File:Ack4d.jpg

Complex map of tetration to base 10.

$u+\mathrm i v=\mathrm{tet}_{10}(x\!+\!\mathrm i y)$

Prepared as Figure 4d for publication in Journal Applied and Computational Mathematics, but not used in the final version

C++ generator of map
Files ado.cin, conto.cin, filog.cin, f4ten.cin, GLxw2048.inc, f2048ten.inc should be loaded in order to compile the code below

// using namespace std; typedef std::complex z_type;
 * 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"

//z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type b=10.; 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 "GLxw2048.inc"
 * 1) include "TetSheldonIma.inc"

z_type f4ten(z_type z){ //NOT SHIFTED FOR x1 !!!! z_type tenZo=z_type(-0.119193073414549, 0.750583293932439); z_type tenZc=z_type(-0.119193073414549,-0.750583293932439);
 * 1) include "GLxw2048.inc"

DB Lten= 2.302585092994046; z_type tenQ=z_type( 0.559580251215472, 1.728281903659204);// =L*Zo+log(L) z_type tenT=z_type( 3.290552906607012, 1.065409768058325); // #include "tenzo.inc" #include"f2048ten.inc" //Aten is defined there int j,k,m,n; DB x,y, u, t; z_type c,d, cu,cd; //      z_type E[K],G[K]; z_type E[2048],G[2048]; DO(k,K){c=F[k];E[k]=log(c)/Lten;G[k]=exp(c*Lten);} //     the initioalization abouve should run at the compillation c=0.; DO(k,K){t=Aten*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.,-Aten)+z)/(z_type(1., Aten)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-Aten)-z)/(z_type(1., Aten)+z) ); return c*(Aten/(2.*M_PI)) +tenZo*cu+tenZc*cd; } DB x1ten= 0.0377406857309657; //#include "figx1ten.inc" DB Lten=log(10.); z_type F4TEN(z_type z){ DB x=Re(z); //                     DB L=log(2.); if(x<-.5) return log(F4TEN(z+1.))/Lten; if(x> .5) return exp(F4TEN(z-1.)*Lten); return f4ten(z+x1ten); } // f4ten end

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=461,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("10.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 10.eps"); system( "open 10.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass{amsproc} \usepackage{graphicx} \usepackage{rotating} \usepackage{hyperref} \newcommand \sx {\scalebox} \newcommand \rme 	 %%makes the base of natural logarithms Roman font %\newcommand \rme 	%%makes the base of natural logarithms Italics font; choose one of these \newcommand \rmi 	 %%imaginary unity is always roman font \newcommand \ds {\displaystyle} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing \includegraphics \usepackage{geometry} \topmargin -94pt %\topmargin -97pt \oddsidemargin -87pt \paperwidth 618pt %\paperheight 216pt \paperheight 212pt \begin{document}

\newcommand \mapax { \put(2,206){\sx{1.2}{$y$}} \put(2,188){\sx{1.2}{$8$}} \put(2,168){\sx{1.2}{$6$}} \put(2,148){\sx{1.2}{$4$}} \put(2,128){\sx{1.2}{$2$}} \put(2,108){\sx{1.2}{$0$}} \put(-6,88){\sx{1.2}{$-2$}} \put(-6,68){\sx{1.2}{$-4$}} \put(-6,48){\sx{1.2}{$-6$}} \put(-6,28){\sx{1.2}{$-8$}} \put(-1,1){\sx{1.2}{$-30$}} \put( 49,1){\sx{1.2}{$-25$}} \put( 99,1){\sx{1.2}{$-20$}} \put(149,1){\sx{1.2}{$-15$}} \put(199,1){\sx{1.2}{$-10$}} \put(252,1){\sx{1.2}{$-5$}} \put(309,1){\sx{1.2}{$0$}} \put(329,1){\sx{1.2}{$2$}} \put(349,1){\sx{1.2}{$4$}} \put(369,1){\sx{1.2}{$6$}} \put(389,1){\sx{1.2}{$8$}} \put(407,1){\sx{1.2}{$10$}} \put(457,1){\sx{1.2}{$15$}} \put(507,1){\sx{1.2}{$20$}} \put(557,1){\sx{1.2}{$25$}} \put(607,1){\sx{1.2}{$x$}} } %\flushright{$b=\mathrm e \approx 2.71$} %\sx{.586} {\begin{picture}(620,212) %%%%%%%%%%%%%%%%%%%%%% %\put(10,10){\ing{tetema}}  \mapax \put(10,10){\ing{10}}  \mapax

%\multiput(110,118)(56.1,10.7){8}{\sx{1.2}{$u\!=\!0.8$}} %\multiput(256,120)(56.1,10.7){7}{\sx{1.2}{\rot{20}$u\!=\!1$\ero}} %\multiput(302,120)(56.1,10.2){7}{\sx{1.2}{\rot{0}$v\!=\!1$\ero}}

%\multiput(189,116)(44.7,10.5){9}{\sx{1.2}{$u\!=\!1.4$}} \multiput(256,124)(33.2,10.7){8}{\sx{1.2}{$u\!=\!-1.2$}} \put(25,108.4){\sx{1.4}{\bf cut}}		\put(296,108.4){\sx{1.2}{$v\!=\!0$}} \multiput(242,92)(33.2,-10.7){8}{\sx{1.2}{$v\!=\!-0.8$}} %\multiput(193,93)(44.7,-10.5){8}{\sx{1.2}{$v\!=\!-1.4$}}

%\put(20,200){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 + 1.3372357014 \,\mathrm i$}} %\put(30, 20){\sx{1.4}{$u+\mathrm i v \approx 0.31813150520 - 1.3372357014 \,\mathrm i$}} \put(20,196){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341+ 0.75058329393 \,\mathrm i$}} \put(30, 20){\sx{1.4}{$u+\mathrm i v \approx -0.11919307341- 0.75058329393 \,\mathrm i$}} \end{picture}} \end{document}