same as the subset value to the corresponding structural class under which it's parent equation is the extraction such that: XI. Extractions for All Sx A. Numerator Exponent Growth Constant A simple pattern that is observed in the numerator is the steady rate of declination that occurs in the expression of the exponent. The constant value that lies within the exponent is the root n where the variable is the value of the subtraction from n. This variable component of the exponent is equal to the value of the subset for sx such that the exponential phrase may be expressed as n-x for all extractions of (dn,sx).   B. Chained Factors in the Numerator If the factored components of the numerator are isolated, once again a pattern develops. Consider the yield of these components for the following early extractions of (dn,sx): (dn,s0) = ...1..., (dn,s1) = ...x-0..., (dn,s2) = ...(x-0)(x-1)..., (dn,s3) = ...(x-0)(x-1)(x-2)..., etc. This pattern always yields results which contain a series of subrtacted integer values from 0 through, but not including, x. This may be simplified as the following expression of the sequence for the factored component of the numerator for all extractions of (dn,sx): Although this sequence yields valid results it remains clumsy in it's derivation and yet requires excessive notation when used to solve for higher extractions of (dn,sx).   C. Simplifying the Chained Factors If the chain were not bounded by a factor n-(x-1) but rather n-(n-1) the sequence of factors could be described as n! since the value of n is multiplied by each successive positive integer that is less than n. To limit the sequence of factors within n! to end at n-(x-1) another calculation must be incorporated into the formula. To find this component an adequate series for each expression will be evaluated within the boundaries of one coordinate equation. In order for the series to be adequate in this derivation it must contain enough elements to sufficiently illustrate the behavior of the values within it's own boundaries. The equation that describes (dn,s4) should work for this illustration: If the chained factors in the numerator are isolated then: (dn,s4) yields the chain of factors: (n-0)(n-1)(n-2)(n-3) If an arbitrary value is substituted for n, that is greater than x to yield a defined coordinate value, such as 6 in this case, the substitution can be expressed as: (d6,s4) yields the chain of factors: (6-0)(6-1)(6-2)(6-3) If 6! were to be expressed as a chain of factors it would appear as the following expression: 6! yields the chain of factors: (6-0)(6-1)(6-2)(6-3)(6-4)(6-5) This expression contains the set of factors that are present in the chain of factors for (d6,s4) but also includes factors that are not. In order to revert the factors of 6! to mimic those contained within (d6,s4) a set of negating factors must be placed in the denominator of the extraction. This may be notated as such: This expression may be applied to the original extraction for (d6,s4) so that: The elements of the denominator notated as (6-4)(6-5) may also be described as (2-0)(2-1) or 2!. The value of the factorial expression is a result of the difference between the subtraction of subset values for n and x, in this case, 6 and 4. This allows the negating factors in the denominator to be notated as the factorial expression (n-x)!. If this is applied to the extraction for (d6,s4) the result is: This relationship of factorials relates to all extractions for (dn,sx), thus the equation serves as a formula for determining all values on the matrix given coordinates (dn,sx). The following equation describes this notation: XII. Synopsis of the Matrix Formula Having extracted the formula for all cases of coordinates (dn,sx) the value of any plot on the matrix may be evaluated given simply the coordinates (dn,sx). It may now be noted that for all cases where n>x the formula generates values that are positive integers. For all cases where n=x the formula generates values that equal one. This is explained bearing in mind that the factorial expression 0! is equal to one. For all cases where n