01/28/2009, 03:38 AM
(This post was last modified: 02/01/2009, 06:13 PM by Kouznetsov.)

Let sexp be holomorphic tetration. Let

.

This has fractal structure. This structure is dence everywhere, so, if we put a black pixel in vicinity of each element, the resulting picture will be the "Black Square" by Malevich; it is already painted and there is no need to reproduce it again.

Therefore, consider the approximation. Let

.

While ,

id est, all the points of the approximation are also elements of the fractal (although only Malevich could paint all the points of the fractal).

As an illustration of , centered in point 8+i, I suggest the plot of function in the complex plane,

in the range ,

Levels are drawn.

Due to more than lines in the field of view, not all of them are plotted. Instead, the regions where are shaded. In some regions, the value of is huge and cannot be stored in a complex<double> variable; these regions are left blanc.

In such a way, tetration gives also a new kind of fractal.

.

This has fractal structure. This structure is dence everywhere, so, if we put a black pixel in vicinity of each element, the resulting picture will be the "Black Square" by Malevich; it is already painted and there is no need to reproduce it again.

Therefore, consider the approximation. Let

.

While ,

id est, all the points of the approximation are also elements of the fractal (although only Malevich could paint all the points of the fractal).

As an illustration of , centered in point 8+i, I suggest the plot of function in the complex plane,

in the range ,

Levels are drawn.

Due to more than lines in the field of view, not all of them are plotted. Instead, the regions where are shaded. In some regions, the value of is huge and cannot be stored in a complex<double> variable; these regions are left blanc.

In such a way, tetration gives also a new kind of fractal.