"The Surrealistic Confluence"
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A mathematical argument supporting the contention that families of patterns |
Confluence: 1) A coming or flowing together, meeting, or gathering at one point. 2) a: the flowing together of two or more streams; b: the place of meeting of two streams; c: the combined stream formed by conjunction.
We view the world at large through many diverse and sometimes conflcting lenses. Incoming information is also filtered through not only culturally induced worldviews, but also through personal conditioning, experiences and learning. Perspective, what we believe life is supposed to be and how it's supposed to be lived, dictates our destiny to a very great extent. Reality, if it can ever be touched and known, lies beyond all such measurements. We, homo sapiens, are the measurers and coordinators of the universe; more accurately, we are homo symbolicus. One problem to be discussed here is taking the measure for the real without question; that is, the acceptance of assumptions and dogma handed down and taken for granted in an attempt to bring order and sense to our world. Through the exploration of the difficulties involved with the problem of discernment by use of mathematical ideas and concepts, the following essays undertake the proposition that it is possible to know the difference between the conceptual overlay and the physical concrete lying just beyond our grasp, between the perceptual and the real. To begin with, we will have to create a mathematical environment suitable to this end.
If we were to place a small cube-shaped object on the center of a table, then step away to look at it, we would see the width, height and depth as possessing a certain quantitative relationship, measured by eye. Say we drew a circle on the floor around the table equidistant from the center of the cube. If we were to then move around the table on this circle, not taking our eyes off the cube, we would eventually get to a spot where the cube would again look the same, as far as relationship of the various dimensions was concerned. Any place in between would render a different set of relationships. Continuing our circumnavigation of the table, maintaining our perspective, we would eventually come back to our original spot. Each position where the cube looked the same is a symmetry spot. Alternatively, if we had rigged a mechanical turntable under our cube, controlled by a button or switch of some kind, then turned the cube while remaining still, the same thing would happen. There would be only four positions giving us the same set of relationships. What we are looking for are the positions of symmetry.
Group Theory studies symmetry and the nature of invariance, what constitutes an invariant property. There are four axioms determining the algebraic structure, or arrangement, known as a Group. They are: Closure, a global concept not inherently attributable to any subset of a figure composed of points, but existing at the level of relationship; the operation, or composition, on any two elements of a group yields a third unique element, also of the group; it delimits and gives definition to space. Without the notion of closure, or limit, we can have no discussion of 'form,' as classification of elemental form is what the closure axiom insures. Associativity is also global by nature; but the existences of an Identity element and a unique Inverse for each element are decidedly local.
More technically:
The standard examples of Groups are Z (integers), Q (rational numbers), R (real numbers), and C (complex numbers) under the operation addition. Under the operation multiplication, the nonzero Q, R, and C also form groups, but nonzero Z does not. The reason Z does not form a group, of course, is that there are no multiplicative inverses, no fractions.
For any element, 'g' of a group not in the subgroup, 'N,' if the relation, 'gNg-1 = N' for all 'g' in 'G,' (gN = Ng) holds, then 'N' is defined as Normal (or invariant). Another way of stating it: N = gNg-1 = {gng-1 | n in N}. This form of the axiom of commutativity is essential for purposes of forming factor groups, and for precisely determining the focal point of actions within a system. A Maximal normal subgroup of a group G is a normal subgroup M not equal to G, such that there is no proper normal subgroup N of G properly containing M. That is, there must be no other normal subgroup of its respective parent group having numerically greater order [number of elements]. The maximal normal subgroup [a subgroup is a group in its own right] is just that set of configurations which the Group as a whole holds in common.
A group is Simple if it has no proper normal subgroups. Every simple group of prime order is cyclic, that is, the elements of the group can be derived by the repeated operation on any generator; and every cyclic group is abelian. For a group of prime order, every element acts as a generator; the general rule is: any element, number, relatively prime to the order of the group is a generator of the group. A simple group is a special form of a Normal Group, a sub-class; more revealing in the present context is its equivalent characterization as Invariant. The individual elements of a simple group are commutative under whatever the group operation is, they are well-defined; any properties shared by the group's different symmetric permutations remain unchanged after group transformation. It is non-divisible; this means that no subgroup of the simple group (other than the trivial one containing only the identity element) exists as a component capable of standing alone; that is, there are no subgroups, or sub-symmetries. Each element of a simple group is necessary for the group's integrity; and no subset of them share a relationship other than their position, and function, as members of the whole. All groups can be built-up through product formation by simple groups; in this respect, simple groups are the prime factors and basic building bocks of complex ones. Although all groups of prime order are simple, it is not necessary for a group to be of prime order for it to be simple; for example, there is a simple group of nonprime order, namely 168, between the alternating groups of order 60 and 360, which are also simple.
All flags are the same, in so far as they are flags; all rocks are the same, for the same reason; all clouds; all frogs, etcetera. A 'map' is a 'function,' and a necessary criterion for the definition of function is that elements in the domain are mapped to one and only one element in the range; only then can it be considered well-defined; that is, no ambiguities are introduced. We could call a map from the set of all flags to the set of all rocks an 'identity map' provided there was an equal number, making the map 'one-to-one' in correspondence, i.e., each flag mapped to one and only one rock. As objects they would remain independent of whatever coordinated space we chose to impose on them.
An Isomorphism is a kind of identity map from one group to another, with certain restrictions. The group-identity map means that the properties and rules of behaviour defining a given group, the operations performable on the elements of the group, its structure, in other words, occurs, or acts to organize the elements around a central design, in the group at the receiving end of the map, the range group. These roles, of domain and range, become meaningless, however, under an isomorphism. Formally, a one-to-one correspondence between elements of the two groups, operating to define an inverse function, constitutes an isomorphism; that is, each discrete element of the two takes on the role of either independent or dependent factor in perfect symmetry of function, alternating with respect to the direction of the map. What we have here is a mutual interplay of the generating and dynamic forces of two interchangeable descriptions. They may be labelled differently but are structurally the same, or isomorphic to the same group of permutations. They can be thought of as being two different representations of the same object. If it were possible to overlay one onto the other, we would expose their common features as invariant properties, as the fundamental geometry, and by so doing define their structure as the space itself. An identity element of a group is not the same as group-identity; group-identity calls attention to structure preservation; the identity element embodies the symmetric opposites, the great annihilator, the quintessential center of symmetry.
More precisely: An isomorphism of a group G with a group G' is a one-to-one function # mapping G onto G' such that for all x and y in G, (xy)# = (x#)(y#). The groups G and G' are then isomorphic. Loosely, an isomorphism maps the identity onto the identity and inverses onto inverses.
We speak of 'function' quite blithely in a mathematical environment as though the concept were universally understood, or understood the same way for everyone reading this. As it is a basic idea of mathematics, the definition given above is worth repeating. A Function or mapping * from a set A into a set B is a rule which assigns to each element 'a' of A exactly one element 'b' of B. We say that * maps 'a' into 'b', and that * maps A into B. And the same applies to the idea of a permutation. A permutation of a set A is a function from A into A which is both one-to-one and onto. In other words, a permutation of A is a one-to-one function from A onto A.
This is all by way of leading up to: "The collection of all permutations of a nonempty set A forms a group under permutation multiplication." If a set A and a set B both have the same number of elements, then the group of all permutations of A has the same structure as the group of all permutations of B. One group can be obtained from the other by just relabeling elements. This idea of the same underlying structure having different presentations may be more subtle than it looks. The elements of the group-set are permutations. All groups are permutations in disguise; that is to say, any group of any kind of element representation is isomorphic, or structurally the same, as some group of permutations. Essentially what we're saying is that there are only certain possible orientations allowable for an element to be a member of a symmetry group.
The Dihedral group, D4, of eight permutations, corresponding to the various rotations and reflections about the vertical and horizontal center lines and the two diagonals (henceforth to be known simply as 'reflections') allowable for a square to maintain symmetry, is an example of the permutations of a nonabelian group, and includes sub-symmetries that don't form subgroups. The set of rotations do, as the set includes the identity map; the set of reflections do not, although they provide a symmetry of their own. Looking at our square, it should be clear that only the zero degree rotation mapping each corner to itself, that is, leaving each corner fixed, can be the identity map; no single reflection can do this. The subgroup of rotations composed of four permutations, curiously enough, is abelian. So here is an example of a nonabelian group containing as a component, an abelian subgroup, a time-reversible sub-symmetry. 'Nonabelian' in the sense that the order or sequence of certain combinations of mappings is unique; when any two are reversed in order, the resulting permutation may not be the same.
For example, imagining a square drawn in front of you, if we do a rotation of 90 degrees counterclockwise, and then a reflection about a vertical line through the center, we end up with a diagonal flip mapping the lower right corner to the upper left and leaving the other two corners fixed. Going the other way, however, that is, first the reflection and then the rotation, we end up with the other diagonal flip, mapping the lower left corner to the upper right. Nonabelian groups can be used to model dynamic processes in the macro world dependent on linear time, cause - effect, unidirectionality. But one can find elements of this group, the dihedral group, that behave abelian-like. The two reflections about center lines, for instance, are commutative; so we have instances of abelian-ness even amongst elements that aren't members of a subgroup. Yet, the group itself, D4, is classified as nonabelian.
A less restrictive map # of a group G into a group G' is a Homomorphism if (ab)# = (a#)(b#) for all elements a and b in G. That is to say, an operation performed on elements of G is mirrored by the operation performed on the images of these elements in G'. The Kernel of a homomorphism # of a group G into a group G' is the set of all elements of G mapped onto the identity element of G' by #. The Kernel of a group is a normal subgroup and contains that set of elements of the group which form the identity equivalence class under the canonical homomorphism. We speak of the 'kernel-of-a-homomorphism' as it comes into being only as the result of this mapping.
It all happens so 'quickly' as to 'appear' simultaneous. That is, given the myriad possible (probable) states the parent group can be in, a normal subgroup of one state is 'chosen' (or observed) by which the parent group is factored into a group of cosets of states or configurations having as identity-form that of the subgroup. This normal subgroup is, of course, the very same kernel, but, its role as identity element of the group of cosets is not realized until the 'instant' of factoring.
Another perspective on the concept of kernel is to consider, for example, a linear equation with positive coefficients in three variables. The solution set is the kernel of the equation. Each set of three numbers that will render the truth of the mathematical statement can be thought of as forming an interdependent system, an essential feature of the equation. Its graph [a plane in Euclidean three-space], under any coordinate system, is a visual representation of its kernel, or solution set. Of all possible sets of three numbers that there are, an infinite number, only those sets of three that annihilate or dissolve the equation, only those that give meaning to the pattern, will fall into the kernel. The kernel as cookie-cutter, root, identity, center of symmetry.
The Fundamental Homomorphism Theorem:
If N is a normal subgroup of a group G, then the canonical (or natural) map #:G --> G/N given by a# = aN for a in G is a homomorphism. A homomorphism of a group is completely determined if we know its value on each element of a generating set, and the number of elements of this generating set is the rank of the group. No element of this set can be derived from any combination of the others; and the rank is unique up to isomorphism. These facts will be important when we consider their parallel in the generation of a vector space by a set of basis vectors.
Let H be a subgroup of a group G, and let a be contained in G. Given a group, 'G,' and a subgroup of it, 'N' (Normal), under the induced operation, a Factor Group of G modulo N, denoted by 'G/N' forms a group of remainder, or equivalence, classes called 'cosets' of N . The left coset, aH of H is the set {ah | for all h contained in H}. The right coset, Ha is similarly defined. The term 'modulo' introduces the concept of remainder or residue; that is, two numbers can be said to reside in the same class if, after division by some fixed positive number, the remainder is the same. The remainder dictates the classification, in other words. For example, 1 is congruent to 6 modulo 5 because the difference, 1 - 6 = -5, is divisible by 5 with zero remainder. Therefore, both 1 and 6 lie in the same coset, in this case, the identity coset. All mutliples of 5 lie in this coset; that is, {... -10, -5, 0, 5, 10, ...}. The number 2 lies in the coset consisting of {... -8, -3, 2, 7, 12, ...}. The difference between the numbers, as can be seen, is the modulus, 5.
The cosets are residue classes (equivalence classes) of G modulo N, such that 'N' is a normal subgroup of G and acts as the identity element of the factor group. Only this identity equivalence class is independent. The other members of the factor group, defined as left and right cosets, are derived from the identity class by composition with those elements of the parent group not included in the identity equivalence class, the subgroup N. For a normal group, left and right cosets are identical. Yet, there are no elements of the parent group which appear in more than one class or coset; each class contains the same number of elements, and this number divides the order of the parent group; and, all elements are spoken for. The induced, or inherited, operation is well-defined. For example, in the group [Z5], all multiples of five, including zero, form a subgroup of the integers [Z] and the identity class of the factor group, Z/5Z.
Let's tear this example apart to see what's going on. The group of integers under addition, Z, modulo all multiples of 5, 5Z, written as, Z/5Z, is isomorphic to Z5. What does this mean? The group Z5 consists of the elements, numbers, {0, 1, 2, 3, 4}. The number 5 is the identity element equivalent to 0; the operation 2 + 4 yields 6 which, in our group, Z5, is identified with 1 on division by 5. The number 38 is identified with 3 on division by 5, and so forth, until all the numbers, an infinite amount by the way, are identified with one or another of the members of Z5. Now, Z/5Z has as identity 'element,' or kernel, the subgroup of Z, 5Z, which contains all multiples of 5. This entire infinite class of numbers is mapped to 0 in Z5; the coset containing {... -14, -9, -4, 1, 6, 11, ...} is mapped to 1 in Z5; the coset containing {... -13, -8, -3, 2, 7, 12, ...} is mapped to 2 in Z5; etcetera; you get the picture. All integers in Z organize themselves into cosets around the identity template, 5Z. There are precisely 5 cosets, infinite in membership, each of which is mapped to one and only one number (element) in Z5. The structure is preserved; addition of cosets is by representative number, any number. For instance, 6 + 7 = 13, divided by 5 gives us a remainder of 3 which puts us in the coset containing 3, that is, {... -7, -2, 3, 8, 13, ...}. Pick another representative, say, 11; adding this to 12 we get 23, the remainder on division by 5 puts us in the same coset. The induced operation, addition in this case, is well-defined.
What we have is five cosets, or members of the factor group, each member of which containing an infinite number of integers. It's as though each number of Z5, {0, 1, 2, 3, 4}, on deeper examination, expanded to an infinite subsystem of the same form, the form prescribed by the identity factor, 5. That is, each number of any given coset differs from another of the same coset by some multiple of 5. Once we know the basic number of a given coset, and the modulus number, uncertainty, as to membership in the coset is eleminated; we can predict the proper form if not the precise number itself. Imagine the entire set of integers arranged randomly on an uncoordinated field, a field that went off to infinity, or as far as we could see. Imagine further that these integers had an organic quality all their own. Momentarily, spontaneously, they acted to self-organize, as through the twist of a kaleidoscope. First, the integer 5 bonded with 0, the annihilator of all opposites, bringing with it its opposite, -5. This action then precipitated the bonding of all multiples of 5 to 0, forming a discernable pattern. Almost immediately, or as quickly as time allows, this pattern expanded to influence the remaining integers to cluster, or bond, similarly, taking their 'rule of organizing' as a blueprint. When this process was over, we would look out over the field to see five clusters, each resolved into identical patterns, the 5-cluster at the center of the other four, equally spaced. The number 1 is opposite 4, 2 opposite 3. Our kaleidoscope has turned. Each cluster is unbounded, yet the entire collection of five is finite. If we peel beneath the 'surface' of any number of Z5, we are confronted by an infinite set.
Another kind of group which, coupled with the concept of permutation group, can further help to reduce the number of ideas concerning group taxonomy to a few is called a Free Group. As any group is isomorphic to some group of permutations, we can forget about labelings and concentrate on underlying structure. A Free Group is distinguished as one which is generated by a basic set of elements; that is, these elements are capable of forming all elements of the group's set under the group's operation. A homomorphism of a group is completely determined if we know its value on each element of a generating set. The members of this set are minimal in the sense that none can be derived from any combination of the others. All free groups are infinite. The importance of this idea is to be found by comparing the concept of "bases generation" to that found in Linear Algebra, where vector spaces are generated by linearly independent basis vectors; and in Topology, where topological spaces are generated by the union of basic open sets of points, that is, a set of open sets of the space forms a base if every open set of the space is a union of sets of the base.
This may be a good place to formally introduce the concept of a Topological Space:
A Topological Space is a set of points, X, in which certain subsets, called open sets, are distinguished; the collection of open sets satisfies the axioms:
To prescribe the open sets is to assign a topology to X. If U, V are two topologies on the set X, then U is finer than V if every set of X which is open in the topology V is open in the topology U. Subsets will always be supposed to carry the induced topology.
A set of linearly independent basis vectors that together span a vector space are equivalent in function to mutually disjoint, basic open sets of points the union of which generates a topological space, and these, in turn, are equivalent to the "smallest set of generators of a free group." In addition, any two sets of generators of a vector space, topological space, or free group have the same number of elements, that is, have the same cardinality. This cardinality is called the rank and is unique.
We look for parallel patterns beneath the symbols of representation. This approach to understanding various fields of mathematics, by consideration of the basic ideas and concepts from which each is constructed and on which each depends as a guiding theme, was practised by none other than the last great mathematical comprehensivist, John Von Neumann; therefore, we humbly attempt to follow suit.
In the nonabelian dihedral group above, D4, factoring it by the normal subgroup of rotations abelianizes it in the form of two cosets, the rotations and the reflections. This subgroup of rotations is cyclic, that is, it can be generated by a single element of the subgroup. Representative elements from each coset don't always produce the same element due to the nonabelian nature of the group, however, the product of any two elements from each coset can be found in the same coset. The original group, D4, is nonabelian because of this feature of unpredictability in the mapping of individual elements; however, by factoring it with the abelian (normal) group of rotations, it changes into an abelian group. This means that regardless of the sequence of symmetrically allowable actions, the end result will be the same, the equations of physics will be time invariant. In the world of the quantum, time-reversibility is built in. In the macro world we have a nonabelian directionality or orientation, quantumizing it, or abelianizing it, we uncover time symmetry.
In forming a factor group of G modulo a normal subgroup N [the nontrivial case], we are essentially collapsing the group structure by putting every element in G, which is in N, equal to the identity element, e, for N forms your new identity in the factor group. However, as we proceed through the series of factor groups, each successive progeny factor group contains the genes of its parent, transmuted and mapped to the identity of the successor. Specific features of the parent group's overall pattern or configuration, merge, or synthesize, into a new pattern, a subgroup of essential features, the identity of the progeny factor group.
Operations are performed on cosets by choosing a representative element. The symmetric form expressed by the group of cosets remains invariant even though there may be an infinite number of possible pattern connections amongst the representatives of cosets. The modulus defines the pattern. This complex system of representative interconnections underlies the constant form on the surface, similar to the underlying interrelationships amongst subsystem vortices of a turbulent system, revealing order in the form of the strange attractor on the surface. We cannot 'know' which specific elements of the cosets of a factor group are 'coming into play' to form, collectively, the overall symmetric structure of the group, but, by isomorphism to a finite group, where each coset is mapped to a single element, or number, a form is discernable and applicable. The partitioning of an infinite set into a finite number of cosets, each of infinite extent, is a very powerful concept. It's as though the infinities were pushed into the background, the foreground giving us the appearance of an absolute, belying the turbulence beneath. And if we further examine 'beneath,' we may find other factors determining patterns of relationships complete and self-sustaining. Wheels within wheels, ad infinitum.
Consider the factor group, Universe modulo Earth. The symmetries of the laws of physics, and in particular, quantum mechanics, to which are associated corresponding laws of conservation, what we have as yet discerned of these based on questions asked, underlined by the principle of evolution, are inherited by the progeny factor group and form the kernel of that group. Both the organic and inorganic aspects of the Earth stand for the cosets of this group.
Factoring further, we have Earth modulo the Biosphere. All the life-forms the Earth has produced, flora and fauna, share certain genes. At the time of the Cambrian explosion, 530 million years ago [mya], all the phyla presently represented and diversely expressed came into existence. Phyla represent the fundamental group plans of anatomy. The stamp was set, on the biochemical level, prior to this diffusion of phyla. The phylogentic relationships among microorganisms implicitly indicates the universal ancestors of all forms of life. A gene is that lenght of DNA script that describes one enzyme or protein. The properties of each protein produced, manufactured, by some combination of nucleic acids, the underlying four, are governed by the 3-D shape into which it folds. There are classes of shapes which categorize a common denominator function, form is function. The cosets of this factor group, Earth modulo the biosphere, stand for the many different species collectively constituting the biosphere, the grand ecosystem. Species is fundamental in this group.
Factoring still further, consider the factor group, Biosphere modulo Homo Sapiens. The repository of invariant, fundamental properties defining the human race as species makes up the kernel. Each unique coset, an individual member of this group, has the 'invariant property' stamp, infused from within, so to speak, like a magnetic pattern, a field effect; each coset represents a collection of talents, capacities and abilities uniquely expressed. The elements of these cosets vary on the surface, but the underlying arrangement, complex as it is, reamins the same. The common stamp, the kernel, runs across the entire family, the factor group, Earth/Homo Sapiens. The differences are appearance only; 99.9% of the human genome is shared by all its members. The differences, physical appearance, unique personalities, lie at the edge, the interface between subjective and objective realities, between consciousness and the Unconscious. This is the locus of adaption and modification.
The factor group hierarchy begins at the classification of kingdom, then to phyla, classes, orders, families, genera, and species. According to Stephen Jay Gould, "Classifications are theories about the basis of natural order, not dull categories compiled only to avoid chaos." Possibly the Earth, through the force of evolution, attempted an untold number of biochemical combinations before striking on the biochemistry of life. Or, the number of possible combinations was limited by some, as yet unknown, conservation law. Whatever the scenario, it seems natural to conclude that the basic template of the universe must include the possibility of Life or, more strongly, the imperative of Life.
In "The Origin Of Species," by Charles Darwin, he speculates that if all the species that have ever existed were arranged with those currently existing, the result would be a continuum of variation; we would be unable to tell where any chosen species began or ended, all the holes would be filled in. Topological spaces are built up, or generated, by a union of basic open sets that span the space. These sets of points are interconnected, or what's called "pathwise connected," forming local neighborhoods of the space; these neighborhoods impinge on one another. If we think of each species of this continuous manifold of organisms as a point in spacetime, an event, topologically, then this manifold can be generated by a set of basic open sets of points [genera], or open-ended patterns, the union of which, across the eons, accounts for all species, under arrangements of permutations, modifications, variations. Each species, each component of the bioshpere, is self-sustaining and self-regulating, yet its plastic nature allows for the possibility, through mutation and adaption to environmental conditions and, most importantly, according to Darwin, competition with other species, of transforming its basic template to another form through continuous topological maps.
A manifold is also a surface, but a surface of what in the present context? Moreover, what, we may ask, stands for the 'interior' of this surface? Certainly this surface is three-dimensional, a hypersurface, the interior would thus be four-dimensional; by comparison, the surface of a sphere in 3-space is two-dimensional; the interior, three-dimensional. What do we 'know' to be four-dimensional? Spacetime, as proposed by Einstein and Minkowski, and captured mathematically by the Lorentz Transformation for relative frames of reference, is understood as four-dimensional. All living things have that in common; it is the 'within' of all Life.
The symmetries of the physical laws, the laws of nature, in the classic macro world and in the micro world of quantum field theory, act as guiding lights for inquiry, as well as being aesthetically pleasing in and of themselves. Richard Feynman, in his book "Six Not-So-Easy Pieces" lists a few symmetry operations in a table on page 24. They are:
The laws of physics are symmetrical across frames of reference in relative motion to one another. If we imagine a continuous spectrum of frames, from one we decide to be fixed or stationary, across the board to one that is approaching the speed of light, each frame will manifest a process or event by a difference of graduation with respect to time and space from our fixed standing point. The faster the relative frame, the slower time will appear, spatial objects appearing shortened or contracted. This works well for deducing the end result of applying physical laws that govern any actions or measurements. Time slows, rulers shorten, an event over time as measured will balance out; the form of the physical laws remain constant under coordinate transformations. Our physical laws are based on and held together by the constants of nature. The speed of light is the same regardless of frame; how it's measured is what alters symmetrically.
But what if these constants were to undergo an evolutionary change. If the principle of evolution is indeed part of the fabric of the universe, part of its template, its identity, its nature, then a change in any particular constant will affect the others. Forces, masses, coherency of atoms and molecules, all of nature, the universe, will undergo change due to the fact that the constants of nature are related nonlinearly. That is to say, for the constants presently known, a relation can be found, mathematically, between any two, that interconnects them into an overriding web of orchestrating parameters. So what would happen to the symmetries of the physical laws? Would the symmetries remain, yet the forms change? What then are the invariant properties? What about the associated laws of conservation? If the constants evolve, so must the forms of the physical laws; these laws, afterall, are descriptions of how the universe operates, its nature. The physical laws, symmetric across a spectrum of relatively moving frames of reference, must undergo an evolutionary process themselves as the universe undergoes variation with modification.
According to evolution, a species tends to vary due to mutations, which may be considered an internal transformation, and from pressures produced by the local environment, most importantly, from relationships and competition with other species. Variation with modification towards an adaption, structural or functional, or both, that is either advantageous or spells death to the species, is the central issue of evolution, the action of natural selection. The question then must be asked: If the universe is evolving, what then is its 'environment'? Contrary to our conventional way of understanding, that to every 'inside' there is an 'outside,' we tend to think of the universe as being 'all there is.' That is, the universe has no 'outside.'
To any particular species, all else, all other species and the physical environment at large, represent the 'objective reality' that it must adapt to and make work for its continuing survival. Natural selection decides which variation best fits the local environmental conditions. If it's to the advantage of a species, it undergoes change in function and/or structure for the good of this evolving species; if not, it puts itself at odds with the immediate environment and tends towards extinction. Evolution feels its way, looking for contingencies, contingencies that are nonetheless governed by physical laws, by biochemical possibilities. The entire ecosystem of living things, flora and fauna, act as one whole organism, with each individual species playing its part as a component, interacting in a directed way in the midst of the assemblage. Each species is a homomorphic image of the genus from which it is descended; the genus, in turn, is a homomorphic member of a family, and so on up the hierarchy of taxonomy. But the action of evolution, its tendrils of sensing and feedback, happen on the species level, embodied in the individuals that make it up. Time is deeply woven within the body of evolution; time and uncertainty are built-in.
There are several theories about the nature and origin of the universe. Einstein's theory tells us that the universe is either open, closed or flat. A closed universe is one that is finite but has no boundaries. It is finite in the sense of energy and mass being conserved. And it is unbounded, as though on a surface of some kind. But there is no 'border' beyond which is something else. An open universe is infinite in extent as it does not close in on itself, and continues to expand 'forever.' A flat universe is one in which the density of matter is exactly equal to the critical density, that is, the density necessary for the universe to put the brakes on and begin to contract. It is also infinite in extent like the open universe; however, the difference is that in a flat universe the expansion eventually becomes so small that it is not determinable. In a flat universe the average curvature is zero, and the geometry is Euclidean. Given that the latter is the case, one conclusion could be that the universe includes within its nature, and structure, a subject/object dichotomy. It 'pushes against' itself, evolving on its own. Or perhaps we do not yet 'see' that aspect of the universe that could be considered the activating part as opposed the acted-on part, its 'environment.' In any case, what would the invariant properties be of such a universe? What is its identity? It is easy to say that the process of change itself is an invariant property. But that is not very satisfying to the intellect. It sounds a lot like the old adage, "The more things change, the more they stay the same." But evolution isn't just about change or transmutation; evolution is a budding force, the continuation of the "Big Bang," a driving impulse, an asymmetric proclivity, Life. The force of evolution sets the stage by allowing certain transformations to occur. Each organism has a history which must be taken into account. The work and effects of evolution, however, are to be found in the details of contingency, in the moment of interaction with the environment. Choices of modification present themselves and can be realized, at least in principle, by a particular organism as it attempts to balance its relationship with the environment; but, and this is significant, time to change is the key factor.
Astrophysicist Fred Hoyle coined the term, "Big Bang,' intending it to be a derisive comment on a theory that was at odds with his own steady state theory of the universe. The Big Bang theory, however, is now generally accepted. He was more successful in helping to explain how the elements came to be synthesized step by step in the stars, starting from hydrogen and helium.
String Theory introduces the concept of a universe composed of ten dimensions, the four of ordinary spacetime, and six others 'rolled up' into a ball compacted to a singularity. A concept similar to factor group in algebra is the notion of 'quotient space' in Topology. We have discussed factor group and will use this valuable concept repeatedly, so, in order to get a feeling for its companion concept in topology, we introduce it formally here. This definition is taken from Analytic Topology:
Let R be an equivalence relation on the points of a topological space, X. Let Y be the set of R-equivalence classes and let k:X-->Y ('k' is a map or function) send each point to its equivalence class. If Y is given the identification topology determined by k we may write Y = X/R and say that Y is the "quotient space" of X by the relation R (with the quotient topology).
These are different lenses having a similar basic functionality. The symmetries of the factor group can be applied to the arrangement of points of the topological space.
As another example of a homomorphism: [bottom of page 109, "A First Course in Abstract Algebra" by John Fraleigh] The map #:R -->C given by x# = Cos(x) + iSin(x) is a homomorphism, where R is under addition and C is the multiplicative group of nonzero complex numbers. Cos(x) + iSin(x) = 1 if and only if (iff) x = 2(pi)n for some integer n. Thus the kernel of the homomorphism is the cyclic subgroup {2(pi)} of R. The Fundamental Homomorphism Theorem shows that R/{2(pi)} is isomorphic to R#, which is the multiplicative group of complex numbers of absolute value 1, that is, the complex numbers on the unit circle. This isomorphism can be visualized geometrically. Every coset of R/{2(pi)} has exactly one representative greater than or equal to 0 and less than 2(pi). Thus the interval between zero and 2(pi) (not inclusive) can be visualized as R/{2(pi)}, and if bent around so that the open end of the interval at 2(pi) joins onto the closed end at 0, it forms a circle. The addition in R/{2(pi)}, viewed in this way as a circle, is just arc length (or central angle) addition, which is exactly what happens if two complex numbers on the unit circle are multiplied.
A Normal (or invariant) series of G is a finite sequence H0, H1, ..., Hn of normal subgroups of G such that Hi < Hi + 1, H0 = {e}, and Hn = G. A Normal Series of a group is a 'Principle' or 'Chief' Series if all the factor groups Hi + 1/Hi are simple. For abelian (commutative) groups, the concepts of composition and principle coincide. By the theorem that states: M is a maximal normal subgroup of G if and only if (iff) G/M is simple, Hi + 1/Hi is simple if and only if Hi is a maximal normal subgroup of Hi + 1. Thus for a composition series, each Hi must be a maximal normal subgroup of Hi + 1. Furthermore, any two composition series of a group G are isomorphic.
A Composition Series of a group renders a panoramic view of the organization of some structure on all levels and dimensions with regard to symmetry of fundamental relations, fixing the identity 'element' at each level as its center. It is a hierarchy of embedded factor groups arranged sequentially, by order of symmetry. Looking at it from the top down, so to speak, if one were to visualize a complex mandala-like tapestry, changing focus it would be possible to choose different sub-patterns at varying scales of intricacy of design, self-contained and complete, yet always symmetrical and embodying the quintessential stamp or signature of the underlying order. If we were to further imagine being able to take opposite design elements and compress them to some 'nearby' centrally oriented design element of a sub-pattern, we would come closer to the actual process occurring in the generation of a factor group. Of course, we can go up the ladder too.
Somewhat differently stated: the normal series, from bottom to top, of an abelian group, consisting of embedded, or nested, factor groups, each of which is simple, is a series of cross-sections of ever broadening identity classes, the union of which contains all essential features of some process, event or thing, capable of being modeled by the group, in various degrees of scale, or phases of operation.
Of singular importance is the fact that all members (Factor Groups) of this series are Simple. We can thus state that the set of all identity equivalence classes, that is, the set of the sets of all elements of the group mapped to the identity class, or kernel of the respective homomorphism for each factor group, exhibits the grand equivalence class of all essential features of the group, taken as a whole, defining the symmetries from every possible level of magnification.
As an example, to be elaborated on in detail later, in Algebraic Topology our groups are composed of 'simplex chains' and subgroups, 'boundary chains.' A chain is composed of links, each link is a vertex. Boundary chains are a dimension less than a respective simplex chain. For instance, the one-dimensional perimeter of a triangle is the boundary of the triangle, considered as a two-dimensional object. Our factor groups thus contain equivalence classes of simplex chains modulo boundary chains, the latter being the identity equivalence class. If a space is defined only as boundary, as a hollow sphere is, then it maps back onto itself as the identity, 'G/G.'
An infinite convergent series can be considered as a sequence of ordered refinements, collectively approximating, by summing over an infinite number of parts [Algebra, Calculus], or through the linear transformation of translation [Geometry, Trigonometry], a curve equal to a nonlinear function set over the same range. This function is a form of structured language dimensionally greater than that of the linear pieces, or sum of pieces; and gives more sophisticated information than the sum total of linear translations represented by the rough-fitting series.
Infinity has many definitions depending on context. We usually think of infinity as some place very far away, or a number of things beyond counting, i. e., no matter how many of something we have, infinity allows us to add one more. The Irrational numbers can be considered an infinity of a sort as each can be represented as an infinitely long non-repeating decimal, each diverges away from an accumulation point. The Irrationals will not be contained, or predicted in form. Does a lack of periodicity construe to mean infinity?
The mathematician, George Cantor, creator of set theory, using only the ideas of set and equivalence, defined transfinite numbers as levels of infinities. The whole numbers, or natural numbers, are considered the base infinite class, any infinite subset of which, such as the even numbers, is equivalent to the whole, the entire set. This level of infinity is dubbed 'countable.' What we're dealing with here is not simply a pairing of numbers, but rather the notion of infinity. Cantor proved, for instance, that there were 'more' irrational numbers than whole numbers. The Integers and the Rationals, which include the Integers, are both considered 'countable'; the Irrationals are, however, of the next level of infinity, there are 'more of them' than the sum of the wholes and rationals, they represent a 'continuum.' And the Real numbers, comprised of both the Rationals and the Irrationals, are also not countable, they too represent a 'next level' of infinity, a continuum.
In this respect, then, we have increasing (or decreasing), nested levels of infinities. As an example, the expression, [0, 0/0), a half-open set of real numbers from zero to infinity, informs us that there is no number which can be considered the largest. Furthermore, sets of elements or numbers that can be put into one-to-one correspondence with the whole numbers, or integers, are called 'countable.'
However, when we speak of infinity, unless otherwise made clear by the context, we will be referring to "the nth degree maximization refining the quality and precision of detail of information perceived," a sound to noise ratio approaching the intuitively understood, S/O. When the parts composing something are no longer perceived as such, and the whole as relationship of these parts emerges independnetly as a 'something' in and of itself, we have passed a threshold of dimensional awareness. For example, when we increase the number of sides of a polygon to the point of infinity, the polygon transforms into a circle. An infinity in one world is but a point of departure in another.
Probably the most important concept in Calculus on which is based most of applied mathematics is that of Limit. Limit is the closure principle of infinity. It stands for that magical transition point or moment when one kind of order passes through a discontinuity [phase shift] into an entirely different realm. This transitional border, this infinitesimally narrow band of quantum-like duality, deserves elaboration and expression towards a deeper understanding of its significance.
Does the ability to discern discrete units become blurred at and beyond this point? In Topology, particular identities of a given structure become irrelevant as, for example, a sphere transforms into a cylinder, a cube, or an amorphous blob. What is of significance here is not simply that each of these elements is a homeomorphic image of the other, but rather that there is encountered a genuine inability to know specific information other than what category they belong to in the topological environment. Due to the many configurations the essential properties can take, this represents a higher-ordered packet of information, giving us a holisitc picture of a complex class, revealing a deeper, broader understanding of the compactified identity. From this perspective, each geometric shape's commonality and specifics are graspable. The kernel (identity class) expels, or filters out, only those features and properties that are subject to change, that are not 'geometric,' that are not part of the identity of the respective form within the context, or matrix, of selected scale.
Nonlinear dynamics recognizes that at a certain point in the life of a system, the initial conditions become irretrievable. The Indefinite Integral is defined in the plane as the limit of the sum of coordinated, mutually adjacent rectangles, or ordered pairs, as their number grows to infinity, or, alternatively, as the common width of the rectangles approaches zero. Because it has no initial conditions to fix it in space, it is called 'Indefinite.' What we have is a general pattern, or family of patterns, a melody without a definite scale, a sort of generic function or family of shapes, separated one from the other by a proportionality factor, a factor of unceratinty. This is also a higher ordered language form than the sum-of-parts. The significance of the 'error factor' adjoined to the solution of an indefinite integral is to free us from restricting the solution set, or kernel, to a local neighborhood; setting before and after conditions closes this uncertainty. Currently it is seen by dynamicists and mathematicians alike as having a central position, influencing and dictating the life history of any given system, particularly organic systems sensitive to initial conditions. This sign of nonlinearity leaps out at the observer of what had previously been considered purely linear in nature.
The binary operation performed on a set of elements includes the notion of closure, as stated above. However, the principle of Closure plays an important part in all mathematics and physics, important enough not to be pushed into the background of ideas. What is the criteria by which closure is established? Some examples: 1)factor group formation; 2)definition of a function or map, how this partitions a set; 3)limit on infinity; 4) the bounding of an open set; and, 5) the first axiom of Group structure. Closure can also act as a 'method' to fine-tune an otherwise error causing procedure. A nonlinear or transcendental function can be represented by a series of linear factors which can approximate that function graphically, that is, against the background of coordinated space.
An infinite series of linear factors of a complex figure (modeling a system), which remains invariant across scales of magnification, briefly and minimally describes what is called a Fractal. In general, Fractals are characterized by infinite detail, infinite length, no slope or derivative, fractional dimension, self-similarity, and are generated by an iterative process. Linear here means that the result of an action is always proportional to its cause, it is superpostionable. Linear models can't possibly hope to capture the essence and predict outcomes of a naturally ocurring nonlinear system. It is not only that all the variables [in the case of physics, 'regulation' by quantum fluctuations] cannot be accounted for as imagined in the Newtonian universe, but that the nature and essential features of the whole, whatever type it may be, completely subsumes the underlying parts, or sum-of-parts, both perceptually and as a self-organizing, autonomous entity in its own right.
A fractal is a static snapshot of an attractor. Considered as a function, a fractal represents a linear approximation of the otherwise indeterminable locus of the resultant of a finite number of nonlinear superimpositions, reinforcements and dampenings. An attractor, as a dynamic entity, can at best render a range wherein it may accumulate as a probability. The nonlinear perturbations and interactions of a dynamic system produce attractors of different dimensions. Point, limit-cycle and Torus attractors represent linear and predictable systems, such as a clock pendulum for a limit-cycle. The more interesting ones are what are called 'strange.' They characterize complex adaptive systems constrained and bounded at the edge of chaos, where order and disorder intermingle.
The difference between 'an infinite series of linear factors' and 'the resultant of nonlinear interactions' is a difference of perception, dimension, and the import at the threshold when delta-time approaches zero. This would be pure sound, maximum density of information content, S/O. Analytically we can look at it this way: the set of tangent lines to a curve, a nonlinear function, may be conceived as ONE tangent line or envelope, under closure. An Envelope is a curve tangent to each of a family of curves. The envelope bounds the attractor by being consistently tangent to all levels of the fractal. Having only one notion to carry around, relating two apparently inconsolable ones, can prove beneficial to understanding.
This gives us a tool by which the following procedure can be accomplished: The Euclidean group of transformations, which define that geometry, collectively transmute into a single operation, a 'knob' or frequency modulator that adjusts only the nonlinear features or aspects of Euclidean geometry. The linear operations of reflection, translation and rotation can be grouped into a 'wave adjustment' operation. This is a manipulation of the whole figure, whatever it might be, from one locus or orientation to another as though the three functions were a composite one, regardless of order, a nonlinear one. It is also possible to focus, using our 'knob,' across dimensions of factors.
What are we dealing with here? Given a figure in a specific dimension, if we apply some sequence of linear transformations, say a reflection, a rotation, and finally, a translation, and assume the product is not an identity map, our figure is going to come to rest in a different orientation and/or a different place vis-a-vis our chosen coordinate system. The same result could be accomplished by 'pushing' the basis vectors of our coordinate frame through the nonlinear 'knob' composed of the specific maps involved. It's important to realize that in order for our 'knob' to have genuine nonlinear characteristics, its linear factors must interact 'iteratively,' that is, we have to imagine that these functions, or maps, are interconnected and feed
back into one another in a continuous loop, affecting the behaviour of the whole. Our initial conditions are our set of basis vectors. And we can rule out any unpredictable activity, caused by the accumulated effect of rounding off nonlinearly infused error factors or unseen information, inherently due to the closure of our frame of reference. Minute changes in initial conditions can give rise to very large global effects; microscopic changes affecting and altering the macro through the interdependency of the 'knob's' composite parts or functions. This is simple to see but partly because our nonlinear function can easily be factored into its constitutent linear parts.
Now suppose we had a collection of such 'knobs' operating on a complex figure which, either over time (as an attractor), or over space (as a fractal), exhibited its complete set of possible permutations simultaneously as a result of the interaction of each 'knob' on the combined effect of the others. We can go one step further by the use of a 'master' knob that would adjust all the others in a way unforseeable. Because our system
is nonlinear [the whole is greater than the sum of its parts], there is no way to determine, except within a probabilistic region, how, when or in what order the underlying parts of our figure are relating; it is also impossible to determine any previous state due to the complexity of these iterative interactions. We are at an information impasse and so are in need of tools. We have to investigate the overall picture from a qualitative point of view, abandoning all hope of quantifying the incredibly complex orchestration of endlessly reiterating pieces and ensembles. It is necessary, at this point, to introduce the notion of 'representative element.'
Geometrically, a representative of the first dimension is a line 'segment.' This is closure on the infinite line considered from the dimension of a 'string of points.' An infinite series of these segments laid parallel, with a definite idea of what constitutes adjacent segments being 'close' so as to insure continuity, would 'fill-up' a bounded, two-dimensional plane and thus be equivalent to it. Likewise, an infinite series of said planes arranged similarly would be equivalent to a polyhedron of four sides. What would the fourth dimension look like?
Consider a three-dimensional representative of a cross-section of a polyhedron, each dimension quantifiably proportional to the others. If we increase the number of sides or vertices of this cross-section, uniformly approaching an infinite number (Limit), we will reach the instantaneous point of discontinuity beyond which emerges a new geometric, cross-sectional identity, a topological sphere. We can think of this as happening over real time as a process or event, precipitating this form by the interactions of subsystems.
This transitional shift from linear to nonlinear brings with it the 'life-blood,' so to speak, of the sum total of all noninteracting linear components by internalizing and initiating their nonlinear capacity to integrate and self-organize, thereby acquiring a global identity. Each separate element of the previous configuration (a side or vertex of the cross-section representing the polyhedron, e.g.) now owes its existence to the fact of being an integral part of the whole. From the perspective of Form: the previous 'parts' that joined to constitute this new geometric shape have no meaning of and by themselves after they collectively 'move' to create a distinct pattern. In effect, the polyhedron evolves, crystal-like, into a sphere. Can consideration of the individual letters, or subset of letters, of a word give us any information as to the word's meaning?
A representative element incorporates all essential identity features [A feature is an essential descriptive element of any geometric model.], minimally expressed, of whatever form it is capable of generating. It does not contain any particular feature or characteristic that extends beyond the general order and balance of the whole. It is scale invariant; and can not be broken down into separate, self-sustaining components. Each sub-part of the representative is mandatory and necessary without which its integrity, and hence ability to generate the whole, would be impaired. That is to say, there are no sub-dimensions [a dimension in phase space is a degree of freedom, a defining feature] composing a subset of the total number of dimensions involved [a component] capable of generating the whole; there are no features that can be derived from any subset of the others, each is independnent. It is complete, maximally efficient, and symmetric in configuration.
HYPOTHESIS: Best representative of a given equivalence class or, on a larger scale, identity-form of a factor group <--> Simple group corresponding in order (number of elements) to the number of independent, mutually balanced dimensions of a given geometric figure <--> an infinite dimensional hyper-sphere <--> indefinite integral as limit of infinite series of ordered points in N-space, as the number of vertices of our N-dron increases (in 3-space our N-dron is a tetrahedron; and in 2-space our ordered points can be characterized as complex numbers equivalent to the Euclidean plane).
In order to prove this hypothesis it is necessary to find a group that is composed of an infinite number of elements and is also simple. Geometrically we need a space bounded by an envelope, and composed of linearly independent points nonlinearly bound. It has to be an indivisible whole or pattern, componentless. This is an important point because in order to draw a parallel between the limit of an infinite series as a higher ordered language form, a basic language form, and a simple groups acting as a model of all essential features and dimensions of a nonlinear process geometrically represented, we must find an acceptable group that is: infinite in order, nondivisible, and normal or invariant.
The infinite abelian group of real numbers, under the operation addition is not simple. There are several subgroups of the reals under this operation. Is there such a creature as an infinite simple group? In order to gain a better insight into the construction and structure of a composition series, the following information
should be introduced.
The fundamental theorem of Abstract Algebra combines the above notions under the heading of a particular kind of transformation or map referred to and defined above as a homomorphism. This map is characterized
by its group operation or structure preserving quality. The most significant fact concerning these maps is that it makes it unnecessary to go 'outside' a given group in order to attain all its homomorphic images. That is, any homomorphic image of any group can be found to be isomorphic to some factor group of the original
parent group, 'the group modulo the kernel, G/K.' In fact, the set theoretic definition of homomorphism is gained through the process of finite intersections of subsets of a given parent set, together with an equivalence relation, reducing, ultimately, to the smallest subset, the identity element of the parent group. The order of each subset divides the order of its respective parent set. Even the idea of 'function' or 'map' partitions the set, thus establishing equivalence classes as solution sets for each function working on the set.
The homomorphisms of a simple group, considered as parent group, either relate each element of the group to an equivalence class containing only that element (essentially forming an isomorphism,'G/[e]'( '[]' - notation for equivalence class), or map all elements to one class,'G/[G].' In this latter case the kernel thus would be the entire group. That is to say, under a homomorphism, either the simple group is left intact with each element displaying its linear property, or it takes on the role of identity equivalence class evidencing its nonlinear, holistic quality, a quantum shift.
Does there exist an infinite simple group?
How do we appraoch this? Possibly by declaring that no part of a given form can exist separate from the whole? We are interested in the notion Form here, not 'collection of elements' exhibiting that form. Is there one best represtative of a homeomorphic and isomorphic form?
The generic template or set of basis elements are characterized as mutually independent or disjoint: vectors of a coordinate system (linear algebra), a set of elements of a group capable of 'manufacturing,' under the group operation, the whole group (modern algebra), open (or closed) sets of points (topology), differentials (differential equations), prime factors (arithmetic), or, a representative Euclidean 'piece' of any finite dimension (calculus). Regardless of the mathematical field, there is always one fundamental source whose orientation to the answer is clearly the root of the question and dissolves the interface. Topology concerns sets for which the idea of when two points are close enough together allows us to be able to define a continuous function. Two such sets of points, or topological spaces, are structurally the same if there is a one-to-one mapping between points of the respective spaces such that both this function and its inverse are continuous. Two spaces that are structurally the same are called homeomorphic (analogous notion to isomorphism in Algebra). Any continuous function, as a member of a topological group, generates a homeomorphic image of any given figure, and is thus this interface dissolver.
First, what we need to do is come up with a shape or form that is topologically equivalent to all other shapes of the same dimension to act as our geometric model, with a few restrictions. For the sake of simplicity and visualization we can limit this form to three dimensions. We also want the surface to be uniform, no 'holes.' Holes introduce the problems of discontinuity and sub-units (neighborhoods, coverings, connectedness (path-wise), etc.). Surfaces are examples of what, in Topology, are called manifolds. An N-dimensional manifold is a space with the property that every point has a neighborhood homeomorphic to Rn [equivalent to Euclidean space of n-dimensions]. A set in Rn is a surface if and only if [iff] it is compact, connected, and satisfies the above property. The adjective "closed" in the context of surfaces, or manifolds, then means compact without boundary. We want it to parallel a simple group construction. The most economically efficient and symbolically agreeable form for the job is the sphere.
The surface of a sphere is two-dimensional. We have, at present, no justification for including the interior of our sphere. It exists in three-dimensional space but, as yet, our choice of metric unit is unimportant; we are only considering its topological qualities, scale has no meaning. In Topology there are criteria distinguishing the surface of a point-set figure from its interior. It may be difficult to visualize a two-D surface without thickness but, if we hold a ball in our hand and look at it, what we see, of course, is just that -- a two-dimensional surface curved in three-D space. This ball, bounded and infinitely dense, is self-contained and is thus closed.
We will briefly touch on two infinite groups in order to develop motivation for the one we're after.
A hollow sphere may be used as our model for the group of positive real numbers under multiplication. The sphere is a curved, nonlinear, finite surface in 3-D. The number 'one' is the center of symmetry, the group identity element. It represents a point equidistant from all sets of mirror images. One is next mapped onto a geodesic curve represented as a circle on the surface of our sphere at the 'equator.' These circles are much like lines of latitude, by choice we can position all reals greater than one on the 'northern hemisphere,' and all reals greater than zero but less than one on the 'southern.' Inverse-pairs are spaced based on symmetry, the coordinate system maintaining the integrity of the group structure. All the positive reals, due to this one-to-one mapping, take on a two-dimensional character. These circles are infinite in number and cover the surface of the entire sphere. The 'north pole' is the limit of infinity, which is never reached, as the 'south pole,' the point of zero, also is never reached. For example, one-fifth would be the same 'distance' from the number one as would five, in the opposite direction, of course. We have the open set, (0, 1), and the half-closed set, [1, 0/0), the union of which accounts for all positive real numbers, (0, 0/0).
Algebraically the positive reals are still one-dimensional; geometrically, they are not. The positive reals, as an infinitely dense collection of circles, 'fill-up' the two-dimensional surface, exposing a hidden symmetry to each real number as curve, localized by sets of points on the surface of our sphere. Our best representative of this group is the circle at center of symmetry, having the power to generate the entire sphere by rotation about the 'hole,' and from which all other circles on the surface can be derived as homomorphic images, members of the family. We have been forced into giving our 1-D group a 2-D character because of the geometry of our model.
A Vector Space consists of an abelian group V under addition and a field F, together with an operation of scvlar multiplication of each element of V by each element of F on the left, such that for all a, b in F and @, $ in V the following conditions are satisfied:
Consider the abelian group {R, +} = R x R xR x ... xR for n factors which consist of ordered n-tuples under addition by components. Define scalar multiplication for scalars in R by: r@ = (ra1, ...., ran) for r in R and @ = (a1, ... an) in Rn [the cross-product of the Real Numbers, equivalent to the Euclidean space of n dimensions]. With these operations, Rn becomes a vector space over R. In particular, R2 = R x R is a vector space over R, that is, all vectors whose starting points are the origin of the Euclidean plane.
The complex numbers form an algebraic structure known as a field. A Field is a commutative divsion ring, more structures. A Ring distinguishes itself by being the first algebraic structure a student of Modern Algebra encounters that consists of two binary operations, defined on the real numbers. As such, it forms an abelian group under addition; multiplication is associative; and for all the real numbers, the left and right distributive laws hold. The concept of a 'division ring' includes the addition of multiplicative inverses for all nonzero real numbers. Along with: 1) an identity element called unity, and, 2) the commutative property of multiplication, we have created the structure known as a field. What we have, in effect, is an algebraic structure that is both an additive abelian group and a multiplicative abelian group together with both left and right hand distributive laws.
A complex number is composed of two parts; its form is a + bi where a and b are real numbers, and i is called an imaginary number; algebraicly this imaginary part is equal to the square root of minus one, allowing for the solution of polynomials whose degree is five or greater. The field of complex numbers is a two-generator system. Considering the real parts as ordered pairs, (a, b), we can arrange them against an orthogonal axes system, or grid, the horizontal can be labelled the real line, the vertical, the imaginary line, the intersection is called the origin. The complex numbers map one-to-one onto the ordered pairs of real numbers coordinating the Euclidean plane, and as such are indistinguishable from this plane.
So far we have amassed a few algebraic structures to act as lenses through which to interpret what we see in an otherwise uncoordinated space of objects, be they recognizable figures or processes or events or simply patterns, they are still objects capable of being independent; they, in fact, create the space wherein they reside. The fundamental structure of Group, the two-operation structure of Field, and Vector Space are our lenses and tools for analysis and perception. What duty they perform comes into play when we wish to see what properties remain unchanged, or invariant, when respective structure-preserving maps operate. For a group, these are homomorphisms, and for a vector space, linear maps. Each algebraic structure brings its own peculiar nature with it; their commonality lies in the symmetry they portray; sets of elements, acting as one, under operations.
What we have with a two-dimensional information source is a limit cycle repeated on an infinity of different scales. If we were to constrain the surface of our sphere, that is what we would have, a limit cycle. But the adjective "closed" in the context of sufaces or manifolds means 'compact without boundary.' Compact is a topological concept. A collection of open subsets of the manifold form what is called a "covering" of it. The manifold is compact if every open cover of the manifold has a finite subcover, that is, if there is a finite number of sets of this collection such that the manifold is contained in their union. Our sphere is therefore finite yet unbounded.
The information offered is of a nonlinear type but, without the further dimension afforded by the lens of a vector space, we have no sense of direction, or orientation, except what can be gleaned by symmetry. Remember, the sphere represents a dynamic system, we need more than symmetry to relieve uncertainties of a linear nature.
In order to ask a real number question of a complex field, the reals must first be put into complex terms for it to make sense, otherwise the information received will be incongruous in this added dimension. This is true regardless of the fact that a real field structure can be found embedded in the complex. Consideration must be given to the imaginary axis induced by the complex coordinates.
The group of positive reals under multiplication is deficient for our purposes; If for no other reasons, it is composed of an infinite number of subgroups, and is incapable of modeling the interior.
The second group to consider as a potential candidate is the group of all real numbers under addition. It too can be overlayed onto our sphere. The positive circles 'fill-up' one hemisphere, the negative, the other, with zero at the center of symmetry. It just so happens that the group of positive reals under multiplication is isomorphic to this additive group by the relationship: Log (xy) = Log (x) + Log (y), (x,y: real variables), so it too fails as a candidate to model, algebraically, a simple infinite group.
Isomorphic means there is a one-to-one correspondence between elements and that the structures are equivalent, and all laws of physics are symmetrically conserved. Technically we are dealing with a map, from one group to the other, which is bijective both ways, i. e., its inverse is also one-to-one.
Our algebraic model consisting of surface only is insufficient; there are an infinite number of locally independent components - subgroups. We must generate an interior in order to access a further dimension; constrained to the 2-D surface we will not be able to uncover its source.
A vector space covering the sphere, besides granting the additional information accorded by direction and orientation, automatically brings with it a linear aspect. In the third dimension, the end points of vectors radiating from the center of our now solid sphere compose a uniform connected surface. When the basis vectors span the delimited space, they generate the interior, as well as radii. The circles on the surface have transformed to include the interior; we now have a solid sphere composed of an infinite number of orientable, concentric discs. Orientation of the surface is imposed by the two-dimensional evironment of the vector space. The interior orientation is induced by a right angle dimensional extension along the radii. The surface's operational environment is a sharing of the isomorphic complementarity afforded by the two-D vector space occupying identically the space of the complex number field, a two-generator system, itself mapping precisely onto the Euclidean surface of our sphere.
This changes our model from a hollow sphere composed of an infinite number of circles, to a solid sphere of concentric discs, the centers of each aligned through an axis. This is because in the process of relabeling our group structure we shifted from a nonlinear environment to a linear one, due to the operation of addition. The elements of this group are discrete, real numbers. The rule of combination establishes an algebraic structure that allows us to travel from one point to another by translation. In this atmosphere only linear creatures can exist. The linear aspects of our geometric model now stand out from the background and are the sum total of the 'kind' of information we have accessible to us.
If, in the other case, the operation is product formation on the positive reals, we must travel from point to point by rotation because our structure bends or curves. Multiplication includes addition as a special case, as a quantumly lower language. This is one of the reasons our model changes from surface-only [non-linear] to solid [linear]. We now have such things as: radii, center point, chords, etc. A radius, e.g., cannot even exist unless there is a geometry to support it. The set of points referred to as the 'interior' gives this.
If a system is graphable, it is linear; otherwise it can onle be assessed as a probability wave or a map. Obviously a 3-D figure extends beyond the confines of a 2-D frame.
A group, as foreground, whether under the operation of multipication or addition, can be considered as a less restricted sub-system of the more complex system of a vector space. A geometric figure can be modeled by a unique vector space with a given basis, and, as algebraic structures are capble of geometric interpretation, the vector space must, by parallel mimicking, induce a metric in the topology of the geometric figure.
Assigning an algebraic representation to each discrete point linearizes its relative position and orientation, inducing a compatible orientation to the whole space. Its identity is still based on relationship, however, but it now has the dual nature of nonlinearity, by virtue of its holistic or global aspect, its place in the scheme of things; and linearity, from its unique point of view.
Geometric points have no reality in and of themselves; they exist only as part of the pattern of relationships. Because our model has a dynamic character, it would probably be better to think of these points as 'states' of what is called 'phase space,' or as 'events' in space-time. A trajectory in phase space is an image of connected points reflecting a particular life history of a system. Obviously the initial, or boundary, conditions are important determiners of future history, as are minor fluctuations within the system itself. Interacting nonlinearly throughout its history with its self and with the dynamics of the environment within which it unfolds, the completed trajectory is only one of an almost unlimited number of possible paths. In Nature, however, balancing gravitational effects with velocity, an object's free-fall motion through spacetime will follow the optimal path, maximizing the time it takes to travel from point to point.
Our solid sphere is three-dimensional, of course. The vector space of R3 (3-D real number space) is isomorphic to Euclidean 3-space. It lends descriptive features (information), definition and detail, conditioning whatever Euclidean group is used to analyze and classify a 3-D geometry (e.g., group theory applied to crystallography has determined there are seven basic crystal forms). The group, in its role, highlights the symmetric and invariant relations of these added features. And, because the general linear topology is the environment of our sphere in the broadest context, we create the necessity of establishing a Euclidean 3-D space as a special case metric.
The symmetries of the geometry and vector space of our best representative are expressed by some group and, in that sequence are the different lenses through which we observe the primitive concept-configuration, perceiving relationships in alternating relief, gestalt-wise, and aquiring information conducive to a deeper understanding of the contextual facets. We can envision, as the number of discs of varying diameters approaches infinity, the environment shifting from linear to non-linear, or, from a series of ordered sets to an envelope, analogous to the sahift, at limit, of a polyhedron to a sphere.
Looking at our sphere through the lens of Algebraic Topology illuminates those idea and concept relationships profiling an algebraic treatment of the relevant topology. One such idea which will be of some use is that of Slice. A slice is an equivalence class of points (represented as 'ordered triples' for our solid sphere model, vector space-wise) sufficiently close so as to be continuous or globally connected. It can take any geometric design, from the combination of a point on the surface plus a radius plus the center point, to that of a circular cone emanating from the center ending at one of the discs composing the sphere whose circular boundary is part of the surface. It can also, by extension, end at the entire sphere. This is what 'allows' a solid, infinitely point-dense sphere to be compacted from surface to center point. The center point is the common one for all slices having discrete points on the surface as boundary members and, is sparated from the surface by radii, infinite sets of points in our topological space.
What is required is that there be no holes, and that the collection of points reflect common properties (or a single property) considered as a unitary sampling of the overall figure under study. Minimally the slice as equivalence class is a line from center to some surface point. The individual points interact nonlinearly,
in this dynamic system, insofar as they owe their meaning to the collective, and without any one of which there would be no slice. The utility of this idea will show itself shortly.
Another idea from Topology which will be of some use is that of what is called a 'Ball.' Simply put, the n-ball En is the set of all points in Rn a distance less than or equal to the radius. If the ball is open, it does not include the surface (in some texts, referred to as the "frontier") whereas a closed ball does include the surface. Thus a closed E3 is what we usually think of as a solid sphere, E2 is a circular region, and E1 is a line segment that includes the end points if closed. The center of our solid sphere is a single point, or singularity, homotopically equivalent to the entire sphere; we shall label this point, zero. 'Zero' here is not the identity element of a given group, but rather stands for the center considered as the annihilation of opposites. The opposite components constituting the ball need be synthesized to a higher ordered embodiment, symbolized by radii as representative or primitive 'elements.'
As an analogy of what we have so far discussed, the center of the sphere may be construed as 'self.' The surface periphery is the ego collective, each element of which has meaning, significance and existence owing to its relative position, oriented in time. The ball as interior minus the surface, as between ego and self, is a vector space of inverse-pairs, opposites joined and integrated, synthesizing the complex conjugate-pairs on one level, and coordinated on a lesser, linear dimension as opposite planar points of a segment, that is, a reflection about an axis. This latter level corresponds to linear time; the deeper level, 'imaginary' time. Imaginary insofar as its axis bisects linear time at right angles. And beyond that, the real part of each complex conjugate is also a synthesis of conjugate-pairs, ad infinitum. So that linear time and nonlinear time are interblended, intermingled, intertwined, joined as one, all through the trajectory of some life-history. The vector space of 2-D fills the same space as the complex number field and as such acts as a planar template, a horizontal slice, perpendicular to the direction of linear time. The linear ego, oriented in time, possessed by negative animus from the unconscious, remains unaware of this cross-section. It must first acknowledge, become aware of, its nonlinear, interactive side vis-a-vis the rest of the family of psychic functions, its functional position; and in the process realize its own identity as the opposite end of a psycho-somatic slice with the self as center and anchor.
Inner knowledge, or "soul substance," is gained when dominance by possessive, unconscious animus is confronted, and 'defeated' allowing for unconscious anima to fuse with ego bringing to consciousness its nonlinear heritage, and, in so doing, acting as conduit to its well-spring, the unconscious. With this, center of consciousness is transformed from periphery or surface to true center - self. The surface closes or bounds the infinite-dimensional ball and represents the linear/nonlinear, fractal-like interface between consciousness and unconscious mind. But this transformation cannot occur without the help of animus intelligence, that introspective light, doing the soul searching. The geometry, the unconscious nonlinear aspect, is there, in place.
What is the algebraic aspect of the model? What is the nature of the model that complements?
The opposite functions must be internally synthesized, including the self in its dual aspects. Identity must stem from the self, with ego as counterpoint and surface-interacting agent of personality. The algebraic aspect of the sphere must have no sub-systems capable of functioning independently, in correspondence with separate psychic functions which can act only as transitory portrayals or pre-transformed parts of the whole when expressing or emphasizing their linearity.
The algebraic model must, therefore, be infinite simple; be compatible with the geometric model; and embody, symbolically, the same psychic relationships as the geometric. The two infinite groups we've looked at do not fit the bill. They are decidedly not simple. We are all the way into the details of our sphere and yet can find no candidate. We have to shift gears.
We are forced into using a 3-D model. That is, our 1-D group of reals under addition, pushed into 2-D characterization by the curvature of the surface, as was the multiplicative group of positive reals, is further pushed into 3-D-hood because of the nature of our geometric model. For the sake of argument, it is important to look at substance as well as form. Also, for the argument, we may treat linear aspects of our geometry, parts of the whole, as basic units, for now. Our keys are the frameworks of linear and non-linear environments. One of the significant formulations of non-linearity is that there is no direct discernable relationship between any subset of parts, or components, of a system and the whole. The linear principle of 'superposition' cannot be applied. It states that the sum of parts equals the whole, and is used to test a system to determine its eligibility as linear.
We're getting away from merely mapping a set of elements to a unique image [e.g., the pair (3, 2), under addition, to 5), and venturing into moving a coordinated point, expressed by ordered n-tuples of real numbers, through a linear transformation to be mapped to another point, possibly itself, in a vector space. The 3-D vector space coordinates each point by an ordered triple. Regardless of basis elements chosen, they can be normalized and reoriented to coincide with the three axes. Each coordinated point, as well as the whole, is simultaneously oriented when the basis is fixed; and each can be represented by its separate components, combinations of scalar multiples of basis vectors, that can be arranged column-wise in a matrix. These basis vectors are linearly independent, that is, they do not interact, none can be derived from any combination of the others. The choices of bases is unlimited affording various orientations, or points of view, which not only reveal contrasting facets, but also the inexhaustive configurations of relationships scaled to order.
What we have constructed is called an 'orthogonal transformation.' All orthogonal transformations can be determined as reflections and rotations on some basis in mutually perpendicular dimensional planes. An orthogonal group has as elements linear transformations in matrix form; these matrices are non-singular (which means the column vectors are linearly independent, and each such matrix has an inverse); and the set of 3-D, non-singular matrices under matrix composition constitutes a group. As matrices of order greater than or equal to two are not commutative, neither is our group. We have a 'unity' in the form of the identity matrix; and each non-zero matrix has an inverse. You may recognize this structure from above as a division ring.
"This classical group, (GL(n), General Linear Group of 'n' dimensions) will, in general, have a center, 'Z,' consisting of those scalar multiples of the identity that are in the group, and in the case of the orthogonal group there may be a normal subgroup with an abelian factor group. Such a normal subgroup (as GL(3)) up to certain identification with its center (that is, modulo its center), will be, in general, a simple group."** The center of a group is the set of all a in G such that ax = xa for all x in G, that is, the set of all elements of G which commute with every element of G. The center of a group is a normal subgroup of the group.
Analogously, the center of our sphere is not a point but a unit-sphere with center concentric with that of the larger sphere model. The unit-sphere, identified with self, is geometrically non-divisible and personifies the union of opposites; and algebraically, as the repesentative element, is capable of generating the entire sphere. The geometric center is strict nonlinearity as it symbolizes the union of both sides of the unconscious; while the algebraic embodies both linear and nonlinear aspects of the self. Although the self is in direct contact with the negative animus of the unconscious, it transcends the power of the animus to control by a union with maternal anima, its feminine half, but on a deeper level than that of the ego. The self, however, must face this terrible force of the unconscious in order to free the ego from its destructive slavery; and the ego, for its part, must recognize the situation and join in the struggle to work for its own liberation.
What is the solidness of a system, an event, a process? This question appears nonsensical; nonetheless, because the notion of solidness is important to this discourse it must be understood in terms of the present context:
By way of summation: Given the infinite set of matrices in 3-D vector space, each one of which includes linearly independent vectors, they form the following algebraic structure: collectively they comprise a subgroup, OT(3), of the general linear group, GL(n), of three by three, non-singular matrices with non-zero determinants; OT(3) is the infinite group of orthogonal transformations under matrix multiplication in three-space; 'Z' is composed of all scalar multiples of the identity transformation [in standard form - (1,0,0), (0,1,0), (0,0,1)]. These basis vectors span the entire space, that is, they generate coordination of the space. The product of any one of these orthogonal matrices and its inverse equals the identity matrix by definition of non-singular. The factor group, OT(3)/Z, looks like the set of equivalence classes whose identity element, Z, is as described above. Each class [coset] contains one distinct element (matrix) together with its scalar multiples; and the set of classes are mutually disjoint. Moreover, Z forms a maximal normal subgroup of OT(3), that is, there can be no larger normal subgroup of OT(3) than Z as Z, by definition, contains all identity elements of OT(3).
The group of orthogonal linear transformations, in matrix form, modulo its center, as defined above, herewith OT(3)/Z [the symbol 'Z' here stands for center, not the set of integers], is a factor group of OT(3) and, therefore, a homomorphic image of it, preserving the structural relations and operative uniqueness of the parent group. OT(3) is that group of continuous transformations leaving invariant the algebraic and topological properties of our model. Topological groups are always infinite. OT(3)/Z reveals all symmetric permutations of any homeomorphic (akin to isomorphic) image of our sphere; and, most importantly, it is simple.
It remains to prove that OT(3)/Z is indeed an algebraic sphere.
Going back now to our solid sphere overlayed with 2-D surface points as closed circles:
In the act of defining this set of identity-multiples as the interior and identity of the fator group OT(3)/Z, we bound on the finite spherical surface an infinitely dense set of points, a globally uniform continuum. The vector space of 3-R imposes order on these points, and the group displays their collective symmetries.
If we view our presently constructed sphere through the eyes of its topological space, it can be pointed out that a 2-D disc in three dimensions consists only of its boundary points. This is because no ball of positive radius lies inside it, a general criterion condition for the existence of solid objects. For a solid to exist in three dimensions, it must be compactible to a ball in order to define the 'neighborhood' of its center point. A 2-D object in three-space obviously cannot be so compacted. Furthermore, the center of symmetry of our topological group is the collection of inverse-pair, identity-multiple maps contained in Z, not a single point.
In a composition series each dimension embeds those less than it, and the diminishing sequence of structures is nested interdependently. If nonlinearity is introduced as a property of the geometry the series emulates, determination as to which dimension one is 'looking at' is not possible, or should I say, predictable. Each factor group structure is scale invariant and independent of whatever coordinate system is applied. For example, in a dynamic system, the third dimension contains all invariant properties of the second integrated as a unity, but the when and where of the emergence of a particular permutation, from the point of view of the third, can only be couched in probabilistic formulae, or through the medium of a strange attractor.
The disc at the center of the grid overlayed onto our sphere has no interior in the topological space as long as it remains limited to two dimensions. But, a three diemnsional disc does, and from this vantage point the entire set of two-D discs becomes a subject of indeterminable nonlinearity, and the sum and substance of our solid sphere in phase space.
Our algebraic sphere is of a dynamic character, a 'sphere' of transformations. It cannot 'exist' unless it is in motion. Rotating our representative disc around 'zero,' its geometric center, we generate a solid sphere. Which disc in particular? The identity disc at center of symmetry on our grid, the one at the equator on our sphere. The edge of this disc is the surface circle, the closed curve of our grid, mapped to zero of our additive group of real numbers. Because its third dimension is composed of all 2-D discs in series, contractible to its center point, it is also all the interior of our solid sphere. Therefore, the three-dimensional version of our representative element is 'Z,' the kernel of our homomorphism from OT(3) to OT(3)/Z.
The infinite set of three-tuples of our vector space on the surface, the 3-D disc, and the center point of our sphere are all of the same homotopy type. Loosely, two figures will be said to be of the same homotopy type if one can be reshaped, compressed or expanded, into the other. So, our center disc is homotopically equivalent to its center 'point,' itself expandable to all the interior of our sphere, and as extension, its surface as well.
We have created an algebraic sphere independent of coordinate systems defining its surface area; its 'boundary' is defined by the group axiom of 'closure.' And, this sphere has the nature of the factor group, OT(3)/Z. If we consider the collection of first derivatives of this bounded surface, we see that the angle between each tangent line and the surface-boundary is the same for each. These tangent lines are related linearly by a proportionality factor, and its constantcy over the entire set of derivatives transmutes the set into a single form, a constant. This 'class of tangent lines' will become important when we examine the boundary of our sphere.
Analogously, the psychic significance and symbolism offered by our various centers of symmetry can be framed in the following manner: a Euclidean geometric point; a 3-D vector space consisting of the three unit-basis vectors; a compacted, topological ball; and the identity element of our factor group, itself a collection of inverse-pair identity maps, the kernel of our homomorphism - 'Z.'
When dealing with form there are only those questions that can be asked that concern kinds-of-things, and no further. A framework is a form which can underlie metaphors labeling a structure. All equivalent structures maintain the integrity of that frame, and preserve the related, invariant algebraic dynamics contained within. A representative element contains a maximum complexity of order, while yet utilizing a minimum amount of necessary, defining, invariant properties to express itself, the least amount of energy to maintain itself in balance and symmetry. It possesses what we may call, molecular autonomy. A solid sphere is the best representative element of the class of all possible, uniformly continuous, homeomorphic images of itself. Therefore, discs and center point 'exist.' But this is only up to the point of Limit on behalf of the integral, and Modulus for the group. The surface of our sphere models most succinctly both our infinite simple group, OT(3)/Z, and the integral as limit of the number of sides of a poyhedra composed of polygons (these are the 'discs' of our solid sphere at limit).
As the number of discs of our solid sphere increases to the limit of infinity, the surface, f(x,y,z) = x2 + y2 + z2 + k, is formed as the general solution of the indefinite integral, equivalent to a family of images of the fundamental surface, separated from one another by the proportionality factor - 'k' (an onion peel effect). Also, if the interior of a figure is homogeneous and merely echoes, congruently, the boundary curve, then it offers no additional group theory information. Our concern is for symmetry and orientation, how the figure behaves under suitable transformations.
Analogously, considering a unit-sphere as metaphor for 'self,' of any dimension greater than or equal to the third, with normalized basis vector orientation, a transformation that maps or identifies one of the bases to the center [collapsing a dimension], affects the formerly balanced self by splitting the dualities and setting askew the nonlinearity of the collective, empowering one or another of the various psychic functions. And, by so doing, denies the self as center of personality, granting it instead to the ego, linearized once again by possessive, unconscious animus.
Each successive identity equivalence class of a normal series, correlating and connecting various defining features in unique ways, delineates the best representative element, or center of symmetry, obtained from the higher-dimensional kernel, in terms of levels of integral part relationships. The dimensions of this series are independent of the modulus effect. The action of the modulus displays and emphasizes the identity of the whole as 'seen' from a perspective that organizes and associates symbols along the lines of a definite, recognizable pattern; a pattern that generates itself amongst the remaining symbols of the parent. It creates a class from formerly embedded characteristics by associating their commonality. The commonality of properties shared amongst the elements of a class, or simple group, is not the same as the properties associated with the whole as such; the global invariant properties of a system are greater than the sum of separate yet common relationships.
We have taken the group of orthogonal maps in three-space and, by defining an algebraic center of orientation, have imposed a collective identity on a specific form of matrix, the infinite set of non-zero real number multiples of the identity matrix. Order, and the establishment of a 'boundary' or 'membrane' porously interfacing and separating the layers of a composition series, is the result of the modulus effect - OT(3)/Z.
We have the composition series beginning with the orthogonal group OT(3). Its maximal normal subgroup is the collection of all scalar multiples of the identity, Z. The factor group, OT(3)/Z is simple. The scalars have not as yet been defined. We can begin with the complex numbers. The introduction of conjugates of Z has the effect of geometrically creating an additional dimension. Z, with complex number scalars, contains the maximal normal subgroup, Z(R), which in turn contains the maximal normal subgroup, Z(Q), consisting of the set of all rational multiples of the identity matrix. Its respective factor group is Z(R)/Z(Q). There are no non-trivial maximal normal subgroups of Z(Q), therefore, Z(Q)/[e] (where 'e' is simply the identity matrix itself) is the final member of the series. We have four members, OT(3)/Z(C), Z(C)/Z(R), Z(R)/Z(Q), and Z(Q)/[e], each infinite in number. However, the reals, composed of the rationals and the irrationals, are of a higher ordered level of infinity than the rationals which are considered 'countable.' The factor group, Z(Q)/[e] is infinite but with only a single element in its kernel. For Z/Z(Q) we find ourselves factoring infinities, or limits, depending upon your point of view, of nonlinear identifications, establishing a transitional border as between absolutes.
Indefinite Integrals, in general, are form-characteristic; Definite Integrals are interior measurements of a specific bounded space; and Partial Derivatives give us a local picture correlating variables of time and space in their relation to one another, and to a whole geometric figure modeling a process. Continuous and uniform acceleration into the present is the second derivative of space with respect to time, i. e., holding time still. The first derivative is velocity, distance with respect to time. Each dimension acts as the generator of the next, and the set of first derivatives in union, the 'class of all tangent lines,' envelopes our geometric sphere. This equivalencs class functions as the generator of the set of all accelerators by holding time still, and in so doing renders a higher-structured language with sublevels of uniform patterns. This class of acccelerators, in turn, dissolves the proportionality factor by focusing all existing pattern probabilites into one comprehensive design or shape, and, in the interior, effects the mutual annihilation of opposites through synthesis.
The leap from polyhedron to sphere or envelope of the series of discs; from addition to multiplication as dimensional shift from linear to nonlinear; from series of normal factors, all simple, to Simple as best representative of a kind, are each expressive of parallel clues, metaphorically exhibiting a natural tendency. The orhtogonal group, OT(3), modulo the set of all inverse pairs (identity under product formation), modeled by the interior of our solid sphere, gives us the infinite simple group we need to justify the assertiion that at the threshold of an infinity a non-linear form, in relation to its parts, emerges as an individual unit.
A Simple Group and the notion of Indefinite Integral modeling, for purposes of information, equivalent descriptions of some globally connected process give us an insight into the probable way that nature itself renders substance and function: in terms of growth and pattern. And, in the actions of generation, evolution, and discovery, an infinity of parts is seen and understood in a new light: as an interdependent ecology of distinguishable features. What is real and perceivable is the quantum jump; what is only imaginable are the preliminary steps.
Applying the knowledge of one system's patterns and permutations, its configuration, to another unknown system, allows us to learn about this other system. But it is not the information, the content, that is applied, but rather the manner by which that knowledge was organized, the interrelationships, the connections across dimensions, the overall shape, all functioning as one whole, a configuration of thought-contents, ideas. We are looking for recognizable pattern relationships, something comparable. But there has to be other configurations possible that connect relevant features, there is no 'one way of seeing things.' Instead of imposing a structure onto an unknown glob, trying to find sets of realtionships within that system that will conform, we can choose to try to understand the other system from its point-of-view. The third choice [there's always a third choice] is to start from scratch, from the ground level, using mental tools of pattern building, those very same tools we originally worked with to generate the ideational forms we come to know in a more advanced state of development. The building tools are discrete and unitary, perceptions on the simplest level. With these, we can construct edifices of the mind, see patterns not seen before, and deconstruct the conceptual overlays through which we perceive reality.
Geometry is a way of reasoning about space. Topology, Affine, Projective, Riemann, and Euclidean geometries, defined by principle groups of transformations on their respective invariant properties [Felix Klein], may be considered as a "normal series of embedded, self-contained, simple factor groups" themselves, offering their unique pictures of the world for purposes of calculation and comprehension. Euclidean allows us to ponder parts of a whole; Topology lends a broad vista, twists and turns shapes, ordering reality into a connected force, mapping objects to one another based on a measure of continuity, and displays groups of forms as its basic units [Algebraic Topology]. The object of Algebraic Topology is to associate to each space certain algebraic invariants such that two spaces will be homeomorphic if they have the same invariants. That is to say, If two spaces are homeomorphic, the groups of one are isomorphic to the corresponding groups associated with the other. Therefore, for spaces to be homeomorphic, it is necessary for their groups to be isomorphic. This doesn't work the other way around, however. Isomorphic groups between two spaces do not insure that the spaces are homeomorphic. What is important to realize is that continuous mappings from one space to another renders homomorphisms from the groups of one into the groups of the other.
Compactification identifies the surface of our solid sphere with its center point. Because of the 'onion peel effect' we have a family of surfaces decreasing in radii from some idealized outer boundary until we reach the singularity, at which the laws of physics, and hence our Reality, break down. This is true, provided we go through the medium of infinite density, or connectivity, which is a 'filling up' of all space. A space is connected if any two points can be joined by a path lying totally in that space. If a space is not connected, then it is split up into a number of pieces, each of which is connected but no two of which can be joined by a path in the space. These pieces are the connected components of the space.
Our integral is equivalent to the orchestration of its parts. Its identity is the pattern, the melody; the group's is its center, radiating to its outer edge; the sphere's is its surface, its outline.
What I wanted to show was: through the process of exploration, in order to determine what, if any, group would best be compatible with the geometric model, and give us the simple group arrangement to parallel the integral, we were forced into using the natural group of orthogonal mappings, by which to factor into a simple form.
Our first group, positive reals under multiplication, was not useful because, even though it was conformable to the surface of our grid, was itself not simple and contained no normal subgroup by which it could factor into simple. We tried infinity of one-dimensional discrete elements under multiplication, then under addition with the entire set of reals, a different atmosphere, similar terrain. But, even with our grid arrangements, we discovered that it is not possible to impose a one-dimensional world onto a two-dimensional continuum expecting, moreover, to model algebraically a three-dimensional object. The collection of three-tuples as coordinated points isomorphic to the vector space of triples of 3-R is also not apllicable for the same reason as the reals.
Trying to create a nonlinear environment so that form and function can exist as a phase space necessitated finding the right group that was not only infinite, but had a maximal normal subgroup by which to factor it into a simple group. Our group of linear transformations in 3-space has a dynamic character, and is concerned with form as function. Families of patterns are the basic units by which we grasp function, by which we think.
"The geometric theorems about sets of points (coordinates) are only a peculiar way of stating algebraic results about homogeneous equations."* The fundamental group is the matrix-represented collection of mappings of an algebraically determined geometry onto itself, its automorphisms [automorphisms are isomorphisms of a group onto itself]. Its factor group series reveals the number of dimensions of the geometry ("... if there are objects called points, grouped into classes of various sizes called lines, planes, etc., then, if these classes satisfy certain relations, this classification gives a geometry the number of dimensions of which depends on the number of different sizes of classes.")**
From an algebraic point of view, considering geometric figures as systems of homogeneous equations, it does not matter how many dimensions are being considered. A theorem on a sphere, or an argument using the geometric notion of sphere, can easily be generalized to one on the hypersphere of any dimension.
The Group is the basic mechanism for determining balance, a global property.
Recall that the first axiom of group theory is Closure. Closure is, in a very definite sense, equivalent in meaning to a negative feedback loop. Positive amplification would be a breaking of this symmetry. As the limit of perfect balance is realized, we generate a purity of form eliminating random effects. But in nonlinear reality, this is not possible in either the physical or psychic realms where asymmetry is intermingled with symmetry as the union of complementary opposites. Probability waves describing the behavior of dynamic systems are thus a direct result of this unpredictability caused by the nonlinearity. Cyclic determines predictability. Nonlinear is random in the small, probabilisticly 'cyclic' in the large. The general locus or state of a dynamic system, nonlinearly interdependent as per function, is apprehendable as a dynamic fractal caught, as it were, between dimensional shiftings. Invariance over scale demonstrates that all homomorphic dimensions of nonlinearly generated sub-attractors are magnifications on a theme.
Therefore, a collection of sub-vortices has a symmetric-constant/asymmetric-variable dual nature. Our infinite simple factor group, OT(3)/Z, is cyclic. The determinant of each matrix of OT(3) is equal to a real number and thus forms a homomorphism between the group of three-by-three matrices and the group of real numbers under multiplication, and this latter group is infinite cyclic. OT(3)/Z inherits this property from its parent, OT(3).
Randomness is seen to have an overall pattern as the result of nonlinear functions intersecting dynamically to produce an attractor. These functions all have the parameter 'time' in common. The error factor of each partial differential equation describing a given state or snapshot of an event represents a hole in space where the symmetry leaks out. The collection of an entire system's confluing error factors symbolizes the inherent instability and elusive locus of a system's signature figure.
Given the right field environment, any equation modeling a system can be resolved into its linear components, thus a coordinated figure can be used to represent a cyclic sub-event. However, randomenss stems from: 1) the inability to predict the particular sequence of those known possible forms; and, 2) not being able to determine the exact instant when a certain collection of indeterminable factors will interact, at what phase, to effect what sub-possibility of all possible patterns of the group taken as model. This nonlinearity is the uncertainty of Quantum Mechanics.
As a group of transformations determines a specific geometry by preserving its invariant properties, its structure, the set of possible loci of an attractor resulting from nonlinear functional interactions of linearly independent pieces likewise establishes a specific geometry, with its own unique, attendant signature group. Seen this way, an attractor may be statically compared to a 'family of patterns of fractal arrangements.' The invariant properties of its geometry, nonlinearly relating in a dynamic frame of reference, underlined and conserved by its respective transformation group, represent linearly independent dimensions, and are the limits and identity (as group) of its general form as function.
It may very well be true that there is always one pattern of a transformation group that turns up most frequently in a dynamic system. This would be the center of gravity.
A probabilistically represented wave/event is a gift of all possibles, simultaneously occurring, within a framework. As an example, if we think of the entire light spectrum as continuously pulsing so that every instant was in fact a new universe, then a perception of 'particle' would have to be considered a dimensional shift in the composition series. This renders a change in definition from an aggregate of separate, discernable elements, to a single entity.
Formally, there are linearly independent sets of concepts collectively charged with the creative power to span all other related concepts by some kind of network association on invariant features. And by 'linearly independent' is meant that the concepts are on equal footing, of the same dimension, yet none is derivable from any combination of the others. Spanning a space by expansion, or extension, of fundamental patterns, pattern-building tools, produces more complex patterns or designs, in the process, embedding those generators within the fabric of the whole.
When the essential invariants are reduced below a certain
critical level, there is implosion, collapse of the wave function,
and loss of functional identity as an abstract geometry, or as an
individual. Variations on a theme are permutations constituting
a group structure. And, if we give the notion of group an additional
orientation, a twist, an asymmetrical character, we can use its
geometric aspect to model, among other things, the instantaneous
experience of the creative self-generating moment.
That is what the following two essays are about.
Surrealism: The principles, ideals, or practice of producing fantastic or incongruous imagery or effects in art, literature, film, or theatre by means of unnatural juxtapositions and combinations.
A Group (G, *) is a set G, together with a binary operation * on G, such that the following axioms are satisfied:
1) The binary operation * is associative, that is: for every g1, g2, g3 in G, g1 * (g2 * g3) = (g1 * g2) * g3.
2)There is an element e in G such that e * g = g * e for all g in G. This element e is an identity element for * on G.
3) For each g in G, there is an element g' in G with the property that g' * g = g * g' = e. The element g' is an inverse of g with respect to *.
The identity element and inverses are unique in a group. The axiom of closure is usually added to the list: G is closed under the operation *, that is, (a * b) is contained in G for all a, b in G. Closure, however, is a consequence of the definition of a binary operation on G. In other words, its addition is redundant, but, as it will come up later, I include it here for reference sake. It is important to realize that a group is not just a set G. A group (G, *) is comprised of two entities, the set G and the binary operation * on G. However, I will refer to a group by the single set symbol, but the operation should be understood. A subgroup is a group in its own right as it inherits the identity and the induced operation.
Let # be a homomorphism of a group G into a group G' with kernel K. Then G# is a group, and there is a canonical (natural) isomorphism of G# with G/K (read: 'G modulo K'; meaning of 'modulo': "up to certain identification with"). This needs some explaining.
The elements of V are vectors and the elements of F are scalars.
Our slices, which, as per choice, have as surface 'points' the circles can be represented in our model as sets of solid discs, their outer edges mapped onto specific surface curves one-to-one. Each has a mirror image. This collection of inverses is infinitely dense as surface points and interior. These discs 'fill in' the interior. The interior is, furthermore, composed of scalar-multiples of matrices and their inverses in the coordinated space imposed on the discs' points.
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Addendum:
As an example: A quotient space of universe, U, of ten dimensions would have as candidates: U/1, U/2, U/5, and U/U. These do not represent a composition series per se, but each as factor group can be considered simple, self-organizing, and embedded in ten-space. It might be worth repeating here that complex groups can be built up by the cross-product of simple groups, the basic building blocks. The order of each quotient-space/factor-group is the dimension of the space:
The group SU(N) transforms N objects into each other. The operation is map composition.
Gravity as curvature is the topological nature and texture of spacetime. Einstein's theory of gravity is based on local symmetries, exact symmetries, and as such is geometric in nature. Electromgnetism, light, a 5-dimensional phenomenon, must have two generators to parallel the operational field of the quintic - the complex field. This is a reflection into a dimension we are not privy to, involving as it does the notion of 'imaginary time,' imaginary in the sense of the second component of a complex number, and in the sense of nonlinear time.
Electromagnetism is realized as the asymptotic action of a spherically represented gravitational generator -- the square of the quintic solution. Perhaps both lower-ordered generators -- nonlinearity and linearity -- are at each of the dual 'right' orientations, perpendicular to each other as vertical and surface slices, intersecting at the center, or the center line, of gravitational symmetry, forming the gravitational envelope.
The strong force and beta decay are 'below the surface' of 3-dimensional reality. Furthermore, beta decay is the asymptotic phenomenon of 2-D strong force. It is responsible not only for the transmutation of one particle into another, but also for the creation of stable intermediate elements to act as midwives for the fusion of heavy elements, elements like carbon and oxygen. Without beta decay, there would be no carbon-based life forms.
With the first one, gravity, each dimension of spacetime can be considered the identity element of the group in a state of superposition. Gravity is all-pervasive, varying locally with time. The graviton, the hypothesized quanta of gravitational force, acts on all particles and their associated fields, including itself.
With the second, electromagnetism, two dimensions adopt the role of identity element essentially collapsing the ten dimensions of spacetime to five, four that we know of, and a fifth in the imaginary time reflection. This fifth dimension does not just add to the known four, but rather increases the complex geometrically, giving each, considered as spatial only, an orientation that collectively can be represented as the envelope of a spherically apprehended universe operating outside the condition of linear time.
The third, the strong force, the force responsible for the interchange of identities between the neutron and proton by the emission or absorption of a quanta of energy -- the gluon -- perculating up, so to speak, from their constituents -- the quarks -- is two dimensional.
Five dimensions of spacetime, represented by the electromagnetic field, take on the role of the identity element. The ten dimensions of a string-theory universe collapse to two. The five dimensions (group elements) of the 'normal group' ['normal' here may refer to time reversibility, or, to the abelian nature of the transformations] collapse the ten dimensions of spacetime in such a way that the universe is seen as a union of pairs of dimensions -- surfaces -- each represented by its own coset. Beta decay -- the weak force -- is one-dimensional, having as identity all of spacetime. All ten dimensions map to the identity element of the group -- a unit of spacetime.
This is all pure speculation on my part, of course.
* "Abstract Geometries," Encyclopedia Britannica 15th ed., vol. 12, pg. 1124.
** ibid.