ALGEBRAIC FRACTALS-FRACTAL VARIETIES
by Florin Colceag
The main technical part in understanding how various structures will form more
complex structures in a fractal way but preserving in the same time the main patterns
starts with elementary geometry. From simple to complex structures several properties
will be transmitted and other properties will enrich the new structural stages.
A feedback contains vertexes and vectors. Vertexes are structures that are formed in a
symmetric way by two sets of generators, any two elements of the first set generating a
new element of the second set. Vectors are transformations of the support space, for
example authomorphisms of the projective space (See Cellular automata algebraic
fractals). WE will see in this paper how several vectors involved in an algebraic fractal’s
feedback cycle will enrich the information contained in knots.
The first step in describing fractal varieties is to see how a transformation moves one
structure into a different but isomorphic structure. Let’s take a triangular structure ABC.
Any vertex will oppose to a side. For example vertex “A” will oppose the side “a”. Any
two vertexes will generate a unique side any two sides will generate a unique vertex.
For a tetrahedral structure, any three plans will generate a unique vertex; any three
vertexes will generate a unique plan. In order to have this generation several conditions
will be required: For the triangular generation, vertexes have to be not collinear, for the
tetrahedral structure, vertexes have to be not coplanar.
If on a triangular structure we apply an inversion we will obtain three circles with
their radical axes. We will replace lines and points with circles and radical axes (lines,
common cords with intersection points). Any two circles will determine a unique radical
axe; any two radical axes with intersection points will determine a unique circle. Here
also we have a condition regarding points on radical axes that have to be co cyclic.
If staring from this new structure we change circles with conics we obtain a similar
structure describing another transformation, the polar transformation that acted on the
triangular structure pattern. Rotations will need two steps to generate triangular
structures. The first step is the passage from triangle with points and lines to three
concurrent lines and angles. Any two lines determine a unique angle and two angles will
determine a unique line. By delimiting these lines we will obtain segments with a
common point. A 60 degrees rotation will generate three equilateral triangles with a
common point, and three pairs of equal segments. These equilateral triangles and these
pairs of segments will replace points and lines.
More spectacular will be the dictionary of transformations for tetrahedral structures.
The easiest example will be a triangle with orthocentrum. Each of these four points
will be orthocentrum for the triangle generated by the other four points. In general case
we can take a triangle with a point that is not on any side or vertex of the triangle, and
Ceva theorem in a projective space perspective given by the anarmonic rapport. This
structure will correspond with a plan projection of the tetrahedral structure.
Considering the triangle with orthocentrum and inversion transformation we will
obtain three circles with equal radiuses and a common point. These circles will intersect
each other in three different points. The circle passing through these three points has an
equal radius with the other three circles. Inversion transformation will preserve the
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history of the first triangular structure. If ABC is a triangle and H is the intersection of its
altitudes (orthocentrum), that ABC, ABH, ACH, BCH circles have equal radiuses. The
structure of circles and points will be isomorphic with the structure of points and lines of
the tetrahedron. Any three points will generate a unique circle; any three circles will
generate a unique point (See Figure 1).
The same phenomena of preserving the previous structure after a transformation will
be found by applying another transformation, the rotation with 60 degrees.
The structure obtained by rotating a tetrahedral structure will be more complex,
because of the complex relationship of rotations with the group of symmetries (See
Figure 2)
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In this figure the first referential given by HA, HB, HC rotated with 60 degrees will
give HAA’, HBB’, and HCC’ three equilateral triangles. These triangles will be in a
generating position with another three set of triangles LIH, HJP, and PKL. This second
group will generate OPR. The first group will generate UST. Triangles HAA’, HBB’,
HCC’, and UST are associated with three circles with a common point H that intersect
each other in three new point U, S, and T. We see that the first two sets of triangles
HAA’, HBB’, HCC’ and LIH, HJP, and PKL are in a triangular relationship, each two
triangles from one set generating one triangle from the other set. The relationship is not
completely symmetrical regarding the fourth triangle of any set, because tetrahedral
structures projected in the plane, doesn’t correlate well with rotations that is a specific
plane transformation. For the plane problem we can notice the same property, the new
structure preserved the history of a previous transformation, the inversion. The two
different and symmetrical structures involved in the problem are described in Figure 3
As we can notice in the Part A we start from two concurrent segments rotated with 60
degrees. And obtain an equilateral triangle with vertexes on new-formed segments (red
segments). The first three segments can be randomly selected by angles and sizes.
Part B will contain a different structure symmetrical with part A. This time we have a
triangle covered on the exterior with three equilateral triangles. What we will obtain will
be three concurrent lines intersecting each other in 60 degrees angles. Randomly selected
will be the middle triangle, and consequently randomly obtained will be the sizes of the
three concurrent lines. The figure is symmetrical in part A and B, concurrent lines
transforming in sides of a triangle, and triangles with a common vertex transforming in
triangles covering a central triangle. All triangles are equilateral excepting the middle
triangle from part B. Imposing the condition that circles circumscribed to covering
triangles will have equal radiuses this last triangle will become equilateral too.
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Rotation transformation will include the transformation of part B specific for
inversion into part A specific for 60 degree rotation, preserving the history of the
previous chain of transformations, but loosing the first one from the triangle with
orthocentrum into a structure formed by four circles.
For 120 degrees rotations we can find a similar phenomena, this time using a different
angle (See figure 4). We will also notice the existence of three segments with a common
end that is rotated this time with 120 degrees (See Figure 4 part A) These tree triangles
will generate a new set of triangles in the interior of three equilateral triangles, with a
similar shape (see Figure 4 part B). We can notice in the part B the existence of three
equilateral triangles covering a random triangle, a trace from a previous transformation,
60 degrees rotation. We can see also the structure of two sets of three triangles, each two
triangles from the first set generating an unique triangle from the second set, following
the pattern of generation already described in other examples.
We can see also “the trace” of the first space. In fact each problem already described
is part of the system of axioms that characterize one kind of space. If instead of lines and
points we have triangles as characterized in 60 degrees rotation example, we have another
kind of projective space with these elements. One transformation of this space
transported the structure of generators into a new space that is generated by 60 degrees
rotations as described in the last example. This transformation preserves “the trace of the
previous space in the structure of generators
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The list of examples is not complete. This list of examples describe a phenomena,
doesn’t create a dictionary. The list can be extremely large if we will consider all kinds of
transformations involved in the formation of algebraic fractals feedback cycles.
. We will notice that feedback cycles will contain two kinds of sequences of
transformations: symmetry- inversion-rotation-inversion-inversion-rotation, and
symmetry-inversion rotation-symmetry inversion rotation. The unique exception is
symmetry-symmetry-rotation-symmetry-inversion rotation corresponding to a feedback
cycle characterizing a symmetry that can be associated with polar transformation. These
feedback cycles have to be assessed separately in order to see which kinds of structures
will transport in knots, how each structure will be defined regarding the previous
structure, and what kind of invariants can we find for the entire feedback. For structures
presented behind anarmonic rapport is such an invariant. All figures described before can
be characterized using this invariant. By preserving the “history” of previous steps we
will obtain an increasing phenomena leading to more complex structure to the last step of
the feedback cycle. This last step will associate point and lines with new structures,
possible connecting classes of points and lines organized using internal rules with the
previous points and lines. These kinds of phenomena are common in algebraic geometry.
The second step of description for fractal varieties will contain the dictionary of
transformations of knots’ structures using all possibilities involved by feedback cycles
created using the authomorphisms of the projective space. These structures will reveal a
more complex structure of information that describes feedback cycles behavior in various
spaces generated by two sets of elements able to generate each other.
From the modeling perspective this increasing complexity will lead to a model of
complexity for the real life. In order to arrive to a new structure in the algebraic fractal
we need to compose various transformations characterizing one space with
transformations characterizing the previous space. Each system of generators will
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characterize a different space with similar configuration. Each space will have the same
kind of transformations that can be characterized as inversions, rotations, and
symmetries. Composing one transformation from one space with the next transformation
from the space generated by the first transformation applied on the first space, and using
a feedback characterized by the algebraic fractal, will lead us to a higher level of
complexity. We will not return to the first transformation because algebraic fractals don’t
respect group structure, developing as a variety. Composed transformations
characterizing various spaces involved in this kind of feedback cycles containing spaces
in knots and transformations in vectors, will give a very sophisticated formula of
transformation containing in its internal structure all the history of the feedback cycle.
These formulas can be associated with phenomena in physics. These physical phenomena
will represent objects able to generate other physical objects (fields) in a similar pattern
with the feedback cycle pattern.
Even if technical details for the entire description of these phenomena are not yet
described, this direction seems to lead to a correct and complex modeling possibility for
complexity including life. Other researches and theoretical approaches will identify easier
ways to deal with this complexity by finding and describing invariants of the entire
structure.
The fractal kind structure that will be obtained by using this procedure is more
complex. Each space obtained by applying a transformation on a different space starting
from its group of transformations will transport the entire set of axioms of the first space
to the next space. Examples given here describe how generating axioms are transported,
creating a new version of projective space with different objects. If the first space will
have a specific metric, it will be also transported using the transformation creating
commutative diagrams. The new space will have the same metric if it is originated from
the group of authomorphisms generated by the metric of the first space. If we use other
transformations from a different group, the metric will change creating a new set of
spaces with a different metric or with various metrics. These spaces will be varieties and
will describe families. The main mathematical object originated from the projective space
will be a more complex algebraic fractal, a fractal varieties theory.
All this complex structure will describe more complex objects, but will preserve
several characteristics as invariants. Among these characteristics are: feedback cycles,
projective kinds relationships among generators of the space, history preservation in the
internal structure of these generators, and of specific transformations describing the way
in which they had been obtained. All these characteristics are common for physics,
chemistry, biology, and sociology description and modeling phenomena.
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