File:ExpQ2plotT.png

Explicit plot of exponential to base $b\!=\!\sqrt{2} \approx 1.414213562373095$

The thick curve: $y=\exp_b(x)$.

The thin line shows the identical funciton, $y\!=\!x$.

The fixed points $L\!=\!2$ and $L\!=\!4$ are solutions of the equation

$\exp_b(L)=L$

for $b=\sqrt{2}$.

Any of these fixed points can be used to construct a superexponential to base $\sqrt{2}~$ ^[1].

References

http://en.wikipedia.org/wiki/Square_root_of_2

http://en.wikipedia.org/wiki/Exponential_function

C++ generator of curves

#include<math.h>
#include<stdio.h>
#include<stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include "ado.cin"

DB B=sqrt(2.);

main(){ int m,n; double x,y; FILE *o;
o=fopen("ExpQ2plot.eps","w"); ado(o,1204,804);
fprintf(o,"602 2 translate 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
for(m=-6;m<7;m++) {M(m,0)L(m,8)}
for(m=0;m<9;m++) {M(-6,m)L(6,m)}
fprintf(o,"2 setlinecap .01 W S\n 2 setlinecap 1 setlinejoin \n");
 for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W .8 0 0 RGB S\n");
M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  system("epstopdf ExpQ2plot.eps");
  system(    "open ExpQ2plot.pdf");            
  getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% file ExpQ2plot.pdf should be generated with the code above in order to compile the Latex document below.
% Copyleft 2012 by Dmitrii Kouznetsov

%