File:Tetsheldonmap03.png

Complex map of tetration to Sheldon base $b=s=1.52598338517+0.0178411853321 i$.

Function $f=tet_s(x+\mathrm i y)$ 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.

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; // typedef 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); 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=601,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("tetsheldonmap.eps","w");ado(o,602,202); fprintf(o,"301 101 translate\n 10 10 scale\n"); DO(m,M1)X[m]=-30.+.1*(m); DO(n,200)Y[n]=-10.+.05*n; Y[200]=-.01; Y[201]= .01; 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<31;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<31;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;q=.5; 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 tetsheldonmap.eps"); system(   "open tetsheldonmap.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \paperwidth 618pt \paperheight 214pt \textwidth 1060pt \textheight 500pt \topmargin -108pt \oddsidemargin -70pt \pagestyle{empty} \usepackage[usenames]{color} \usepackage[utf8x]{inputenc} \usepackage{hyperref} \usepackage{graphicx} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \ing {\includegraphics} \newcommand \rme {\mathrm{e}} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \parindent 0pt \begin{picture}(202,211) \put(10,10){\ing{tetsheldonmap}} \put(2,206){$y$} \put(2,188){$8$} \put(2,168){$6$} \put(2,148){$4$} \put(2,128){$2$} \put(2,108){$0$} \put(-6,88){$-2$} \put(-6,68){$-4$} \put(-6,48){$-6$} \put(-6,28){$-8$} %\put(0,8){-10}\put(261,0){$-4$} \put(-1,0){$-30$} \put( 49,0){$-25$} \put( 99,0){$-20$} \put(149,0){$-15$} \put(199,0){$-10$} %\put(222,0){$-8$} %\put(242,0){$-6$} \put(252,0){$-5$} %\put(262,0){$-4$} %\put(282,0){$-2$} \put(309,0){$0$} \put(329,0){$2$} \put(349,0){$4$} \put(369,0){$6$} \put(389,0){$8$} \put(407,0){$10$} \put(457,0){$15$} \put(507,0){$20$} \put(557,0){$25$} \put(607,1){$x$} \multiput(24,180)(118,-1){5}{$v\!=\!1.2$} \multiput(70,180)(118,-1){5}{$u\!=\!2$} \multiput(112,158)(118,-1){5}{\rot{-50}$v\!=\!1$\ero} \multiput(102,142)(118,-1){5}{\rot{-35}$v\!=\!0.8$\ero} \put(336,107){\rot{-11}$v\!=\!0$\ero} \multiput(206, 92)(119,-22){4}{$v\!=\!-1$} \multiput(70, 84)(119,-22){4}{$u\!=\!1.2$} \multiput(110, 76)(119,-22){4}{$v\!=\!-1.4$} \put(470,200){$u+\mathrm i v \approx 2.0565+1.1445 \,\mathrm i$} \put(30,20){$u+\mathrm i v \approx 2.2284-1.3508 \,\mathrm i$} \end{picture} \end{document}