File:Ack4dFragment.jpg

Fragment (with improved resolution) of image http://mizugadro.mydns.jp/t/index.php/File:Ack4d.jpg

Complex map of tetration to base 10.

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

Prepared as Figure 3d 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

//#include "tet2f4c.cin"
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 7) define z_type std::complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 1) include "conto.cin"


 * 1) 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 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, cu,cd; //z_type Zo=z_type(.31813150520476413, 1.3372357014306895); //z_type Zc=z_type(.31813150520476413,-1.3372357014306895); int M=641,M1=M+1; int N=402,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("tet2m2.eps","w");ado(o,1604,804); // FILE *o;o=fopen("tettenm2.eps","w");ado(o,1604,804); FILE *o;o=fopen("amsfig4dFragmen.eps","w");ado(o,1604,804); fprintf(o,"802 402 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n"); DO(m,M1)X[m]=-8.+.05*(m-.3); DO(n,200)Y[n]=-4.+.02*n; Y[200]=-.01; Y[201]= .01; for(n=202;n-99999. && p<99999. && q>-99999. && q<99999. ){ g[m*N1+n]=p;f[m*N1+n]=q;} d=c; for(k=1;k<31;k++) { m1=m+k*20; if(m1>M) break; //               d=exp(a*d); d=exp(d*log(10.)); p=Re(d);q=Im(d); if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }          d=c; for(k=1;k<31;k++) { m1=m-k*20; if(m1<0) break; //             d=log(d)/a; d=log(d)/log(10.); p=Re(d);q=Im(d); if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }       }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=20;q=1; for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .9 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .9 RGB S\n"); for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n"); for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .6 0 .6 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 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( "ggv fig3.eps"); system("epstopdf amsfig4dFragmen.eps"); system( "open amsfig4dFragmen.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass{amsproc} \usepackage{graphicx} \usepackage{rotating} \usepackage{hyperref} \newcommand \sx {\scalebox} \newcommand \rme 	 %% %\newcommand \rme 	%% \newcommand \rmi 	 %%imaginary unity \newcommand \ds {\displaystyle} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing \includegraphics \usepackage{geometry} \topmargin -94pt \oddsidemargin -70pt \paperwidth 1666pt \paperheight 856pt \textwidth 1900px \textheight 900px \begin{document} \parindent 0pt

\newcommand \mapax { \put(18,820){\sx{5}{$y$}} \put(18,730){\sx{5}{$3$}} \put(18,630){\sx{5}{$2$}} \put(18,530){\sx{5}{$1$}} \put(18,430){\sx{5}{$0$}} \put(-14,329){\sx{5}{$-1$}} \put(-14,229){\sx{5}{$-2$}} \put(-14,129){\sx{5}{$-3$}} \put(-14, 29){\sx{5}{$-4$}} \put(14, 0){\sx{5}{$-8$}} \put(114, 0){\sx{5}{$-7$}} \put(214, 0){\sx{5}{$-6$}} \put(314, 0){\sx{5}{$-5$}} \put(414, 0){\sx{5}{$-4$}} \put(514, 0){\sx{5}{$-3$}} \put(614, 0){\sx{5}{$-2$}} \put(714, 0){\sx{5}{$-1$}} \put(844, 0){\sx{5}{$0$}} \put(944, 0){\sx{5}{$1$}} \put(1044, 0){\sx{5}{$2$}} \put(1144, 0){\sx{5}{$3$}} \put(1244, 0){\sx{5}{$4$}} \put(1344, 0){\sx{5}{$5$}} \put(1444, 0){\sx{5}{$6$}} \put(1544, 0){\sx{5}{$7$}} \put(1634, 0){\sx{5}{$x$}} } %\flushright{$b=\mathrm e \approx 2.71$}

{\begin{picture}(1620,850) %%% \put(50,40){\ing{amsfig4dFragmen}}  \mapax \put(114,660){\sx{8}{$b=10$}} \put(60,434){\sx{4}{\bf cut}} \multiput(128,460)(330.2,108){4}{\sx{4}{$v\!=\!0.8$}} \multiput(120,402)(330.2,-108){4}{\sx{4}{$v\!=\!-0.8$}} \multiput(374,390)(330.2,-108){4}{\sx{4}{$u\!=\!0$}}

\put(80,796){\sx{4}{$u+\mathrm i v \approx -0.11919307341+ 0.75058329393 \,\mathrm i$}} \put(80,90){\sx{4}{$u+\mathrm i v \approx -0.11919307341- 0.75058329393 \,\mathrm i$}} \end{picture}} \end{document}