## TYPOLOGY AND FRACTALS

### Fractal

In the mid-seventies of 20th century, thanks to the possibility offered by strongly  expanding computer technique, a new mathematical discipline was established – THE THEORY OF CHAOS, which encompasses with its application the phenomena in physics, chemistry, meteorology, even biology. One of the basic notions of this theory is FRACTAL. Fractals were introduced by an American mathematician, a Jew immigrant from Poland, Benoit Mandelbrot, who defined them as a geometrical object showing a structure rich in details, regardless of how much the structure is magnified. In other words, by magnifying a detail of this structure we always find ever new details (therefore it is called a structure with infinite number of details). Viewed from a mathematical standpoint, these structures originate through a definite series of transformations of the starting geometrical figure. However, the main characteristic is that the number of these transformations is fixed and limited, and that the series of transformation is applied to every newly obtained figure (Fig. F–1). In that way the most different geometrical shapes have been obtained, e.g. leaves of various plants, surface of mountain ranges, clouds and other intricate, curly, wrinkled, strange (apparently chaotic) structures, not obtainable thus far.

 a b c d e f
 Fig. F–1. An illustrative example of getting a linear fractal (so called Koch's curve). One starts from a segment (of a line) a) whereon the operation b) is applied. Then the same operation is applied on each segment of b), so that c) is obtained. The process goes on, theoretically, ad infinitum. Characteristic of such fractals is that for every sufficiently small segment of the obtained curve there is a same such segment in the extension of the curve, and a very similar segment (but lessened) in the depth of the curve.

Depending on the type of transformations, the fractals are divided into determinist and stochastic ones. With the determinist fractals the cited series of transformations is unchangeable, while with the stochastic ones it depends on one or more stochastic (that is accidental, unpredictable) perimeters. Both these kinds of fractals are, again, divided into linear and non-linear ones, depending on the type of transformations applied.

One of the main properties of these structures is self-similarity, with which when beginning from a detail, by further magnification, after certain number of steps one comes to the same (or rather very, very similar) detail (Fig. 2).

 a b c d e f g h
 Figure 2. An example of so called exponential fractal (Mandelbrot). After successive magnifying (the framed) details, the initial figure (a) is got in the step g (although in a different context).

Considering that, for the present, the fractals represent sole mathematical objects with that property, and that it has been established that Biblical types (which, as we shall see, have a general character) also have the property of self-similarity, the conclusion is: THE TYPES are FRACTALS.

### The Bible and Fractals

On the basis of typological analyses cited in the following parts of the book, here a hypothesis is made that the character of all existing (including also spiritual realities) is represented by the system of fractals, in terms of time, space, events, and even beings. Indeed, the notion FRACTAL in natural sciences is corresponded in its form by the notion of TYPE in theology. In that way, fractals become a link between natural sciences on one side, and theology (and literature) on the other.

Let us start from nature and space. All known fundamental structures have similar centrally rotational structure. In an atom, the electrons move by definite orbits around the nucleus. In the Solar system, planets move around the Sun by their orbits, and it is probably similar also with other stars and their planets. In galaxies, the stars move around the centre of the galaxy.

As to the time, we have first the system of sevens (weeks), which is, on the basis of the expounded in this book, similar to a real fractal. Magnification (or contrary, shrink), whereby one comes to a new seven is 360 000.

With respect to events the correspondence of the types of events is seen both within the scope of characteristic sevens and history in general.

As to the creatures, there are also characteristic types. The clearest types are the beings similar to God. They are, on one side Adam, and on the other the 144,000 (Rev. 7:4), viewed as a whole. Further in Adam’s week, we have the animals "according to their kinds" (Gen.1:21.24), clean and unclean ones, while in human history we have those saved "of all nations, tribes, peoples, and tongues" (Rev. 7:9), as well as, unfortunately, those others who were not found written in the Book of Life (Rev. 20:15).

### Kinds of Biblical Fractals

Similarly to real fractals, the biblical fractals also can be divided into two groups. The first includes the units of the same kind that have similar form, and therefore correspond to the linear fractal (i.e. types). The second group includes the units belonging to different forms, but similar in a metaphorical (that is spiritual) sense. A biblical metaphor has not only a literary-esthetical value, but also a deep cognitive-spiritual message. These structures are called here exponential types (fractals).

Thus in the part TYPOLOGY OF TRINITY it is inferred that in every man there is a hidden trinity, which is as a rule under both God’s and Satan’s influence. On the other hand, entire communities of humans can represent, too, a type of some person of the trinity, and thus we have the lamb’s bride as a trinity (Sun, Moon and stars), beside three beasts from Apocalypse representing Satan trinity and finding their fulfillment, as we shall see, in monotheistic religious-political organizations on the Earth.

In the part TYPOLOGY OF DUALITY one may delve into the most pervasive and the most universal fractal - duality

Same as Adam was a crown of the creation described in the first pages of the Bible, so we have the 144 000 as a crown of the creation described in the last pages of the Bible.

Concerning time, we have ordinary weeks (of seven days each), which are similar mutually, as well as characteristic (Creation) weeks whose ratio is 1 : 360 000, which are similar in a spiritual sense (see TYPOLOGY OF SEVEN).

Linear fractals can also be called quantitative ones, while exponential can be called – qualitative ones. Common to both linear and exponential fractals is their Archetype, which can serve as a link between the two. Thus Christ is an archetype both for Adam and for all people who live according to God’s will (linear fractals), and also for the 144,000 (who are an exponential fractal relative to Adam). The seven God’s spirits are an archetype both for both common weeks, of course if they are led through according to God’s will (linear fractals) as well as for the Creation weeks.

### Literature:

1. Hartmut Jürgens, Heinz‑Otto Peitgen und Dietmar Saupe, Fraktale - eine neue Sprache für komplexe Strukturen, SPEKTRUM DER WISSENSCHAFT, sept. 1989, str. 52-64.

2. David J. Wright, Dynamical Systems and Fractals Lecture Notes.

3. Paul Bourke, An Introduction to Fractals.

Last updated 16.1.2006