Informational Fields, Structural Fractals

INFORMATIONAL FIELDS, STRUCTURAL FRACTALS

By Florian Colceag

The recent theories and many modeling attempts in the field of sciences which originated in the XXth century put forward the idea of informational fields as an objective necessity when it comes to explaining numberless phenomena having non- standard behaviour. Although the idea of informational fields and of the grammars and vocabularies associated to these fields, which enable information quantification was often made conspicuous by experimental or theoretical research, up until the present, a correct, coherent and adequate modeling of the phenomenon was not reached. The present article aims at displaying an approach able to cover this theoretical necessity. The existence of informational fields is conditioned by information transmission and by the informational feedback. For this reason the basic instrument that we are going to use within this modeling is informational feedback defined by accurate means. Even if its name is the one given by electronics, structural feedback has specific characteristics that differentiate it from the feedback defined by Norbert Viener.

Definition

Algebraic feedback is an assembly of six elements A1, A2, A3, B1, B2, B3 having the property of forming a self –generating model.(Figure 1)

Figure 1 Informational feedback cycle Self-generating signifies that any two elements in a row generate a single element from the second row. Thus 4 and 2 generate 3, 1 and 3 generate 2, etc. From a functional point of view such a field has six nodes noted by: Data entrance, entered data processing, using entered data, exit data, processing data, using exit data, which are in fixed positions numbered from 1 to 6.

The behaviour of such a field is dependent both on the nodes as well as on the vectors, the feedback circulating according to the following algorithm: In data entrance there is a data-selection filter. The latter selects part of the data, the rest being sent in an additional memory. The selected data are sent to the second level of entered-data processing. Here, part of it is passed through the algorithmic filter specific to the node, and the rest, which cannot be passed through this filter is sent in an additional memory. The algorithm-developed data reach using “entered data”. Here the data is compared to the data existent in the system and positioned according to the already existent categories. The data unable to select categories compatible with their own structure are sent in an additional memory. The categorized data are subsequently passed in a fourth node, of data exit. This exit is done according to a specific language that can generate a specific meta-language. The data which cannot find an expression in the language structure pass in an additional memory. The data linguistically expressed are subsequently passed to the fifth node where they are assessed according to the already existent value or referential structures. The data that cannot be assessed pass in an additional memory. Finally the data already assessed pass to the 6th node where they enable the generating of metastructures, metaconcepts, etc. The data which cannot be used in this process pass in an additional memory. From the 6th node the data pass again to 1, but they are accompanied by an improved entrance filter thanks to already developed metaconcepts. Thus, part of the data entered in the additional memory from 1 will be filtered and sent to 2, where processing algorithms will be themselves improved, etc. The cycle is capable of constant self-improving. From a mathematical point of view such a cycle can be completed under the following conditions: The existence of a structure on the nodes of six elements preserving the feedback form with generating structures, and the existence of a system with informational vectors also preserving the six-element structure with generating structures. These two types of structures will be treated separately in this material. An example of six-element structure with self-generating model on nodes is the triangle-like structure. In this structure, any two vertexes generate a unique side and any two sides generate a unique vertex. Other such structures can be obtained by their transportation by bijective functions, or by making complex the structure given by the application of vectors. An example of six-vector system yielding self-generating is given by three successive functions f, g, and h and their converses f’, g’, and h, if fogoh=1f. In this case fog=h’, f’og’=h,m etc. Observation

It is noticed that under these conditions the successive composition of any three successive functions on the feedback chain yields function 1f. For example if the first system of generators is given by f, g and h in this order, then fog’oh=1f, g’ohof’=1f, etc.

Algebraic fractals

A particular example of informational structures given by informational vectors is created by the self-generating structures given by auto-morphisms of the projective straight line. These structures are adequate for the theoretical treating of informational fields thanks to the fact that they treat the element-positioning relationship from a linear structure between themselves, not being connected to an affine-type metrics characterizing only an instrumental projection on the informational field. Also, being non-commutative structures, they give logic to the explanation of the informational phenomena related to time. The cycles of the auto-morphisms connected by generating relations whose composition yields 1f algebraic feedbacks or, by language extension, feedbacks, the cycles whose composition yields the auto-morphism noted by f5 will be called feedforward, and the ones yielding the auto-morphism noted by f4 will be called feedbefore. (Figure 2)

f1 identity f2 symmetry f3 inversion f4 60-degree rotation f5 120-degree rotation f6 polar line circular permutation Figure 2 algebraic feedback-algebraic fractal

In figure 2 a feedback cycle is noted 235 where the two rows are the systems of generators. 635 In cyclic representation, the order of the auto-morphisms is given by the chart in figure 1. The main characteristic shown by figure 2 is the fractal-type informational- structures generability of algebraic feedbacks. This property is illustrated by the existence of two composition tables, the first for the auto-morphisms of the projective line, the second for algebraic feedbacks, in which the composition rules follow common patterns, differing from one another only as regards the modalities of expressing the composition of feedbacks. The second table of the composition of algebraic feedbacks shows that the feedback structure does not form a group; instead it forms an algebraic variety called fractal variety. For the functional differentiation of auto-morphisms from the point of view of the information conveyed, each of these auto-morphisms was associated to a geometrical transformation, association specific to the closure of the projective line and to its projection as infinite straight line of a projective plan. The ways of associating feedbacks on composition show a structure of the information, which can be associated to an informational-spin phenomenon, which is not treated in this article in extenso (see Florin Colceag Algebraic Fractals, Fractal Varieties www://austega.com/florin/). The composition of algebraic feedbacks takes place in accordance to rules common to the types of feedbacks, there not being any common pattern for all feedbacks. The result of the composition is also non-unique, several feedbacks in a class may respond to composition. These characteristics, which point out the composition of the classes of algebraic feedbacks in place of separate feedbacks are the characteristics of operations on algebraic varieties and also characterize the fractal varieties. (Table1).

234234 234264 634234 634264 234224 334234 364364 364324 264364 264324 364334 664364 624624 624634 324624 324634 624664 224624 235235 235265 635235 635265 235225 335235 365365 365325 265365 265325 365335 665365 625625 1 625635 325625 5 325635 1 625665 225625 1 5 1 235234 235264 635234 635264 235224 335234 365364 365324 265364 265324 365334 665364 625624 625634 325624 325634 625664 225624 234235 234265 634235 634265 234225 334235 364365 364325 264365 264325 364335 664365 624625 5 624635 1 324625 1 324635 4 624665 4 224625 4 1 1 4 4

224224 334334 664664 225225 335335 665665 5 5 225224 335334 665664 224225 334335 664665 1 1 624264 234324 364634 625265 235325 365635 5 625264 235324 365634 624265 234325 364635 1

Table 1 The classes of feedback cycles, feedforward or feedbefore, under the linearized form read on a cycle.

The way in which different classes of feedbacks can behave as elements in a high-level algebraic feedback is illustrated in figure 3.

Figure 3 second –degree feedback cycle in which each element is a first-degree feedback.

In the case of second- degree algebraic feedback, each feedback which characterizes a node in a simple feedback is also a simple feedback. The cycles which are formed by the connection of data entries, by instance, are informational cycles which are not tantamount to feedbacks, feedforwards or feedbefores, since they do not have the role of informational exit, but only the one of internal information-rolling systems. For instance a circuit module in table 1 can act as a feedback of second degree, refining the specific information induced by the module on different informational directions of action. Such example is supplied in Figure 4.

in linear writing in writing in the following fractal stage (a possibility in the semantic cone)

in writing under the form of feedback in the matric tables in the third fractal stage.

The significance of the three types of massages is mostly of the same type, but semantically differentiated at superior levels. The second stage of semantic differentiation corresponding to the second-degree cones comprises a semantic-interpretation cone which becomes restricted in the third fractal stage in a single meta-significant highly nuanced in its own structure.

Figure 4 Semantic refinement of a superior-degree feedback Drawing a conclusion on the described properties, we can notice that the feedback, feedforward and feedbefore cycles can be interconnected in order to create superior-level cycles. Also, a cycle of any level can be expressed to the exterior by a specific automorphism obtained through the composition of the elements of the cycle(1, 4, or 5 for simple cycles), or by a cycle having the degree inferior by one unit (comprising the elements having a degree inferior by one unit in the superior-degree cycle). These properties illustrate the unlimited possibility of informational fields to differentiate information until it can apply to a fixed context. On fractal varieties containing informational varieties with vectors and nodes, this property allows refinement of the information conveyed by vectors so that it can cover the structure of the nodes. The cycles of second degree or of a superior one can be represented in several ways, since they are complex units of information. For calculations, linearization by parastrophic (lattice- like) tables is the most appropriate, but the most adequate informational aggregation form is the form of the anchor ring or of the hyper anchor ring. This form observes both the different configuration of cycles (feedback, feedforward, feedbefore or internal feedback), and its structuring on different structuring directions characteristic of the string form (figure 5).

Figure 5 representations of high-level cycles

We can notice the similarity with the representations of the solutions in the Calabi Yau equations which mathematically configure the physical theory of strings, theory which converges to the same representation of an elastic and reactive universe as the theory of informational fractals.

The parastrophic (lattice-like) representations can correspond to topological sequences from such complex strings made up of clusters of superior degree obtained via operational rules. According to the theory of parastrophic (lattice-like) automata these units manifest themselves as computational unit. (Figure 6) Figure 6 parastrophic (lattice-like) automaton

If the edges of a cube or of a hypercube are completed by algebraic feedbacks, there will be obtained a mathematic object in which there is a tendency of completing also feedbacks on each dimension of the cube or of the hypercube. This being impossible, gaps will be obtained (informational faults) which, being fixed in such a way as to obtain feedbacks, they will move on another cycle, which they will unbalance. Fix positions are thus obtained in the cube which act as Turing machines, having the capacity of choice only between two possible modification options and thus generating computational capacities for the parastrophic (lattice-like) automaton. This phenomenon does not occur when the fibers of the anchor ring containing the automaton are of different nature produced by the algebraic-feedback superior-degree cycles, but when the information is introduced under restrictive conditions. This type of behaviour can suggest explanation line of the quantic non-determinist behaviour correlated to the informational processing capacity of each object in physical space. The phenomenon also supplies an image on the manner of conduct of the informational field enunciated as hypothesis at the beginning of the article. It can be visualized by dedicated computer programs.

Operations with algebraic cycles

Informational cycles can be close (feedback, feedforward, feedbefore), or open (cycles which on composition yield the auto-morphisms f2, f3 or f6) and which characterize inner processes in the superior-order feedbacks or in the component lattice-like (parastrophic) automata. The feedback cycles have a number of characteristic operations. One of the operations is inner, that is the operation of projection on a new set of generators. (Figure 7). Figure 7 The operation of projection on a new set of generators

By this operation three interrelated cycles are obtained. For instance, G appears as generated by A and B in the perspective of D and E and will be expressed through a bi-dimensional lattice containing the cycles A and B on the lines and D and E on the columns.

Other operations are external relating the informational cycles. (Figure8)

Figure 8 Coupling operations of the close or open cycles

These operations enable the formations of cycles yield the possibility of information passing and modification through the informational network forming the space-support of informational layers. The nodes of the network can be auto- generative structures, cellular automata, or units comprised of these two categories of objects, that is clusters of cellular automata connected as auto-generative structures.

Following on finite examples the way of action of the information, it can be observed that an informational metabolism of informational feedback cycles is formed which initially generates other feedback cycles of different informational type, and then feedforward or feedbefore cycles which in their turn generate open cycles. This can be correlated with the informational metabolism characterizing local phenomena. This phenomenon in correlation with the generating of superior-degree feedbacks and of lattice-like (parastrophic) automata appearing under restrictive informational conditions enables modeling of the phenomena characterized by a set of initial orders given on a finite space, having little informational contribution (matter, energy, etc. ) This type of phenomena is present in the cycles of cellular biochemistry, global economy, of an organism’s metabolism, or in the behavior cycles of the solutions of a Calabi Yau equation. These instruments are correlated both with the process dynamics in complexity theories (chaos, catastrophes, dissipative systems, synergetic, etc.) as well as with the computational behavior (cellular automata, Turing machines, etc.) in models both structural as well as dynamic. Modeling using these instruments is conditioned by the entered data and can be performed by computer programs.

Fig. 10 – Informational metabolism on a network-type structure

The network-type structures and their behavior can be associated by using these instruments with the specific languages described via orders of the type: feedback, feedforward, feedbefore or open cycles having diverse degrees of complexity. This enables the discovery of natural languages of informational transfer between various information or logic structures.

Classic fractals and algebraic fractals

The classic fractals described by Mandelbrot depict form modification through reiteration of a basic law. Algebraic fractals describe condensation and diversification of the information through reiteration of other basic laws given by the behavior of informational cycles. Between these two types of fractals there is the connection between form and content. With classic fractals, if using a roughly assessing instrument, one will obtain measurable curves. With algebraic fractals, if such roughly working instruments are used, simple cycles made of auto-morphisms are obtained. For each of these fractals refinement of the instrument enables refinement of the form and respectively refinement of the information. A projection of these properties in the physical space shows that from an informational point of view passing to a lower assessing scale gives the possibility to discover hidden dimensions, which were not expressed at the less refined scale. This was discovered initially in quantum mechanics, became the working basis in the theory of superstrings. If the analytical approaches using the concept of measure operate a cleavage of the fractal reality of the universe enabling a description of the latter at a certain scale via equations, the structural approaches enable an analogous situation, describing the informational reality by matrix-like tables which are correlated with equations, for instance tables with rational solutions of certain geodesics in the fields of an equation solutions.

The algebraic fractals increase however the understanding capacity of natural phenomena through the introduction, by means of quantified information via close or open informational cycles, of an informational grammar enabling the understanding of not only the expression forms of the universe, but also of its informational content.

Parastrophic (lattice-like) automata enable in addition the understanding of the way in which information evolves in dynamic processes and the synergy of the informational universe. The introduction of the hypothesis regarding communication between the different components of the universe by means of dialog forms with grammars and vocabulary differentiated by levels of communication, can facilitate the understanding of many phenomena in nature which can only be described by classic mathematical techniques. From this perspective Mandelbrot’s fractals displaying different forms in different reiteration stages give an overall image on information differentiation at these levels.

Algebraic fractals, by being expressed only in 3 ways by: f1, f4 and f5 where f1 represents the lack of an exit outside the system, actually give the possibility of an informational-complex output reaction’s passing into basis 2 via f4 and f5. This passing into basis 2 can be transferred into the analytical working method, reiterations being transparent at different complexity levels of the algebraic fractal, and in the analytical form it can be coupled directly with Mandelbrot’s fractals describing this form. Transfer can be performed via computer programs able to reiterate an algorithm several times. It can be also performed by the refinement model of information which is coupled with the exits of the cycles by the f1, f4 and f5 tables. Not all the physical phenomena can be obtained by this simplified reiteration procedure of a unique algorithm. Most of these phenomena can be described informationally as coupled with the network laws of parastrophic (lattice-like) automata, there being possible to strategically choose the way of expressing the data in the composition table of feedbacks and of the computational behavior of parastrophic (lattice-like) automata. This fact enables the understanding of Heisenberg’s indetermination principle and of unusual behaviors described in quantum mechanics showing sensitivity of the quantum environment to information.

For these reasons, the application of algebraic fractals and parastrophic (lattice-like) automata can be performed at any level of information structuring, being able to form a trans-disciplinary working method. The modeling specific techniques involve informational transfer from the data of the problem in current language to the language specific to algebraic fractals, the description of fix laws and of dynamic phenomena in this language and integration of the information in parastrophic (lattice- like) automata structures with computational capacities.