File:Tetsheldonzoo.jpg

Complex map of tetration to Sheldon base, zoom-in from the central part of figure http://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png

$b=s=1.52598338517+0.0178411853321 i$.

$f=tet_s(x+\mathrm i y)$ is shown in ths $x,y$ plane with levels $u=\Re(f)=\mathrm{const}$ and levels $v=\Im(f)=\mathrm{const}$; thick lines correspond ot the integer values.

Usage
This image should be second picture of figure 18.3 of English version of book Superfunctions

C++ generator of map
Files ado.cin, conto.cin, filog.cin, GLxw2048.inc, TetSheldonIma.inc should be loaded in order to compile the code below // using namespace std; typedef std::complex z_type; z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type a=log(b); z_type Zo=Filog(a); z_type Zc=conj(Filog(conj(a))); DB A=32.; z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd; int K=2048; //#include "ima6.inc" z_type E[2048],G[2048]; DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);} c=0.; //z+=z_type(0.1196573712872846, 0.1299776198056910); z+=z_type( 0.1196591376539, 0.1299777213955 ); DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; return c;}
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "filog.cin"
 * 1) include "GLxw2048.inc"
 * 1) include "TetSheldonIma.inc"

int main{ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d; //int M=161,M1=M+1; int M=201,M1=M+1; int N=701,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("09.eps","w");ado(o,2030,2020); FILE *o;o=fopen("tetsheldonzo.eps","w");ado(o,2030,2020); fprintf(o,"1010 1010 translate\n 100 100 scale\n"); DO(m,M1)X[m]=-10.+.1*(m-.5); DO(n,400)Y[n]=-10.+.025*n; Y[400]=-.001; Y[401]= .001; for(n=202;n-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} d=c; for(k=1;k<11;k++) { m1=m+k*10; if(m1>M) break; d=exp(a*d); p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }          d=c; for(k=1;k<11;k++) { m1=m-k*10; if(m1<0) break; d=log(d)/a; p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }       }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.6;q=.7; for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n"); for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n"); // y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)} // fprintf(o,".07 W 1 .5 0 RGB S\n"); // y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)} // fprintf(o,".07 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tetsheldonzo.eps"); system( "open tetsheldonzo.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 2090pt \paperheight 2080pt \textwidth 2090pt \textheight 2090pt %\textwidth 700pt \usepackage{graphics} \newcommand \sx \scalebox \newcommand \ing \includegraphics \parindent 0pt \topmargin -104pt \oddsidemargin -54pt %\usepackage{rotate} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(2064,2056) %\put(0,0){\ing{04}} %\put(0,0){\ing{tetshelim}} \put(50,40){\ing{tetsheldonzo}} \put(10,2024){\sx{6.2}{$y$}} \put(10,1832){\sx{6}{$8$}} \put(10,1632){\sx{6}{$6$}} \put(10,1432){\sx{6}{$4$}} \put(10,1232){\sx{6}{$2$}} \put(10,1032){\sx{6}{$0$}} \put(-36, 832){\sx{6}{$-2$}} \put(-36, 632){\sx{6}{$-4$}} \put(-36, 432){\sx{6}{$-6$}} \put(-36, 232){\sx{6}{$-8$}} \put(-16,  -8){\sx{6}{$-10$}} \put(200, -8){\sx{6}{$-8$}} \put(400, -8){\sx{6}{$-6$}} \put(600, -8){\sx{6}{$-4$}} \put(800, -8){\sx{6}{$-2$}} \put(1049, -8){\sx{6}{$0$}} \put(1249, -8){\sx{6}{$2$}} \put(1449, -8){\sx{6}{$4$}} \put(1649, -8){\sx{6}{$6$}} \put(1849, -8){\sx{6}{$8$}} \put(2020, -8){\sx{6.2}{$x$}} %\put(1000,1000}{\sx{6}{\rot{-8} $u\!=\!0$ \ero}} \put(1070,940){\sx{8}{\rot{82}$u\!=\!1$\ero}} \put(1300,930){\sx{8}{\rot{82}$u\!=\!2$\ero}} \put(1566,900){\sx{8}{\rot{76}$u\!=\!3$\ero}} \put(1300,1516){\sx{8}{\rot{-19}$v\!=\!1$\ero}}% \put(1330,1008){\sx{8}{\rot{-11}$v\!=\!0$\ero}} \put(1330, 618){\sx{8}{\rot{19}$v\!=\!-1$\ero}}% \end{picture} \end{document}