File:Sqrt27u.png

Check of the approximate symmetry of the explicit plot of tetration to base $\sqrt{2}$ in figure http://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg

$y=\mathrm{devi}(x)=\mathrm{tet}_{\sqrt{2}}(x) + \mathrm{ate}_{\sqrt{2}}(-x) ~$ (dashed line) and

$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{ate}_{\sqrt{2}}(x))+x ~$ (solid line)

Usage: this is figure 16.6 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

This image is used also as figure 7b in the article .

C++ generator of the curves
Files ado.cin, sqrt2f21e.cin, sqrt2f21l.cin should be loaded in order to compile the code below.

typedef std::complex z_type; // #include "conto.cin" //#include "tq2e.cin"
 * 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.)
 * 1) include "ado.cin"
 * 1) include "sqrt2f21e.cin"
 * 2) include "sqrt2f21l.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("sqrt27b.eps","w"); ado(o,104,63); fprintf(o,"21 41 translate\n 10 10 scale\n"); M(0,-4.06)L(0,2.06) M(-2.06,0)L(8.06,0) fprintf(o,".014 W S\n"); for(n=-2;n<9;n++){if(n!=0){M(n,0)L(n,-.1)}} for(n=-4;n<3;n++){if(n!=0){M(0,n)L(-.1,n)}} fprintf(o,".02 W S\n");
 * 1) define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);

/* M(-2,-4)L(-2,2)L(4,2) M(-2,2)L(4,-4)

M(-2,-4)L(-2,2) fprintf(o,".006 W S\n");

fprintf(o,"1 setlinejoin 0 setlinecap\n");

for(m=0;m< 99;m+= 2){x=-1.9999+.01*m; z=x; c=F21E(-F21E(z))+z; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)} for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(-F21E(z))+z;    y=Re(c);y*=100; L(x,y)} fprintf(o,".01 W 1 0 0 RGB S\n");

//for(m=0;m< 99;m+= 2){x=-1.8+.01*m; z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)} //for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(z)+F21L(-z); y=Re(c);y*=100; L(x,y)}

//for(m=0;m<101;m+=2){x=-1.738+(.0012*m*m); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)} for(m=0;m<101;m+=2){x=-1.738+(.00097*m*(m+2)); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)} fprintf(o,"[.1 .1] 0 setdash .03 W 0 .8 0 RGB S\n");

/* for(m=0;m<101;m+=5){x=-1.85+(.0012*m*m); z=x;c=F21L(-z)/F21E(z)+1.; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)} fprintf(o,"[.1 .1] 0 setdash .02 W 0 0 1 RGB S\n");

for(m=0;m<102;m+=2){x=-1.998+(.0012*m*m); z=x; c=F21E(z); d=-F21L(-z); c=(c-d)/(c+d); y=Re(c); y*=100.; if(m==0) M(x,y) else L(x,y)} fprintf(o,"[.05 .05] 0 setdash .006 W 0 0 0 RGB S\n"); //                            c=TQ2E(z); //                            p=Re(c); // #include "plof.cin" fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt27b.eps"); system(   "open sqrt27b.pdf"); //for LINUX //     getchar; system("killall Preview"); // For macintosh }

Latex generator of the labels
\documentclass[12pt,a4paper,oneside]{book} %\newcommand \EN[1]   	% make the English version \newcommand \EN[1] –     	% suppress the English version \newcommand \RU[1]   	% make the Russian version (in this document not supported) %\newcommand \RU[1] –       	% suppress the Russian version %The Japanese version is not yet supported. While \JP is used to suppress several lines at once. \newcommand \JP[1] –    	 % ореsuppress some text %\usepackage[space]{cite }% If exist. \usepackage[utf8]{inputenc} \usepackage[T2A]{fontenc} \usepackage[russian]{babel} \usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating} \usepackage{hyperref} \usepackage{wrapfig} \usepackage{geometry} \paperwidth 145mm \paperheight 86mm \textwidth 550mm %\oddsidemargin -76pt \oddsidemargin -70pt \topmargin -97pt \textheight 457mm \pagestyle{empty}

\newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \parindent 0pt \begin{document} \sx{4}{\begin{picture}(100,60) \put(0,0){\ing{sqrt27b}} \put(17,58.8){\sx{.36}{$y$}} %\put(60,48){\sx{.4}{$y=F_{2,1}(x)+F_{2,1}^{-1}(\!-x)$}} %\put(48, 8){\sx{.4}{$y=F_{2,1}(-F_{2,1}(x))+x$}} \put(46,48){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(x)+\mathrm{ate}_{\sqrt{2}}(\!-x)$}} \put(47, 8){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{tet}_{\sqrt{2}}(x))+x$}} %\put(184,600){\sx{3.3}{$2$}} \put(12.3,50.0){\sx{.33}{$0.01$}} \put(10.0,30.0){\sx{.33}{$-0.01$}} \put(10.0,20.0){\sx{.33}{$-0.02$}} \put(10.0,10.0){\sx{.33}{$-0.03$}} %\put(160, 0){\sx{3.3}{$-4$}} \put( -1,37.4){\sx{.33}{$-2$}} \put( 9.0,37.4){\sx{.33}{$-1$}} \put(30.2,37.4){\sx{.33}{$1$}} \put(40.3,37.4){\sx{.33}{$2$}} \put(50.3,37.4){\sx{.33}{$3$}} \put(60.3,37.4){\sx{.33}{$4$}} \put(70.4,37.4){\sx{.33}{$5$}} \put(80.4,37.4){\sx{.33}{$6$}} \put(90.4,37.4){\sx{.33}{$7$}} \put(100.,37.6){\sx{.36}{$x$}} %\put(603,377){\sx{3.4}{$4$}} %\put(60.0,42.0){\sx{.42}{$x$}} \end{picture}} \end{document}