Difference between revisions of "File:Sqrt2tetmap.jpg"

Fig.16.2 from page 220 of book «Superfunctions» ^[1], 2020.

This image appears also as Рис.16.2 of the Russian version «Суперфункции»^[2]

This image is used also in figure 2 in the article ^[3], 2010 (top left map)

The figure shows the Complex map of tetration to base \(\sqrt{2}\):

\(u\!+\!\mathrm i v = \mathrm{tet}_{\sqrt{2}}(x\!+\!\mathrm i y)\)

C++ generator of the First map

/* Files ado.cin, conto.cin, sqrt2f21e.cin should be loaded in order to compile the code below.*/

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
#include<complex>
typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
DB T22=-8.5715740896774235522;
DB T42= 9.6180745210214273558;
#include "sqrt2f21E.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=201,M1=M+1;
 int N=401,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("sqrt2f21ema.eps","w");  ado(o,0,0,214,212);
 fprintf(o,"112 110 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-10.+.1*(m-.5);
  DO(n,N1) Y[n]=-10.+.05*(n-.5);
 for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
 for(n=-10;n<11;n++) {M(  -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n");
 for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);}
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-10.7,  9.5) fprintf(o,"(y)s\n");
 M(  9.6,-10.8) fprintf(o,"(x)s\n");
*/

// M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n");
 //z_type tm,tp,F[M1*N1];; 
 DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                        f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; 
 DO(n,N1){y=Y[n]; z=z_type(x,y);
        if(abs(z+2.)>.04)
                {       
                //c=TQ2L3(z); 
                c=F21E(z);
                p=Re(c); q=Im(c);
                if(p>-9999 && p<9999) g[m*N1+n]=p;
                if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
                }
        }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");

DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) } 
fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-10.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some printers

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt2f21ema.eps"); 
        system(    "open sqrt2f21ema.pdf"); // for LINUX 
        getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels

\documentclass[12pt]{article}
\paperwidth 422px
\paperheight 418px
\textwidth 1394px
\textheight 1300px
\topmargin -94px
\oddsidemargin -76px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\parindent 0pt
\pagestyle{empty}
\begin{document}\parindent 0pt
\sx{2}{\begin{picture}(204,204) \put(0,0){\ing{sqrt2f21ema}}
\put(6,206){\sx{.8}{$y$}}
\put(6,188){\sx{.8}{$8$}}
\put(6,168){\sx{.8}{$6$}}
\put(6,148){\sx{.8}{$4$}}
\put(6,128){\sx{.8}{$2$}}
\put(6,108){\sx{.8}{$0$}}
\put(-1, 88){\sx{.8}{$-2$}}
\put(-1, 68){\sx{.8}{$-4$}}
\put(-1, 48){\sx{.8}{$-6$}}
\put(-1, 28){\sx{.8}{$-8$}}
\put(24,2){\sx{.8}{$-8$}}
\put(44,2){\sx{.8}{$-6$}}
\put(64,2){\sx{.8}{$-4$}}
\put(84,2){\sx{.8}{$-2$}}
\put(110.5,2){\sx{.8}{$0$}}
\put(130.5,2){\sx{.8}{$2$}}
\put(150.5,2){\sx{.8}{$4$}}
\put(170.5,2){\sx{.8}{$6$}}
\put(190.5,2){\sx{.8}{$8$}}
\put(208.6,2){\sx{.8}{$x$}}

\put(82,182){\rot{90}$u\!=\!3$ \ero}
\put(23,163){\rot{-37}$u\!=\!3.8$ \ero}
\put(16,142){\rot{-9}$u\!=\!4$ \ero}
\put(16,107){\bf cut}
\put(16,73){\rot{9}$u\!=\!4$\ero}
\put(23,49){\rot{37}$u\!=\!3.8$ \ero}

\put(69,18){\rot{86}$u\!=\!3.2$ \ero}
\put(82,18){\rot{90}$u\!=\!3$ \ero}

\put(176,193){$v\!=\!0$}
\put(176,150){$u\!=\!2$}
\put(176,107){$v\!=\!0$}
\put(176,65){$u\!=\!2$}
\put(176,21){$v\!=\!0$}

\put(68,147){$v\!=\!1$}
\put(68,128){$v\!=\!2$}
\put(65,68){$v\!=\!-1$}

\put(110,161){\rot{65}$u\!=\!2.4$\ero}
\put(123,158){\rot{49}$u\!=\!2.2$\ero}
\put(122,114){\rot{40}$v\!=\!0.2$\ero}
\put(109,84){\rot{36}$u\!=\!1.6$\ero}
\put(120,80){\rot{30}$u\!=\!1.8$\ero}

\put(88,43){\rot{27}$v\!=\!-0.4$\ero}
\put(98,32){\rot{22}$v\!=\!-0.2$\ero}
\end{picture}}
\end{document}

References

Keywords

«[[]]»,

«Base sqrt2», «Complex map», «Exp», «Exponential», «Superfunction», «Superfunctions», «Sqrt2», «Tetration»,