Algebric Fractals - Fractal Varieties

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 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)