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
\( F= \{ z\in \mathbb{C}: \exists n\in \mathbb{Z} : \Im(\mathrm{sexp}(z+n))=0 \} \).
This \( F \) 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
\( F_n= \{ z\in \mathbb{C}: \Im(\mathrm{sexp}(z+n))=0 \} \).
While \( \mathrm{sexp}(z+1)=exp(\mathrm{sexp}(z)) \),
\( F_n\subset F_{n+1} \) 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 \( F_8 \), centered in point 8+i, I suggest the plot of function \( \Im(\mathrm{sexp}(z)) \) in the complex \( z \) plane,
in the range \( 7\le\Re(z)\le 9 \), \( 0\le\Im(z)\le 2 \)
Levels \( \Im(\mathrm{sexp}(z))=0 \) are drawn.
Due to more than \( 10^{100} \) lines in the field of view, not all of them are plotted. Instead, the regions where \( |\Im(\mathrm\sexp(z))|<10^{-4} \) are shaded. In some regions, the value of \( \Im(\mathrm{sexp}(z)) \) 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.
\( F= \{ z\in \mathbb{C}: \exists n\in \mathbb{Z} : \Im(\mathrm{sexp}(z+n))=0 \} \).
This \( F \) 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
\( F_n= \{ z\in \mathbb{C}: \Im(\mathrm{sexp}(z+n))=0 \} \).
While \( \mathrm{sexp}(z+1)=exp(\mathrm{sexp}(z)) \),
\( F_n\subset F_{n+1} \) 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 \( F_8 \), centered in point 8+i, I suggest the plot of function \( \Im(\mathrm{sexp}(z)) \) in the complex \( z \) plane,
in the range \( 7\le\Re(z)\le 9 \), \( 0\le\Im(z)\le 2 \)
Levels \( \Im(\mathrm{sexp}(z))=0 \) are drawn.
Due to more than \( 10^{100} \) lines in the field of view, not all of them are plotted. Instead, the regions where \( |\Im(\mathrm\sexp(z))|<10^{-4} \) are shaded. In some regions, the value of \( \Im(\mathrm{sexp}(z)) \) 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.