Curs Algebric Fractals - Fractal Varieties

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.