File:Sqrt2tetmap.jpg

Complex map of tetration to base $\sqrt{2}$:

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

Usage: this is figure 16.2 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

This imahe is used also in figure 2 in the article . (top left map)

C++ generator of the First map
Files ado.cin, conto.cin, sqrt2f21e.cin should be loaded in order to compile the code below.

typedef std::complex z_type; DB T22=-8.5715740896774235522; DB T42= 9.6180745210214273558; 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");
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x.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;;
 * 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}