File:Sqrt2atemap.jpg

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

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

Usage: this is figure 16.3 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 right 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; // #include "tq2e.cin" // #include "tq2L.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=211,M1=M+1; int N=201,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("sqrt2f21lma.eps","w"); ado(o,0,0,214,212); fprintf(o,"112 110 translate\n 10 10 scale\n"); // DB sy=10.1/sinh(N/2./100.); DO(m,M1) X[m]=-11+.1*(m-.5); DO(n,N1) Y[n]=-10+.1*(n-.5); // DO(n,N1) Y[n]=sy*sinh((n-N/2.+.5)/100.); 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( 9.6,-10.8) fprintf(o,"(y)s\n"); M(-10.7, 9.5) fprintf(o,"(x)s\n"); // M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n"); // z_type tm,tp,F[M1*N1];; DO(m,M1)DO(n,N1){     g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; DO(n,N1){y=Y[n]; z=z_type(x,y); if( abs(z-2.)>.2 || abs(z-4.)>.2) {      c=F21L(z); p=Re(c); q=Im(c); if(p>-99 && p<99) g[m*N1+n]=p; if(q>-99 && q<99 && fabs(q)>1.e-18) f[m*N1+n]=q; }}} //p=2; q=1.1;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 1) include "sqrt2f21L.cin"
 * 1) include "plofu.cin"

// M(-13.2,0)L(2,0) fprintf(o,".05 W 0 .8 0 RGB S\n");

M(2,0)L(10.1,0)fprintf(o,".05 W 1 1 1 RGB S\n"); DO(n,27){M(2+.3*n,0)L(2+.3*(n+.5) ,0)} fprintf(o,".1 W 0 0 0 RGB S\n");

//M(2,0)L(10.1,0)fprintf(o,".1 W 0 0 0 RGB [.19 .19] 0 setdash S\n"); //M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); // fails at some printers fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt2f21lma.eps"); system(   "open sqrt2f21lma.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{sqrt2f21lma}} \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(26,198){$v\!=\!0$} \put(18,152){\rot{-2}$u\!=\!-2$\ero} \put(26,107.4){$v\!=\!0$} \put(176,108){\bf cut} \put(18,62){\rot{1}$u\!=\!-2$\ero} \put(26,17){$v\!=\!0$}

\put(96,172){\rot{20}$v\!=\!0.2$\ero} \put(106,158){\rot{20}$v\!=\!0.4$\ero} \put(112,148){\rot{20}$v\!=\!0.6$\ero} \put(121,139){\rot{20}$v\!=\!1$\ero}

\put(156,167){\rot{37}$u\!=\!-3$\ero} \put(178,164){\rot{50}$v\!=\!0$\ero}

\put(74,95){\rot{-23}$u\!=\!-1.8$\ero} \put(87,28){\rot{54}$u\!=\!-2.2$\ero} \put(113,21){\rot{76}$u\!=\!-2.4$\ero} \put(132,21){\rot{87}$u\!=\!-2.6$\ero} \put(137,64){\rot{-79}$u\!=\!-2.8$\ero} \put(151,51){\rot{-43}$u\!=\!-3$\ero}

\put(175,51){\rot{-48}$v\!=\!0$\ero} \end{picture}} \end{document}