File:Tet5loplot.jpg

Graphical search for the real fixed points of tetration:

$y=\mathrm{tet}_b(x)$ for various $b$ and line $y\!=\!x$.

Notations:

$\mathrm e\!=\!\exp(1)\!\approx\!2.71$ is base of the natural logarithm,

$L_{\mathrm e,0}\approx -1.8503545290271812$ is the only real fixed point of the natural tetration, $b\!=\!\mathrm e$

$\tau\!\approx\! 1.63532$ is crytical base; at $b\!=\!\tau$, tetration has 2 real fixed points:

Regular one at $L_{\tau,0}\!\approx\! -1.7$ and exotic one $L_{\tau,1}\!\approx\!3.087$

At smaller base $b$, function $\mathrm{tet}_b$ has 3 regular real fixed points; in this sense, variety of supertetrations is richer, than that of superexponentials.

An $b$ decreases, approaching the Henryk constant $\eta=\exp(1/\mathrm e)$, the biggest fixed point runs to infinity; and at $1\!<\!b\!\le\! \eta$, function $\mathrm{tet}_b$ has two real regular fixed points.

C++ generator of curves
Files ado.cin and fit1.cin should be loaded in order to compile the code below:

//using namespace std; typedef std::complex z_type; //b=10 //#include "f4ten.cin" int main{ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d; FILE *o;o=fopen("tet5loplo.eps","w");ado(o,708,708); fprintf(o,"204 204 translate\n 100 100 scale\n"); fprintf(o,"2 setlinecap\n"); for(m=-2;m<6;m++){if(m!=0){M(m,-2)L(m,5)}} for(n=-2;n<6;n++){if(n!=0){M(-2,n)L(5,n)}} fprintf(o,".006 W 0 0 0 RGB S\n"); M(-2,0)L(5.1,0) M(0, -2)L(0,5.1)            fprintf(o,".01  W 0 0 0 RGB S\n"); M(0,M_E)L(1.,M_E)                            fprintf(o,".006  W 0 0 0 RGB S\n"); fprintf(o,"1 setlinejoin 1 setlinecap\n"); DO(m,300){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.63532),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,300){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(x>5.1 || y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,300){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W 0 0 .7 RGB S\n"); DO(m,300){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .8 0 0 RGB S\n"); DO(m,340){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>5.) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 0 RGB S\n"); M(-2,-2)L(5,5) fprintf(o,".01 W 0 0 0 RGB S\n"); x=3.087; M(x,0) L(x,x) L(0,x) fprintf(o,".001 W 0 0 0 RGB S\n"); DB Lt=-1.8503545290271812; M(Lt,0) L(Lt,Lt) L(0,Lt); fprintf(o,".001 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf tet5loplo.eps"); system(   "open tet5loplo.pdf"); //mac getchar; system("killall Preview");// mac }
 * 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.)
 * 1) include "fit1.cin"
 * 2) include "ado.cin"
 * 3) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 4) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 5) define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} % See geometry.pdf \geometry{letterpaper} % ... or a4paper or a5paper or ... ?? \usepackage{graphicx} \usepackage{amssymb} \usepackage{hyperref} \usepackage{rotating} \usepackage[utf8x]{inputenc} \usepackage[english,russian]{babel} %some packages are not used \usepackage{color} \definecolor{red}{rgb}{1,0.1,0.1} \definecolor{black}{rgb}{0,0,0} \definecolor{white}{rgb}{1,1,1} \definecolor{yellow}{rgb}{1,.93,0} \definecolor{bluedark}{rgb}{0,0,.87} \paperwidth 712pt \paperheight 716pt \topmargin -98pt \oddsidemargin -72pt \textwidth 810pt \textheight 870pt

\newcommand \sx {\scalebox} \newcommand \ing {\includegraphics} \newcommand \tet {\mathrm{tet}} \newcommand \pen {\mathrm{pen}} \newcommand \bC {\mathbb C} \newcommand \fac {\mathrm {Factorial}} \newcommand \rme {\mathrm e} \newcommand \rmi {\mathrm i} \newcommand \ds {\displaystyle} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}}

\begin{document} \parindent 0pt {\normalsize \begin{picture}(702,700)\put(0,0){\ing{tet5loplo}} \put( 172,680){\sx{4}{$y$}} \put( 172,590){\sx{4}{$4$}} \put( 182,467){\sx{4}{e}} \put( 172,390){\sx{4}{$2$}} \put( 172,190){\sx{4}{$0$}} \put( 194,164){\sx{4}{$0$}} \put( 394,164){\sx{4}{$2$}} \put( 594,164){\sx{4}{$4$}} \put( 686,166){\sx{4}{$x$}} \put(344,570){\sx{4}{\rot{80} $b\!=\! \rme$ \ero } } \put(522,574){\sx{4}{\rot{66} $b\!=\!1.7$ \ero } } \put(600,603){\sx{4}{\rot{64} $b\!=\!\tau$ \ero } } \put(550,492){\sx{4}{\rot{42} $b\!=\!1.6$ \ero } } %\put(574,422){\sx{4}{\rot{13} $b\!=\!1.5$ \ero } } \put(580,438){\sx{4}{\rot{14} $b\!=\!1.5$ \ero } } \put(592,400){\sx{4}{\rot{7} $b\!=\! \eta$ \ero}} \put(574,345){\sx{4}{\rot{2}$b\!=\!\sqrt{2}$\ero}} \put(130,504){\sx{4}{$L_{\tau,1}$}} \put(500,170){\sx{4}{$L_{\tau,1}$}} \put(-2,219){\sx{4}{$L_{\mathrm e,0}$}} \put(202,6){\sx{4}{$L_{\mathrm e,0}$}} \end{picture}} \end{document}