File:Penmap.jpg

Complex map of natural pentation,

$u\!+\!\mathrm i v=\mathrm{pen}(x\!+\! \mathrm i y)$ in the $x,y$ plane.

This image is copied from http://mizugadro.mydns.jp/t/index.php/File:Penmap.jpg It is used as figure 7 at the article about holomorphic ackermanns ^[1]

References

C++ generator of curves

Files ado.cin, conto.cin, fsexp.cin, fslog.cin should be loaded to the working directory in order to compile the code below <poem> #include <math.h> #include <stdio.h> #include <stdlib.h> #define DB double #define DO(x,y) for(x=0;x<y;x++) #include <complex> typedef std::complex<double> z_type; #define Re(x) x.real() #define Im(x) x.imag() #define I z_type(0.,1.) #include "conto.cin" #include "fsexp.cin" #include "fslog.cin" z_type pen0(z_type z){ DB Lp=-1.8503545290271812; DB k,a,b; // k=1.86573322821; a=-.62632418; b=0.4827; k=1.86573322821; a=-.6263241; b=0.4827; z_type e=exp(k*z); return Lp + e*(1.+e*(a+b*e)); } z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.)); DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;} return z; } z_type pen(z_type z){ DB x; int m,n; x=Re(z); if(x<= -4.) return pen0(z); m=int(x+5.); z-=DB(m); z=pen0(z); DO(n,m) z=FSEXP(z); return z; } int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=401,M1=M+1; int N=801,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("penma.eps","w"); ado(o,828,828); fprintf(o,"422 420 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-4.+.02*(m-.5); DO(n,N1) Y[n]=-4.+.01*(n-.5); for(m=-4;m<5;m++) {M(m,-4)L(m,4)} for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,"2 setlinecap .004 W 0 0 0 RGB S\n"); 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); // c=pen0(z); // c=FSEXP(pen0(z-1.)); // c=FSEXP(FSEXP(pen0(z-2.))); c=pen7(z); // d=FSEXP(pen(z-1.)); // p=abs((c-d)/(c+d)); p=-log(p)/log(10.); p=Re(c); q=Im(c); if(p>-9999 && p<9999 && fabs(p)>1.e-11) g[m*N1+n]=p; if(q>-9999 && q<9999 && fabs(q)>1.e-11) f[m*N1+n]=q; }} // #include "plofu.cin" fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2;q=.5; for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".002 W 0 .6 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".002 W .9 0 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".002 W 0 0 .9 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".012 W .9 0 0 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".012 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".012 W .6 0 .6 RGB S\n"); for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".012 W 0 0 0 RGB S\n"); DB t2=M_PI/1.86573322821; DB tx=-2.32; M(tx,t2)L(4.1,t2) M(tx,-t2)L(4.1,-t2) fprintf(o,"0 setlinecap .03 W 1 1 1 RGB S\n"); DO(n,64){ x=tx+.1*n; M(x,t2) L(x+.04,t2) } DO(n,64){ x=tx+.1*n; M(x,-t2) L(x+.04,-t2) } fprintf(o,"0 setlinecap .04 W 0 0 0 RGB S\n"); //conto(o,g,w,v,X,Y,M,N, ( 1. ),-99,99); fprintf(o,".12 W 1 .5 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); printf("pen7(-1)=%18.14f\n", Re(pen7(-1.))); printf("Pi/1.86573322821=%18.14f\n", M_PI/1.86573322821); system("epstopdf penma.eps"); system( "open penma.pdf"); } </poem>

Latex generator of labelw

<poem> \documentclass[12pt]{article} \paperheight 832px \paperwidth 846px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -80px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} {\begin{picture}(824,820) %\put(12,0){\ing{24}} \put(12,0){\ing{penma}} \put(8,808){\sx{3}{$y$}} \put(8,709){\sx{3}{$3$}} \put(8,609){\sx{3}{$2$}} \put(8,509){\sx{3}{$1$}} \put(8,409){\sx{3}{$0$}} \put(-12,309){\sx{3}{$-1$}} \put(-12,209){\sx{3}{$-2$}} \put(-12,109){\sx{3}{$-3$}} \put(-12,9){\sx{3}{$-4$}} \put(4,-8){\sx{3}{$-4$}} \put(104,-8){\sx{3}{$-3$}} \put(204,-8){\sx{3}{$-2$}} \put(304,-8){\sx{3}{$-1$}} \put(427,-8){\sx{3}{$0$}} \put(527,-8){\sx{3}{$1$}} \put(627,-8){\sx{3}{$2$}} \put(727,-8){\sx{3}{$3$}} \put(821,-8){\sx{3}{$x$}} \put(50, 747){\sx{4}{$v\!=\!0$}} \put(50, 578){\sx{4}{$v\!=\!0$}} \put(760, 580){\sx{4}{\bf cut}} \put(50, 409){\sx{4}{$v\!=\!0$}}% \put(50, 240){\sx{4}{$v\!=\!0$}} \put(760, 241){\sx{4}{\bf cut}} \put(50, 71){\sx{4}{$v\!=\!0$}} % \put(326, 638){\sx{4}{$v\!=\!-1$}} \put(340, 520){\sx{4}{$v\!=\!1$}} \put(326, 298){\sx{4}{$v\!=\!-1$}} \put(336, 182){\sx{4}{$v\!=\!1$}} % \put(250, 352){\sx{4}{\rot{90}$u\!=\!-1$\ero}} \put(348, 362){\sx{4}{\rot{90}$u\!=\!0$\ero}} \put(448, 372){\sx{4}{\rot{90}$u\!=\!1$\ero}} \put(522, 372){\sx{4}{\rot{90}$u\!=\!2$\ero}} \end{picture} \end{document} </poem>