"Spontaneous Space"
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What art thou, Life? The shadow of a dream: The past and future dwell in thought alone; The present, ere we note its flight, is gone; And all ideal, vain, fantastic, seem. Whence is thy source? And whither dost thou tend? Death surely ends at once the dreamer and the dream. from: Sonnet On Life, by Sir Brooke Boothby |
Spontaneity, from Mister Webster: "Proceeding from natural feeling or tendency
without external constraint; spontaneity applies to acts that come
about so naturally, are so unselfconscious and unaffected or unprompted by ulterior motive or purpose that they seem totally unpremeditated." The manifestation or perception of the condition of 'spontaneity' is a qualitative measurement, the degree to which the mixture of free will and deterministic behavior blend themselves functionally to evoke meaning at the boundary of self and environment. Spontaneity, as Mr. Webster defines it, emphasizes the negation and transcendence of limits established by the conscious ego, allowing for the free form expression of the dance of self/environment. What is the range within which this proportion conspires to present itself as definitively 'spontaneous'; and by what means or mode is the associated phenomenon judged to be so?
It's as though, suddenly, and with no apparent warning, we slip
into a hyper-dimensional or parallel universe where action and reaction are taking place so instantaneously quickly that we don't
notice that familiar pause, accompanying all deliberate acts and
reacts, the earmark of 'civilized living.' We want control, we want
order, we want predictability; we do not want uncertainty and chaos.
What are the transpiring and transforming inner realities
taking place within an individual who finds himself in a completely
alien situation and surrounding, and must act without the benefit
of previously learned parameters or reference points?
When the conventional abstractions we believe are 'real'
and by which we navigate through life no longer suffice as either
values or ordering principles; or when the sense of restraint, conditioned and suffocating, communicates a survival message to our
inner selves, our outer incapacities, then the time has come, through
catharsis or some other inducement, to enter the realm of Chaos.
A peculiar and invigorating awareness grows in the mind
at this transitional border. For the first time, possibly, we are independent, attempting to make sense from what information we
perceive, and negotiating the moment to moment dove-tailing with
our immediate environment, on many scales and mediums simultaneously.
Unless we withdraw out of fear or a loss of confidence, our minds can insinuate their presence into and make contact with an inner sensibility of previously assumed disparate and external alien worlds.
We may discover ourselves buoyed and sustained by the pervasive and
liquid-textured emotional and physical consciousness-sensation of
being completely immersed in the pathways and resonances of all
about us deemed 'other' and 'disconnected' in some parallel dimension.
There is a definite feeling of having been let out of a very tight-fitting enclosure, of discorporating while paradoxically never having
known such solidity, of accepting one's Self unconditionally; and, of
the security that comes with relinquishing that insidious and peculiarly human sense of aloneness in an impersonal and uncaring universe.
The scientific, theological and philosphical effort to reduce qand analyse all phenomena to its radical identifiable segments (the smallest building block of this or that) has been superseded by a change in consiousness. The historical 'thing-in-itself' has become 'part-of-a-pattern,' a 'unity-in-variety.' As such any individual component's role takes on meaning and sense only in terms of 'relationship-to-the whole.'
Free Will can only be realized at the moment of departure
from recent history (the 'sum of our experiences'), a bifurcation. Determinism, on the other hand,
is a function of continuity and predicatability. The symbiotic relationship between "individual" and "earth" is modulated by the
interplay of these complements expressed as adaptability and stability,
respectively. When these are in balance, the mind can form questions
of meaning that go beyond what is answerable by the fact of existence.
Our sphere has two complementary aspects, both occupying
the same space-time ('linear' time for algebraic, 'nonlinear' for
geometric). These aspects are fundamentally inseparable; the schism
lies within. The algebraic as consciousness elucidating, articulating, ordering, reflecting, utilizing; the geometric as unconscious mind,
hereditary, instinctive, conservative, immutable, nature-oriented,
creative source of raw material, fuel for the soul.
'Structure' is a linear concept. It captures the set of
variable relations of a system or process in a typical or symmetric
state. A structure of linear transformations, however, interacting
nonlinearly in a dynamic, is a 'process' subject to occasional bursts of instability and symmetry breaking, the result of an accumulation of excess energy, out of phase with the identity of the whole, reaching some critical value. This positive amplification effect violates the integrity and closure principle of the group of transformations describing the algebraic sphere, shifting it into another phase space entirely.
Accordingly, Group Theory is an extremely helpful tool for any exploration involving ideas associated with symmetry, pattern, orientation, invariance, dimension, and the hierarchy of embeddded factors. a group, specifically, OT(3)/Z, in the context of a geometry, the geometry of the
sphere, is not considered an entity in itself, however. A geometry
or an algebraic structure (vector space of 3-D, in our case) is seen to reveal its peculiar symmetries and factors of symmetrical relations
when seen through the kaleidoscopic eyes of Group Theory. By applying group theory, in general, we bring into relief detail and render
information with respect to families of invariant properties, indicating essential features of a coordinated, or coordinatable, figure.
Geometry itself is not a 'description of space' as though
it were some empirically independent, graspable thing. Rather, it
is a way of 'reasoning about space.' It is a point of view by which we are able to organize our subjective experience of 'form' as an abstraction of and by itself, as a foreground against a suppressed background (Gestalt), or as a given state of a process or event in its relational capacity. The imposition of a group of transformations overlayed onto an existing structure (an undefined yet coordinated space) renders a geometry [Felix Klein]. The end result is an ordered orientation and interpretation of the relations of spatial forms and, the determination of a criteria delineating invariant properties.
For example: under the 'affinitive group', a circle and an ellipse are taken as equivalent since a circle can be transformed into an ellipse; under the 'projective group', there is but one conic section because any two are transformable into a circle and hence into each other; under the 'topological group', there is no distinction with regard to 'shape,' two given point-set figures are equivalent provided the mappings relating them are continuous.
Separate identities dissolve or transmute in this alchemical world, having little or no meaning as separate. Nonetheless, there is rhyme, reason and rules of behavior. An object's peculiarity, those characterisitics which give it an asymmetrical specificity, do not reveal its geometric consequence. Passing through the membranous transformation group, a figure must submit to bringing only those indelible identification marks which the group/geometry decrees as acceptable and necessary, and to leaving the rest behind as irrelevant and meaningless in this new realm. In other
words, we emphasize as 'geometric' only those properties that remain
unaffected by transformation, those which are indispensable for
definition.
Formally: a geometry is the study of those properties of
a space (point-set) which remain invariant under some fixed subgroup
of the full transformation group [definition by Felix Klein].
The least restrictive group of transformations is afforded by what was originally called 'rubber sheet' geometry. A simple example taht falls within the governance of this systme would be the mapping of a flat riangle onto a curved surface such as the bumper of a car. Its official name is Topology
Topology is the study of those properties of space which remain invariant under a group of all possible one-to-one continuous transformations. Example: sin (x) = sin(x + 2pi * n) ['n' stands for any integer] where sin(x) ['x' - a real variable] has the same value under a transformation by any multiple of the infinite cyclic subgroup (2pi) of the transformation group of all real numbers.
As continuity is the fundamental criterion for the group maps, the notion must be precisely defined. The basis for any topological space is represented in terms of sets of points (figures), specifically, open sets. In order to be able to define a continuous function we must devise some way of knowing when two points are sufficiently close, an idea of 'nearness.' For figures considered homeomorphic,
points which are 'near enough' must also be so after transformation; if not, the function is not continuous. The introduction of a hole or cavity in the 'receiving' figure would preclude continuity, certain near points in the 'sending' figure (domain) would be separated around the tear in the range.
A transformation commonly represented in matrix form, of a given set of coordinated points that is a one-to-one function or correspondence of this figure onto itself, is called an automorphism. Except for the identity transformation, all such mappings reveal a new orientation. Automorphisms are of special importance in Group Theory. A group of automorphisms of a given figure is a unique signature for that figure (configuration, pattern, arrangement, constellation), and collectively displays all possible symmetric centers, all possible phase states.
Having established the collection of all possible symmetric orientations of a given figure as a group, the next step is to determine when two such groups are considered to be the same.
Algebraically; A one-to-one mapping of all elements in a domain to those in the range such that the inverse mapping is also one-to-one, therefore bijective, is called an isomorphism.
Topologically: Two sets of points are equivalent representations of a figure if therre is a bijection that is both directions continuous. This is called a homeomorphism.
The combination of Algebra and Topology yields, curiously enough, Algebraic Topology. In this system the basic unit is called the simplex. These have corresponding dimensional criteria. Two points of a figure form a 0-simplex; a line is a 1-simplex, a triangle, a 2-simplex, and so forth. A figure is partitioned, or overlayed, by simplexes according to a certain set of rules.
A space divided into simplexes is a simplicial complex. Continuous maps from one space to another give rise to isomorphisms from the groups of one space to the other.
The fundamental idea, the identification and classification of patterns as coordinated sets of points in space, is important for argument development, as it gives us a mathematical and somewhat definite means to associate disparate pictures, and a means whereby we may correlate and equate aggregates of related features.
Horizontally (space), so to speak, these features are defined in terms of components operating on the same dimension, and therefore are understood in terms similar to that of the parent group.
From Euclidean Geometry; If A is a set on which an idea of distance is defined, a transformation, p, of A is an 'isometry' if d(x, y) = d(y, x) [distance between point x and point y], that is, if p preserves distance. This subset of the full transformation group consisting of all isometries of the given set is a subgroup.
A 'rotation,' y(p, #), is a function which rotates the plane about the point p counterclockwise through the angle #, [where 0<= # <=2pi]. In order to form a subgroup of the isometries, the rotation composition must be restricted in definition to: y(p, #1) * y(p, #2) = y(p, (#1 + #2 modulus 2pi) ['*' is a common symbol for map composition].
So here we have the Euclidean transformation group, a subgroup of it consisting of all isometries, and a subgroup of this consisting of counterclockwise rotations. Furthermore, this group of isometry rotations has cyclic subgroups of finite order [number of elements of the set] due to the modulus restriction.
Another Euclidean subgroup of the group of isometries is composed of 'translations.' A translation of the plane is a transformation [map, function, rule] which moves each point a fixed distance in a fixed direction: T(a, b) * T(c, d) = T(a+c, b+d) [a,b,c, and d are points in the plane].
Therefore, groups and their subgroups explicitly allow us to see into the inner workings of superstructures in terms of components having specific algebraic requirements for their existence. That is, a subgroup must be a group in its own right, and its order must divide that of the parent group without remainder.
Defining the first only, given a group, G, and a subgroup, N, we form a factor group, G/N, composed of what are called equivalence classes as elements, and such that N is the identity element (kernel) of the group. Only the identity equivalence class is independent. The other memberrs of the factor group, defined as right and left cosets, are derived from the identity class. Yet, there are no elements of the parent group which appear in more than one class (a coset is also a class); each class contains the same number of elements; the number of classes of the factor group (its order) is a divisor of the order of the parent group; and all elements are spoken for.
In a Normal group, left and right cosets are identical. That is, for any element 'g' of a group not in a subgroup N, if the relation gNg-1 = N. for all 'n' in N, i.e., gN = Ng holds, then N is defined Normal. This form of the axiom of commutativity is essential for purposes of forming factor groups, and for precisely deterrmining the focal point of actions within a system.,p.
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. 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, for example, then it maps back onto itself as the identity, G/G.
Moreover, for a normal subgroup defined to be Maximal, there is no other normal subgroup of the parent group having numerically greaterr order.
Theorem: M is a maximal normal subgroup of G if and only if (iff) G/M is Simple. To repeat the definition, something that perhaps can't be done too may times, a simple group is defined as one which is normal and, most importantly, has as order one which is relatively prime to any integer up to its order. It has no component capable of standing alone; that is, there are no sub-symmetries.
The fundamental theorem of Abstract Algebra combines the above notions under the heading of a particular kind of transformation, or map, referred to as a Homomorphism. This map is charcterized by its group operation or structure-preserving quality. The most significant fact concerning these maps is that is makes it unnecessary to go outside a given group in oreder 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, reducing ultimately to the smallest subset. The order of each subset divides the order of the parent set. It is important to recognize that even the idea of function or map partitions a set, thus establishing equivalence classes for each function working on the set.
The Kernel of a group is a normal subgroup containing that set of elements of the group which form the identity equivalence class under a specific homomorphism. We speak of the 'kernel of a homomorphism' as it does not arise, or come into existence, except as the result of this mapping.
Our factor group, OT(3)/Z, defines the geometry of the
sphere; and a sphere, in general, is a prominent archetypal symbol
of the individuation process, a mandala. Hologram-wise, the whole
is to be found in any part of a hologramic image (the totality
of parts makes up the whole). This wholeness in each separate part
accounts for the nonlinear chaoticness of a given system or process.
It shows itself in the form of noise or, more generally, as positive
amplification at some discontinuous threshold point.
Fractal self-similarity is a repeating of the shape of the
whole at different scales (a composition series arrangement insofar
as the configuration relationships across scales are preserved). Therefore, under a suitable group of transformations, the invariant features of a family of archetypal symbols can be 'sifted out' from the various cultural colorations. The subjective description of the geometry of the psyche, in terms of groups of transformations, is, of course, less than half the story explaining the dynamics of spontaneous space.
Each and every group of transformations yields its own special
set of invariant qualities, linked nonlinearly in as much as they
mutually support one another's credibility and, as an aggregate,
present information of a holistic nature.
The composition series of each of these groups represents
a unique set of wave-lengths (factors of common behavior), sequentially ordered across scale, nonlinearly interconnected within the
confines of each simple factor group, discontinuous at some critical
value, reforming as another phase/factor group, symbolized geometrically as a series of concentric spheres, one within the other, like
a nested hierarchy of chinese boxes.
1) A composition series is a hierarchy of embedded, simple factor groups arranged sequentially, by order. Each factor group is a homomorphic
image of the parent group; and, the identity of each is called the
kernel. The parent group modulo its maximal normal subgroup (kernel), this subgroup modulo its maximal normal subgroup, and so forth down to the smallest normal subgroup modulo the identity element of the group usually symbolized as 'e'. This series is also referred to as a Normal series. Of singular significance is the fact that all members of this series are simple.
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]), 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, or takes on the role of 'identity element,' a quantum shift.
It should be clear that at the orthogonal intersection of 'horizontal' and 'vertical' components, space and time, the group G is isomorphic to its factor group, G/[e].
As a model of a dynamic system, the elements of each factor group, being nonlinearly related (negative feedback loop), remain within the limits of the group, until at some critical value positive amplification pushes the system into an entirely different mode or phase-state. This condition is represented in our series by a
shift to another factor group, with kernel that is unique to its set of relationships yet maintaining the integrity of the whole.
2) A fractal is a sequence of dimensionally ordered, self-contained, self-similar shapes produced by the cumulative effect of repeated iterations of a set of rules. At some critical, threshold value in the iterative process, unpredictable for nonlinear reasons, we enter a transitional zone, phase shifting to another dimension or scale. The rules defining the eventual outcome, after so many combinations., are the invariants depicted by the resultant shape, complex as it may be.
Regardless of differing cultural labels, we can find isomorphisms existing among the underlying archetypal configurations of otherwise superficially diverse hero myths. That is to say, there is a family of geometric components attributable to each myth which is homeomorphically similar to other families, and which forms a self-contained, nonlinearly related composition series.
'Self-containment' is the essence of the closure axiom; and closure of the algebraic sphere accounts for its 'surface.' This surface restrains the overall system giving support to a negative feedback character shared by the elements of each simple group of the series.
Describing this process using group theory: a maximal, normal subgroup of a fractal is the next lowest scale's self-similar duplication of ordered chaos, sans boundary. Once this subgroup, which is the kernel of the homomorphism, merges with its parent to form a factor group, it loses its independence, except as identity element, because the factor group is simple therefore-- non-divisible. It, the kernel, forms a nonlinear relationship with the remaining elements of the larger group (now arranged in equivalence classes, each
equal in 'order' to that of the kernel), an interdependency.
The fractal basin is similar to the kernel insofar as the elements therein share the same pattern, defining and orienting the symmetry; and more than that, the basin is topologically inconsiderate of scale.
The property relations shared among the elements of the kernel are what hold each coset together.
3) A dynamic system which receives as well as dissipates energy has a long term behavior pattern describable by an attractor, the end result of some basin range. Most systems can have any number of attractors, each with its own basin. Given that each attractor is stable, it is in equilibrium in phase space. The boundary between
these phases is where chaos, the intermingling of order and disorder, lives and rules the global picture.
The initial conditions affect the eventual shape of an attractor and a fractal and also, for a given group of transformations taken as initial conditions (geometric environment), the identity classes of homeomorphically related shapes, their common defining features. 'Initial conditions' are not the same as 'rules of interactions' or the operation of a group. The I.C. allow us to 'pick out' certain characteristic information by creating the environment where those particular aspects stand out, as it were, in relief; rules of behavior are the pumps driving the iterative process to such an extant that continuity itself is no longer predictable.
The I.C's. get lost, that is, the 'beginning state' or environment is not determinable; we can no longer (if we ever could) determine what features we perceive are temporary and which, invariant.
"Sensitivity to initial conditions" applies only when they undergo change; motion on a stable limit cycle, or attractor, is independent of initial conditions. This is true because any point of phase space on the attractor can be thought of as certain "initial conditions."
4) Our specific group, OT(3)/Z, has as identity the interior of the algebraic sphere. The identity of a fractal, on any given dimension, and that of an attractor, its equilibrium state(s), are their interiors. The interior of our sphere, Z, is defined as a collective equilibrium and balance of opposites as it contains
all inverse-pair identity elements (transformations in matrix form).
The order/disorder confluence exists in a state of continuous becoming, ready to leap into discontinuous phase transition mode at the slightest, unpredictable provocation, to cross over, as it were, into another state of temporary equilibrium, taking with it the essential seed of its previous (time scale?) history in terms of invariant possibilities, to reform as a self-organized, self-similar shape or process.
This disorder/order chaos lives as and at the boundary, the border between two (or many, possibly infinite) realms, as the surface of our onion sphere, or as the interface between (one scale at a time) the interior (subjective identity) and the exterior (objective environment) [or interior environment and exterior identity].
This "boundary" symbolizes, paradoxically, both where Self, in its aspect as ego, conjoins, organizes and orders, through consciousness, external reality in all its myriad, dynamic interactions, moment to moment, in terms of patterns of various dimensions; and also, the Source, by its admixture of order and disorder, Intellect and Nature, positive anima and negative animus, creativity and the individuation process.
OT(3), the orthogonal group of transformations in R3 [equivalent to Euclidean 3-space], is our universal environment, ordering and defining space with its own peculiar invariant behavior patterns, above and beyond the axioms of the group itself. Its elements are continuous and map points to points spanning the space. It can, therefore, represent any group defining a particular geometry. Depending on which role it takes, certain qualities are highlighted while others are suppressed.
OT(3)/Z inherits this capability. Moreover, any transformation in this group which is not a multiple of the identity induces an orientation askew from that defined as 'symmetrical,' that is, as a synthesis of opposites and an equilibrium of forces. Any and all askew transformations instigate disorder and asymmetry against the
background of identity-multiple orientations of symmetry.
The nonlinear invariant property is the holistic property. Self modulo ego. Ego as antithesis issuing 'out of' Self as a substructure capable of apparent independent action. Ego-unawares is possessed by negative animus of the unconscious (or 'his' unconscious). To realize the holisitc property and thus break free from the Terrible Mother archetype, the anti-transforming unconscious, ego must form the factor-group-like structure/process, SELF/ego, and at the same time, shift conscoiusness orientation by forming a nonlinear relationship with Self.
This would be a mutual consciousness with 'center' being holisitc-ness itself. The anima/animus balance infuses ego with the synthesis of its duality similar to order/disorder, differentiation/undifferentiation, consciousness/unconscious. The tension between the elements of these and like complementary pairs is the
essence of nonlinearity and the creative potential inherent in spontaneous space.
Linearity 'comes out' of nonlinearity. It appears as independent, autonomous linearity but, it is only a function; it is only the orderly way chaos presents itself to consciousness. But it is not a delicate balancing of opposite forces; it is a dynamic interchange of energy, incoming and outgoing, maintaining an orchestrated arrangement of order and disorder, symmetry and asymmetry.
Linear time runs perpendicularly to nonlinear time. Orientation highlights the linear axis. Imagining this time axis going through the center of our composition series, we have a telescoping effect; we pass from factor group to factor group, adjusting magnification, resolving focus, redefining terms and properties as we go. Perpendicular nonlinearity defines the holistic quality at each dimension of simple factor-group series. What we have is a space-time continuum/discontinuum, (horizontal/vertical), with nonlinear time being more spatial than temporal.
The sphere of Self, concentric with ego sphere, is in the paradoxical situation of 'containing' the ego sphere yet identifying ego as the interfacing agent (surface) with Self symbolized as 'center-point' -- polar complementarities.
Topologically the question of 'size' is, of course, completely irrelevant; surface can be compacted to center point, the singularity 'where' both linear and nonlinear time live in intersecting harmony. The horizontal slice is nonlinear time; vertical time represents the 'depth' of psychic levels.
The role of generator must go to Self, and ultimately to the unconscious, when symmetrically oriented anima/animus-wise.
Simple groups are also basic building blocks. The cross products are capable of forming all groups (considered, of course, as permutation groups). This complex nature models the growth, evolution, and stable states of both organic and inorganic systems, from the least complex to the most sophisticated, depicting an eco-plastic sensibility, across varying ranges of time and space. For instance, a complex, nonlinear system of turbulent behavior may be considered to be composed of an uncountable (or countably infinite) number of separate vortices interrelating in a dynamic the effect of which, that is, its state at any given moment in time, is unpredictable. Each vortex can be likened to a simple factor group of a particular composition series. As these separate, yet interconnected, simple groups vary 'up and down,' so to speak, the hierarchy of their respective composition series, we can easily imagine, given an almost infinite number of such vortices, coming into being and extinguishing themselves over time, the enormous number of possible configurations and permutations produced by the interrelationships. The end result, and all we can apparently 'know,' is the resultant attractor, its form and total energy, its phase state as it undergoes dramatic and sudden transformations, phase shifting. Both the global and local identifications of these entropically controlled structures likewise vary, dependent as they are on orientation, fundamental assumptions, and degree and kind of desired information.
This 'process' of development, from the 'sum of parts' to the 'whole' can be characterized, for our purposes, in the following manner: In Analytic Geometry, the loci of the ellipse, circle, parabola, and hyperbola (the conics) are the end result of a co-evolutionary development of respective generators progressing through a continuum of discrete values dependent on intrinsic algebraic conditions and limits. These generators are variable, being satisfied by all points of any given locus, much like the phase-space points that conform an attractor; nonlinear, iterative chaos generating from all points simultaneously. In Analysis, the limits or boundary conditions give 'particular' or 'singular' solutions.
This metamorphosis from "infinite collection of" to "some whole composed of this infinite set" transpires from dimension to dimension as each incorporates within it the dual nature of at once appearing to be the generator, and then again as that which has been generated, the boundary, outline, interface. We're presented with a picture of continual, tug-of-war tension between becoming and being, a generator and that which has been generated, a gestalt.
For example, an infinite collection of Cartesian coordinates may generate the two-dimensional figure of a circle, obeying the rule: x2 + y2 = r (radius)2. Extended to three dimensions, a circle rotated about an axis, or its center, will generate a sphere. At each phase shift, the 'interior' of the N-Dimensional locus expands to the continuum 'surface' of the generated N+1-dimensional configuration, with the introduction of ideas supporting a subject/object-dichotomy dissolution at the interface. This is quite natural using the factor group concept of Algebraic Topology.
As an example, let's take the cube in Euclidean three-space. Its group is composed of all possible permutations, a subgroup of the general Euclidean group. The fundamental element here is the cube itself.
Now, consider the cube as an infinite aggregate of bounded planes the vertices and edges of which are defined by the variable generators of the cube itself, spanning the planar space which in turn spans the cubic space within a coordinate environment. The basic element here is the identity class of all planes superimposed so as to appear to be one plane, with centers coincident at the itersection of the mutually orthogonal axes. The cube is understood as a set of planes, cube modulo planes.
A cube is, again, an infinite collection of line segments; those that fall on the facial planes constitute its outline, or form, and as such offer one-D information about cubeness. This is the factor group of planes modulo the lines composing them, these in turn acting as identity equivalence class, the kernel. The fundamental element is a class of parallel, coplanar lines. The cube is now seen as a set of lines, .
And finally, a cube can be represented as an infinite assemblage of points, contained and limited by the generating rules and invariant Euclidean principles of cubeness. This is the factor group of lines modulo the center point. The identity element is this point, the center of symmetry. And the cube is seen as composed of an infinite collection of points, a continuum or manifold. These infinities are real variable infinities. The cube as one whole form can be perceived, in turn, through these nested lenses.
The cube as one whole form 'shrinks' to a set of planes, the planes to lines, and the lines to points, a series of dimensional shifts. We can choose which dimension to operate on depending upon what kind of information is required. That is, on the third , the identity is the whole cube; on the second, it's the six planes or faces; on the first, it's the six axes lines (the axes generate the twelve lines that form the boundary of the cube); and on the zero dimension, the identity is the center point or origin (this point generates the eight vertices on the surface). Each consecutive factoring reveals the opposites from the dimension above, i. e., the terms with which the cube is understood.
Topology can be redefined as the study of those properties of space which remain invariant under homeomorphisms. The general problems in this system are of global nature. Essentially they involve determining the homeomorphic type of an arbitrary space. Considering only invariants of a space reduces problems in Topology and Geometry to those of Algebra, hence, Algebraic Topology. This system associates two given spaces as homeomorphic, classifying them as equivalent, if their repective groups of simplicial complexes are isomorphic. It is not true, however, that a homeomorphism between two particular spaces, or kinds of spaces, automatically gives rise to an isomorphism between their congruent simplicial complex groups.
There are two important types of spaces in Topology: Sphere and Cell. The n-sphere [hypersphere] is the set of all points a distance of one unit from the origin in (n + 1) - dimensional Euclidean space [a Euclidean space is a special case of a Topological space when it is placed under what is called a 'metric,' that is, a definition of distance]. The n-cell or n-ball, E to the nth, is the coordinated set of all points embedded isomorphically in the space of Real numbers, R to the nth power, a distance less than 1-unit from the origin.
The cube is homeomorphic to the sphere. S2 is what is called the surface of the sphere in R1 [equivalent to the Euclidean plane]. If we consider a solid sphere with S2 as boundary and interface, and E3 as the interior ball of this
sphere compactable to its center point, the boundary points collectively
forming the surface of the sphere can be said to generate the three-D ball (Homotopy). And it is also true that, given a metrically defined radius, the ball, or interior, generates the surface in a reverse process.
Each point on the surface mapped to its center, and all the interior points 'in between' any such mapping, form what is called a Slice. A Slice is a kind of equivalence class which grants to its members the extra dimension of being a part of a whole, embodying both aspects simultaneously, a single entity. An oreintation of a complex figure is an equivalnce class of orderings of its vertices. This idea of Slice comes from Algebraic Topology.
Intuitively one can imagine a pie slice, cut at any angle, emanating from the center of a flat disc, or, a hyperslice with the same general identity characteristic, linearly defined algebraically as a Hypertrace of an appropriate set [could be only one element] of equivalent orthogonal matrices of coordinated vector space.
A Trace is the sum of the diagonal members of a matrix. The Canonical, or symmetrically oriented, form of a matrix resolves all rows and columns to the diagonal. If A and B are two matrices related by a similarity transformation, that is to say - are equivalent, so that: for a third matrix, C, if B = C-1 * A * C, then Trace A
= Trace B.
This is an important idea for equivalence of orientations, from one set of bases to another, of our coordinated system.
The Slice defines the common features and dualitites of center point to all surface points 'when' the interior, as pairs of opposites fuse, connects the onioned layers of surfaces through the antithetical linearization of the moment. This moment is at once a spontaneous symmetry breaking and symmetry forming. Orientation here is an equivalence class of orderings of the 'surfaces' of the sphere representing our horizontal union and subsequent integration of consciousness and the unconscious.
Linear time, linear existence, is built-up on a continuum of discontinuous, nonlinear existences. Anima must be won moment to moment; it is not a one-time deal. The attempt to surgically abstract linear time, and by so doing distill determinism from its fusion with unpredictability, from its perpendicular source and timeless identity, is like trying to abstract wetness from liquid water.
Orientation in linear time: In the act of perceiving order, in/of something which is intrinsically nonlinear, we, the perceivers, are part and parcel of the fabric of this nonlinearity. Each and every 'something' is both an input and an output for each and every other 'something.' We perceive order thereby we create linearity
by defining, in the act of instantaneous perception, the manner in which we understand cause and effect. It is a wholly syntropic experience. The difficulty arises, more often than not, when we try to impose order from without according to a map or paradigm we carry in our heads. In the process of overlaying such static,
nonlinearly-empty projections onto a dynamic, interrelating 'environment,' from which we abstract our Selves, we shove the nonlinear wildness, and simultaneously our contact with Self, under the carpet or cloak of a deterministic universe. We construct a prison for the Self; and try to consign Mother Nature (Great Mother archetype) to the role of "Thing."
A 'Path' is a continuous function operating on a topological space; its image is a curve. The relation between a curve and its path is analogous to that between a simplicial complex and its transformation group, oriented n-chains generated by the oriented n-simplexes of this n-space. In 3-D, a simplex is an ordered sequence of four vertices of a solid tetrahedron (homeomorphically spherical). A 2-simplex is a triangle; and an n-chain is a finite sum of n-simplexes with integer coefficients. The binary operation on the group is addition.
In order to determine when two given spaces are equivalent, that is, homeomorphic, what we want to do is compare, in the static, stable case of ordered geometry, lists of equivalence classes of closed paths (circles, triangles, etc.) under some appropriate relation. These are our algebraic invariants. For example, an even permutation means an even number of transpositions; this is called a 'two-cycle.' All two-cycle closed paths can be considered as members of the same class. But in a dynamic complex system, the temporarily stable behavior mirrored in its shape, as an end result of infinite, random combinations of simplexes, may be all that is ascertainable.
The composition series of this group of n-chains looks like a fractal of nested cyclic subgroups, factored into a scaled array, expressing discrete, discontinuous, integral jumps from one energy level to another. This series is unique up to isomorphism between respective factor groups of any other series of n-chains; that is,
all homomorphic images are spoken for by any single group of n-chains; there is only one family per dimension. Vertically, this series describes linear growth; horizontally, expansion to nonlinear gestalt when taken as a whole.
The understanding of this relationship between a myriad of possible patterns, transformed through the medium of a particular group and its accompanying factor group, and the reassigning of 'spatial' points to their respective roles in the interconnected picture, with appropriate label-change properties, is fundamentally important as it infers the existence of an ordering principle at once capable of generating and appreciating invariant forms, and of identifying these forms with conceptual layers. The Functional Instrument is the 'self,' and the Function happens in the 'act' of perception and in the 'act' of experiencing.
The idea of identifying center point of a sphere with its surface point(s) through the mechanism of the Slice has as formalistic determination the definition of Homotopy Type.
Two topological spaces, X and Y, are said to have the same homotopy type if there exists continuous maps, f: X -> Y and g: Y -> X such that g * f = identity of X, and f * g = identity of Y. In this case, 'f' is called a 'homotopy equivalence.' This definition supports the following Lemma: R [real numbers] to the nth power is equivalent to D [a 'solid disc,' a circle plus interior, of 'zero' width] to the nth power is equivalent to center point. Informally, two spaces will have the same homotopy type if one can be shaped into the other without introducing any 'holes.'
A map from a simplicial complex (conceptual overlay, symbol systems, schemes of interpretation) to an underlying topological space (environment, both internal and external physical and psychic noumena, the "raw material") orders that space, and integrates sets of interconnected simplexes of one dimension into forms of attractor-like complexes of the next dimension. And, dynamically, this mapping is a rational arrangement and coordination of phase-space points by a factor of observation, rather than an imposition of an image, or set of images, onto a plastic point-set. What effect does this overlay have on the coordinated space?
If we were to plot two distinct points on a Cartesain coordinate system of Real axes, and were to define a metric (a precise definition of 'distance') between these two points, an orientation of the Real hyperplane would therefore be defined as 'an equivalence class of mutually orthogonal, ordered bases with respect to the prevailing metric.' In general, we could define an orientation of an N-dimensional vector space as an equivalence class of ordered basis vectors. A given orientation at the origin induces a likewise orientation on the whole of the N-dimensional hyperplane.
If we have an N-dimensional field, we need have only one dimension in the opposite direction to have opposite orientation. Considering the field of Complex Numbers as an extension field of the Reals, therefore forming a vector space, the characterization of complex number pairs (real and imaginary parts) as ordered Cartesain pairs is a most natural framework for any discussion on the graphability of any physical system represented by sets of partial differential equations, thus focusing on orientation in terms of factor group alignment [the identity equivalence class of each factor group nested in a series, and taken together as an 'equivalence class of orderings'].
An ordinary differential equation expresses an 'instantaneous equilibrium' of some physical quantities (forces, moments, etc.) considered at the same instant. One can say that a differential equation always deals with the present. Partial differential equations retain the history of past environment, readily seen in a vector representation. The difference equation belongs to another class of functional equations in which the past exerts its influence on the present.
For simplification, a polynomial can be written as a sequence of its coefficients. The ordering is what we are interested in for purposes of coordination. For example, 3x5 + 5x3 + 9x2 + 8 = 0 can be written as (3, 0, 5 ,9, 0, 8). This is also true for linearly independent sets of first or second order, partial differential equations describing physical systems. These ordered coefficients as basis vectors span a vector space, each dimension corresponding to a degree of freedom of the system.
From Galois Theory we learn of the fifth degree polynomial, 'the quintic,' and its unsolvability under the field of Real numbers. The brilliant, nineteenth century mathematician, Evariste Galois, who died in a duel at the age of 20, uncovered a realtion between a solvable polynomial, under a given field, and the group of permutations of its coefficients. It was known at his time that, beginning with equations of of the fifth degree, there was no method of reducing them to their linear factors by the usual means of root extraction.
Besides his highly imaginative perception of the precise connection between a group of automorphisms of the coefficients of a given polynomial to specific coefficient fields adjoined by roots of the equation, arranged diagramatically and in inverse relation, he proved it necessary that the solution set of an irreducible polynomial can be obtained directly only by use of the field of complex numbers, that is, a two-generator field.
A group is solvable if it can be broken down into its constituent, prime-order factors, a Composition Series; that is, each factor group must be a simple component. The group of permutations of the roots of the quintic contains 120 elements. Its maximal normal subgroup of order 60 is also the smallest simple group contained therein. Therefore, it cannot be factored further. So, on the fourth dimension we have an autonomous group, one that is self-generating; thus, there is no single-generator field capable of solving the quintic polynomial. Or, another way of looking at it: there is no fifth degree symmetry of the type with only one central mecahnism, one genrator.
We have an asymmetric configuration for which we need an imaginary reflection in order to uncover the Kernel (solution set), to align the series of embedded shells so they conform, are congruent, and vertically superimpose axes oreintation. In the case of the quintic, the fourth shell or screen will not 'line up' root-wise between the first three and the fifth without the introduction of a two-generator environment.
Using slightly different phraseology, the structure of Field as compared to Vector Space, it can be said that the Complex field is a two-generator symmetry for each real root (eigenvalue), producing as it does the conjugate opposites for all numbers of a field having only one generator. While the real part is responsible for the asymmetry, the imaginary part contributes the reflective orientation isoperimetrically, a tandem companionship similar to the collective natures of the Many, and that of the One that infuses and interpenetrates the Many on a higher plane of being. Herein lies the Kernel of the quintic, and higher degrees.
Not only is factor group alignment mandatory for precipitating the identity class of any given polynomial, but, it underlines directly, through its ability to represent geometry and physical systems, the ideational role of orientation. For a phsyical system not to be graphable means that it represnts a quantum mechanical, unpredictable, probabilistic phenomenon. Its position and velocity, or motion, cannot be determined except within certain bounds of possibles, a group.
As we have seen, the complex plane can be represented by a two-dimensional vector space quite naturally. Technically, a two-D Euclidean space, R2, is isomorphic to a two-D vector space of ordered pairs, which in turn is a characterization of the complex plane. There is nothing to stop us from extending this vector space to four dimensions by assigning to each real number along both its real and imaginary axes a complex plane of its own. Each point would thus act as an origin for a perpendicular vector spanning, by rotation or by cross producting with its respective real number, the two-D space of its plane. The vectors generating these planes are mutually linearly dependent along the continuum of each linearly independent axis.
Each axis now reveals itself as a fractal-like, multi-tiered construction of indefinite listings of complex conjugates, emerging in finer and finer detail as each axis of each plane transforms into an infinite series of complex planes. The dimension of the vector space expands by cross product operation on the previous
dimension.
Each real axis stands for linear space; each imaginary, for linear time.
The complex planes emanating from each point on each axis stand for nonlinear complementarity, impinging on and communicating with the linear continuum. Each intersecting point (union of linear and nonlinear) symbolizes the negative feedback, interdependent relationships ongoing within the holistic nonlinear aspect; the
unpredictable amplification and subsequent discontinuity signifying the breaking of symmetry (bifurcation); and the engendering of creativity, insight and the reformalization of chaotic patterns.
Theoretically, at linear space-time of zero orientation (the origin of our linearly independent axes), the state (or, 'nonstate') at the well-spring of these two axes is stripped of cause/effect relations leaving the union of the nonlinearities-- nonlinear space-time. The simultaneity and chaotic intermingling of self-reciprocating, infinitely detailed, 4-D Universe dissolves into a sea of holistic, aecausal, horizontal slices of Eternal Now-a singularity.
There is a natural uniqueness to a given orientation as the effect of choice of basis vectors onto a coordinate system. Furthermore, it is this induced 'collection of orientations' of points of the underlying coordinated space which ought to be called an 'oreintation' of this space; and which gives 'orientation' an equivalence class characteristic. A set of different questions rendering the same answer, couched in different terms for each, must be a system of questions referencing a common point of view.
The philosopher, writer and mathematician, Max Ernst, was able to picture immediate, objective reality in terms of controlling mechanisms or observables, or, what I consider, invariant features. These he identified with horizontal component members (subgroups) of a group considered as a set of "all possible ordering orientations" (within a given framework]. Regardless of the choice of groups we impose on our surroundings, there is only one central organizing and controlling center of symmetry for each system which arrests our point of view. It is the independent part, the generator or initiator of inframotion, as well as the receiver of information and communications from the remaining conglomeration. Our perception changes as we
change the overlaying group and/or key on a different identity. We are forming recognizable sets of relationships, or intelligible patterns, out of previously randomized aspects with each new look.
Einstein's special theory of relativity is the modern, all encompasing version of the principle of invariance. Stable states in physical systems are the horizontal slices depicting conditions of a dynamic process at time-change zero (a fractal snapshot of an attractor). The physical invariants of two 'identical' events
happening in distinct yet relativistically connected frameworks are independent of measuring units. The dimensions of time and space are orchestrated for simultaneity by Lorentz's equations, a factoring
or linearization (graphability), facilitating the passage from one Max Ernst group to another.
All laws of physics are left invariant under the Lorentz Transformation. Two frames of reference (coordinate systems with different bases) of two observers in relative motion to one another reveal that time change and space measurement from these different orientations are symmetric; and that therefore, simultaneity is
relative and cause-effect is in the eye of the beholder.
The principle of symmetry with respect to physical laws may have been the most crucial idea of his theories as it served to connect and unite space and time into one continuum.
For Newton's law of gravitation to remain invariant under the Lorentz transform, Euclidean space had to be viewed as a special case of Riemannian curved space-time. This forces a four dimensional metric 'surface' on the events within an inertial frame of reference, introducing the geodesic (a geodesic is a Path) law of motion in contradistinction to the Newtonian view of straight line propagation (until some force redirects).
The Einsteinian Universe 'includes' the Newtonian, as the nonlinear Universe includes the linear, as a special case, a local phenomenon.
Sets of nonhomogeneous, partial differential equations can describe nonlinear systems over time. What I am interested in for present purposes, however, are answers to the question: what does an event look like, what is happening or apears to happen at instantaneous time - when time stands still?
Briefly, the solution to a nonhomogensous differential equation is equal to the general solution of its homogeneous counterpart, plus a particular, or singular, solution, arrived at through the associated boundary values. There is a geometric interpretation for this latter idea, the Envelope. An envelope is a family of curves tangent to a curve defined algebraically by parametric equations, a polynomial or a homogeneous partial differential. For each variable, a partial differential equation can be imagined as a family of tangent lines, the solution being the curve they trace. The envelope satisfies the differential equation because the slope of the envelope is the same as that of the general solution curve at each of its points. This system of tangent lines of the general solution is equivalent to the tangent lines of the envelope. It is important to note that the particular solution of a nonhomogeneous differential equation accounts for a certain displacement or translation from its homogeneous counterpart.
Informally, an Asymptote can be a straight line or a curve (the latter includes the former, actually). It is described as a 'line' bordering, from a 'distance,' a curve in such a way that as a point on that curve approaches infinity, the tangent lines, or slopes, of both the asymptote and the curve 'at that point' approach identity. Not all figures have real asymptotes.
Topologically, the sphere is the best representative of all homeomorphic images of itself. It is a globally, uniformly-connected, and minimally-surfaced, geometric model for that necessary and sufficient collection of invariant properties held in balance and symmetry. It also is an agreeable backdrop, as a hypersphere of four dimensions, for statements concerning relativity of coordinate frames, physical systems and events. It does not have a real asymptote; but beginning with the fourth dimension, the hypersphere has an imaginary asymptote. That is, 'an orientation in the opposite direction.'
The solution orientation of the quintic forced the entrance of a two-generator system - the Complex Numbers. The imaginary part allowed the fabrication of a 'Vertical Slice' symmetry and thus the annihilation of fifth, and higher degree, equations. The identity equivalence class is undefinable without relabeling roots against the background of the two-generator symmetry system.
The meta-Kernel, the Vertical Slice, can be conceived geometrically as the hyper-equivalence class of spherically layered, colinearly-oriented [through the superimposition of sequentially arranged identity classes] Horizontal Slices. Now, each of these slices can either stand for: the possible arrangement of the unique, invariant, defining features of a 'thing' or 'self' or physical system held in stasis (an impossibility in a relaitivistic world); or, as levels of identity of these events with concomitant quantum transformations in understanding, from the extremes of locally conditioned specificity, to those with only global considerations (here, global has no limit), and form the transitory to the universal.
The predictability of any given horizontal slice, considered the first way, at any given specific time is always unknowable, being nonlinear, and so, quantum mechanical. Considered in the second manner, the continuous shifting in view is a purely 'subjective' capability and experience limited, theoretically, only by the degree to which the 'self' has simultaneously dynamically identified with an external pattern, regardless of mode, and given up a more restricted panorama. Growth first expands vertically to a point, then, spreads horizontally until all is absorbed and known. This is its nature; for it to be denied is to handicap fulfillment and 'self' realization.
A Max Ernst-styled composition series can be imagined as containing an unlimited or unbounded array of interwoven factor groups, oriented geometrically, multilayered symmetrically in the environment of a two-generator field, in such a way as to have one end-point at the center of the hypersphere, the other on the boundary, interface, 'surface.' This slice concatenates all pssible phase-state-permutations of both simple group constellations and products thereof, through centers of symmetry, the point-of-view-identity of the relativistic 'self.'
It should be remembered that this Vertical Slice runs along the linear time axis; and because of the commutativity of the full linear group, if for no other reason, is reversible. Along this dimension, independence of choice of frame of reference, or any expression of 'self' within the boundary values of capacity, has contained within it the seeds of instability. Self here is what is realized at the moment when identity is inverted from skin-encapsulated, separate ego to what is being presently experienced; when the images and maps roughly corresponding to the outside world are suspended or transcended allowing for direct contact (isomorphism) with reality at the interface. Were the instability factor not an integral necessity of the fabric of momentary existence, the creative impulse, and hence Life Itself, would lose its essential asymmetric character.
How is this Slice seen from the center of symmetry of our hypersphere? Looking out from the center of sphere towards the interface, the Slice, taken as a whole, is apprehended as a point, one-dimension higher than the members composing it. The time-line perceived 'end-wise,' or three-dimensional time (linear) oriented at right angles to the center axis of each and every Horizontal-Slice event, symbolizes the generation of the frequency of both Intuition and the instantaneous Now of every act and perception. The shifting of view is a subjective capability and experience, limited and conditioned only by the degree to which self has dynamically identified and interpenetrated external patterns, regardless of mode.
The inherent instability and non-linearity pervading all dynamic systems, and hallmarking not only one's ability to shift identity-view but to act creatively, is manifest in the idea of the Asymptote. The imaginary asymptote, rendering us an orientation at a right angle to the real dimensions (symmetrically balanced and static), or, providing a higher-ordered symmetry for each dimension, however one wishes to understand it, has slope equal to that of any of the elements of the envelope at any given point on the surface, but at a discrete 'distance' from the surface.
But, in a General Linear Topological Space, displacement is not recognized. That is to say, limit is not recognized; limit at the fractal boundary becomes unstable chaos, expanding to reorder in a higher magnitude. The boundary is what is fractal. Each point of an attractor signfies an equivalence class, and so represents another dimension. Fractal boundaries are composition series phenomena, as far as type. A boundary represents the interrconnectedness of all points [phase space] contained therein; the surface is simple, a higher ordered information set.
Nonhomogeneous equations describing physical systems, be they organic or inorganic, do not differ from their homogeneous counterparts either in solution (kernel) or in geometric appearance. All phases of a process/event/system/view-point are herein considered to be one and the same; the envelope as particular solution no longer exists. The asymptote now acts as the surface tangent; induced to be so, in this topological space, by instantaneous dissolution between a static subject and a dynamic object, a factor.
In Nonlinear Oscillation Theory, a singular or particular solution of a differential system represents equilibrium, or stable state, of that system. The motion on a stable limit cycle is independent of the initial conditions, that which accounts for the particlar solution. Unstable limit cycles have no physical meaning; they appear as watersheds separating the zone of attractors of stable limit cycles. Limit cycles are always due to the presence of nonlinear terms in the differential equation(s) that define the system as a whole.
The delta-time convergence of a probability wave describing a dynamic system's behavior, or locus, at the interface between subjective and objective realities commutes, or translates, all 'possible' phases into a single, multi-leveled, ordered family of invariant properties - a higher ordered pattern. This comes about by the act of observation. The integrated dynamic, of which we are an integral part, is transformed, revealing a dual nature. Deterministically, we have a bounded, finite array of possible configurations any one of which can, unpredictably and suddenly, appear. And, within the context of random nonlinearity, the act of recognizing objective patterns circumscribing and impinging on the 'surface,' there is the experience of self-generation and self-arrangement. Asymmetry and symmetry are interwoven into the material of both the sensate-feeling world, and the domain of cognition-intuition. Naively, therefore, these two complementary opposites must have some underlying, connecting, mutually influencing Slice.
Summarily, without the envelope, the asymptote takes on a special significance as it now, in this topological space, 'touches' the interface. In a dynamic Universe, the event at the surface changes instantaneously. Whenever a specific pattern is formed (symmetry), it is immediately broken (spontaneous symmetry breaking), to reform within the confnes of all plastic possibles. The 'point' (Ernst) on the surface of the sphere at the end of the Slice can now be thought of as the 'point' at which the asymptote touches the surface.
The identity-self existant at vertical center-line, topologically identified with this asymptote point, assumes the dual nature of simultaneously accelerating into the Now (asymmetry), and intersecting at this point with the Horizontal Slice. Besides presenting the opportunity to generate the intuitive mode of perception, the opposite orientation afforded by the imaginary nature of the asymptote allows the perceiving individual to hold and comprehend all points (possible states) of the slice, while at the same time, appreciate the instability inherent in the creative nature of patttern formation from apparently, or previously, random, uncorrelated elements in the immediate surroundings.
Intuition and Rationality, two modes of perceiving, are not, fundamentally, of the same dimension. In the present context, Intuition is a qualitative apprehension of a group of transformations representing, symbolically, one's spherically formulated environment. And Rationality is a quantifying appraisal of patterns as collections of parts or sections, framing one's immdeiate surrroundings in separate terms, transitory properties. And, what is transitory is not geometric.
The dimensional shift is that of time. Position is linearly related to velocity. Acceleration is the realtion of position to the second order of time and is thus nonlinear. Continuous acceleration is reflected in the Intuitive mode. The act of factoring a group induces perception of form. This process requires a corresponding phase shift in mental frequencies, from the center of symmetry of the 'externalized' Ernst group, to the center of symmetry of the individual, self-identity, the kernel.
Similarly, the shape of the boundary: fluid, layered, indefinite, isolating subjective experience from objective empirical environment, bends, from a concave sense of being encompassed or surrounded spherically, or of setting limits on the spatial expanse of consciousness, creating an 'out there/in here' dichotomy, to an active, convex, permeation of the texture of mind as it asserts itself into the fray, bringing to consciousness that which appeared to be previously distant, distinct and 'other.' That is, identification with the invariant aspects of our world, while simultaneously grasping those pictures, images and ideas we use to define, describe and make assumptions about it, requires the manifestation of a higher dimensional mode of consciousness than is needed to merely react to 'outwardly' imposed patterns of behavior.
Mind conscious-izes this gestaltic, writhing confluence of nonlinear chaos and linear-ordered definiteness. Linear time, as a function of subjective, psychosomatic apprehension, gauges unit segments to conform to the curvature of local mind, sometimes accelerating to the second order, thereby appearing to be nothing more than a function of present, unself-conscious intensity.
************* FILTER SYSTEMS *************
G/G or G/[e] are the two possible forms of a Simple group. The composition series has as members simple groups. So each of the factor groups can be placed into one or the other of these two forms, that is: If 'N' is a normal subgroup of the parent group 'G,' then the two forms G/N can take are: G/N/G/N or G/N/G/[e]. This is clearly the case with all groups if we consider the isomorphic form of G, that is: G/[e]. The 'denominator' stands for the 'unlimited' (ONE), and the 'numerator,' for the 'limit' or scope of possibilities (MANY).
The two-generator field finds its analogue in the two possible forms of a simple group. These forms are contained within the group itself, and their union composes the set of invariant features each of which is itself a combination of 'limit' and 'unlimit.' The 'unlimit,' seen as factor of 'limit,' gives the stable state, and the analogue of Simple, an inherent instability or asymmetric proclivity.
(A) represents the unlimited or outgoing factor (asymptote); (B) stands for the limitation of self considered as separate from the surrounding 'other,' itself composed of separate, interacting forms. G/G delineates the occurence when the entire spherical surface [interface] is identified as those invariant properties common to the complete aggregate of the MANY. This common denominator is that which binds the Universe of multiplicity and gives it impetus from a higher dimension (an infinity of 'points' shifting to 'line'). G/G ---> union at interface of Mind, or individual self, with its ultimate principle. The extinguishing of the Envelope is likewise a dissolution of the filter/interface so as to allow complete sublect-object harmonization, integration, fusion, and permeation.
'Horizontally,' a group capable of generating a Series may symbolize, through reference to its non-trivial subgroups, if any, differences in vantage points, or depth of penetration. Each individual can be cast as a unique collection of multiple identifications, ranging from the most general to the most restrictive or separated. The individual brings into consciousness (or it is prompted by impinging circumstances) some level of factor group arrangement that is the most appropriate to cover a specific set of related externals, be they physical or psychic. The resulting understanding not only hinges on what terms, and to what degree, are being incorporated within the frame, but how the individual orients and participates in the event at hand: through the third dimension, that which intersects vertically with a Horizontal Slice on the same plane of being (Analytic); or, through that of the fourth, the perspective at right angles to the third, whereof one sees or understands one's self as part of a general pattern (Intuitive).
At each level of identification, idependent of time due to the reversible nature of the Vertical Slice, we have a simple group arrangement, a specific identity point-of-view based on the invariance of essential features, composed of a mixture of two inseparable constituents, the One and the Many. Any two observers may, potentially, perceive similar sets of invariants corresponding to an event, be it physical or psychic, relativistically. At each of these horizontal slice levels there is the inherent mixture of asymmetry [represented geometrically by the dynamic nature of the Asymptote], and symmetry (balance, completion). These are represented respectively by: G/G and G/[e], where G is simple. Crumbling and restructuring: each contains the seeds of the other in our dynamic, living Univerrse.
The surface of our sphere model [actually what we have is an indeterminable number of sphere surfaces, the union of which is a 'solid' sphere] is of the same homotopy type as the center point, through compactification. These and all points 'in between' belong to an equivalence class of orientations.
The empathetic melting of separation barriers accompanies the act of self definition, as well as the leaving behind of those particular and distinctive features of self not allowed to pass through the geometric screen of the Topological Group.
The products of simple groups form all possible groups of permutations. As components, these simple groups symbolize members of an aggregate, acccounting for all horizontal point-of-view identities (worlds) of a given individual. And, in order for all groups to have a composition series, it is necessary to introduce a two-generator linear space in order to embed the transformation group of permutations in an hospitable environment.
Recognizing the relativity of an arbitrary assignment of a template ordering of orderings, identification with the general property relations of our world requires utilization of that mode of consciousness bringing together the complements of Intuition and Rationality. The cross product of phases, a Moirre effect, creates nonlinear linkages shaping and vitalizing the present.
Self-actualization dissolves the illusion of the interface, in its negative connotation of 'barrier,' forestalling regression into the past (or, 'a' past), that is, ego-infusion by unconscious influences; there is no middle ground, only illusions.
The ingrained quality of tension modulating this vast complex interplay of complementary opposites, oscillating mind, enfolding with and within itself through time, fuels the system giving it life, shape, and an asymmetric proclivity.
In a participatory, relativistic universe, the objectively understood Ernst slice not only affords one an identity, but also determines which faculty of perception, or combinations thereof, is brought to bear on it.
Detail of information intensifies at surface/interface until point-density maximization (continuum of awareness) effects a critical state, a threshold. At this gateway, one enters the new, the Now, symmetry is broken, expansion and growth ensue, and the creative experience of the spontaneous moment redefines the parameters of "separate selfhood." Universe modulo the human species, species being fundamental, individual uniqueness notwithstanding, right down to the very local phenomenon, is the very essence of the human experience.
Spontaneous Space is the natural environment of Universe.
The dynamic interface between subjective and objective realities, being a union of order and disorder, represents that critical zone wherein a Material Universe of Creativity and Self-Identity bubble forth simultaneously. Furthermore, at the interface between the three dimensional world of things and the metaworld of its quintessential Cause, there is a mode of perception that precipitates the realization of a dual nature, which in turn affords the opportunity to suspend the identity constraints of the third and lesser dimensions.
B) G/[e]/[e] is isomorphic to G/{e} <--> MANY