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DescriptionOmega-exp-omega-normal.pdf	English: Matchstick representation of ordinal numbers up to $\omega ^{\omega }$ . To save horizontal space, the number line has been coiled up to a spiral. One turn corresponds to the mapping $f(\alpha )=\omega (1+\alpha )$ . Since $f$ has $\omega ^{\omega }$ as a (least) fixed point, larger ordinal numbers cannot be represented in this image.
Date	8 February 2023
Source	Own work, inspired by File:Omega-exp-omega-labeled.svg
Author	Jochen Burghardt
Other versions	File:Omega-exp-omega-normal.pdf * File:Omega-exp-omega-normal svg.svg

LaTeX source code

\documentclass[12pt]{article}
\usepackage{pstricks}
\usepackage[paperwidth=110mm,paperheight=95mm,textwidth=110mm,textheight=95mm]{geometry}
\setlength{\parindent}{0cm}
\psset{unit=1mm}
\pagestyle{empty}

\makeatletter
\def\ite#1#2#3{\setbox\@tempboxa\hbox{#1}%
\ifdim\wd\@tempboxa=0cm#3\else#2\fi\relax}
\makeatother

%         1      2    3    4         5        6      7
% \spiral{ALPHA}{BOT}{TOP}{RGBspoke}{RGBtext}{ALIGN}{TEXT}
\newcommand{\spiral}[7]{%
        \definecolor{JBspoke}{rgb}{#4}%
        \definecolor{JBtext}{rgb}{#5}%
        \psline[linecolor=JBspoke](#2;#1)(#3;#1)%
        \ite{#7}{\rput[#6](#3;#1){\textcolor{JBtext}{#7}}}{}%
}

%         1      2   3    4    5   6            7        8      9
% \Spiral{ALPHA}{AL}{MID}{TOP}{WH}{RGBtriangle}{RGBtext}{ALIGN}{TEXT}
\newcommand{\Spiral}[9]{%
        \definecolor{JBspoke}{rgb}{#6}%
        \definecolor{JBtext}{rgb}{#7}%
        \pstriangle*[gangle=#2,linecolor=JBspoke](#3;#1)(#5)%
        \ite{#9}{\rput[#8](#4;#1){\textcolor{JBtext}{#9}}}{}%
}%

\begin{document}
\begin{pspicture}(-50,45)(60,50)%
\psset{linewidth=0.1pt}
        %\rput(-50,-45){$+$}
        %\rput(60,50){$+$}

% !}texfractal -spiral 50.000 0.650
%%%%% output of C program "texfractal" omitted, for now %%%%%

\rput(0,0){$\scriptscriptstyle\omega^\omega$}
\end{pspicture}
\end{document}

C source code

#include <stdio.h>
#include <math.h>
#include <string.h>

#define Path            "texfractal"

/* ***************************************************************** */
/* ***** matchstick spiral **************************************@@* */
/* ***************************************************************** */

/* return y such that y0 : y : y1 corresponds to x0 : x : x1 */
static inline double intpl(
    double y0, 
    double y1, 
    double x,
    double x0, 
    double x1) 
{
    double const y = ((x-x0) * y1 + (x1-x) * y0) / (x1-x0);
    return y;
}

static inline const char * angle2align(
    double a)
{
    return   a <= -22.5 ? "<?"
           : a <=  22.5 ? " l"
           : a <=  67.5 ? "bl"
           : a <= 112.5 ? "b "
           : a <= 157.5 ? "br"
           : a <= 202.5 ? " r"
           : a <= 247.5 ? "tr"
           : a <= 292.5 ? "t "
           : a <= 337.5 ? "tl"
           : a <= 382.5 ? " l"
           :              ">?";
}

static inline void normalizeAngle(
    double *a) 
{
    while (*a < 0)
        *a += 360;
    while (360 < *a) 
        *a -= 360;
}

/* segment of matchsticks */
static inline void mSeg(
    double *r0,         /* spiral start/end radius (in/out) */
    double qr,          /* spiral radius quotient per 360 deg */
    double *a0,         /* start/end angle (in/out) */
    double da,          /* angle increment (for 1) */
    double af,          /* total angle incrment (for infty) */
    double h0,          /* start spoke height */
    double qh,          /* height quotient */
    int m,              /* label count */
    char * lb[m],       /* spoke labels */
    double R0,
    double G0,
    double B0,
    double R360,
    double G360,
    double B360,
    int n,              /* spoke count */
    double aSpoke[n])   /* spoke angle (output only) */
{
    int i;

    double const qa = (af - da) / af;
    long double const lnr0 = logl(*r0);
    long double const lnqr = logl(qr);
    double a = *a0;
    double h = h0;
    double sda = 0.0;
    printf("%% mSeg");
    printf("  %5.3f %5.3f",*r0,qr);
    printf("  %5.3f %5.3f %5.3f",*a0,da,af);
    printf("  %5.3f %5.3f",h0,qh);
    printf("  %d\n",n);
    for (i=0; i<n; ++i) {
        if (aSpoke != NULL)
            aSpoke[i] = da;
        long double const r = expl(lnr0 + lnqr * sda / 360.0);
        double const Rs = intpl(R0,R360,a,0,360);
        double const Gs = intpl(G0,G360,a,0,360);
        double const Bs = intpl(B0,B360,a,0,360);
        double const Rt = intpl(Rs,1,i,0,m+1);
        double const Gt = intpl(Gs,1,i,0,m+1);
        double const Bt = intpl(Bs,1,i,0,m+1);
        const char * const align = angle2align(a);
        printf("\\spiral");
        printf("{%7.3f}",a);
        printf("{%7.3Lf}",r-h/2);
        printf("{%7.3Lf}",r+h/2);
        printf("{%4.2f,%4.2f,%4.2f}",Rs,Gs,Bs);
        printf("{%4.2f,%4.2f,%4.2f}",Rt,Gt,Bt);
        printf("{%s}",align);
        if (i < m)
            printf("{$%s$}",lb[i]);
        else
            printf("{}");
        printf("%%\n");
        h *= qh;
        a += da;
        sda += da;
        normalizeAngle(&a);
        da *= qa;
    }
    long double const r = expl(lnr0 + lnqr * sda / 360.0);
    *r0 = r;
    *a0 = *a0 + af;
    normalizeAngle(a0);
}

/* segment of triangles */
static inline void tSeg(
    double *r0,         /* spiral start radius */
    double qr,          /* spiral radius quotient per 360 deg */
    double a0,          /* start angle */
    double h0,          /* start spoke height */
    double qh,          /* height quotient */
    int m,              /* label count */
    char * lb[m],       /* labels */
    double R0,
    double G0,
    double B0,
    double R360,
    double G360,
    double B360,
    int n,              /* triangle count */
    double const aSpoke[n])
{
    int i;

    long double const lnr0 = logl(*r0);
    long double const lnqr = logl(qr);
    double h = h0;
    double a = a0;
    double sda = 0.0;
    printf("%% tSeg");
    printf("  %5.3f %5.3f",*r0,qr);
    printf("  %5.3f %5.3f",h0,qh);
    printf("  %d\n",n);
    for (i=0; i<n; ++i) {
        long double const r = expl(lnr0 + lnqr * sda / 360.0);
        double const Rs = intpl(R0,R360,a,0,360);
        double const Gs = intpl(G0,G360,a,0,360);
        double const Bs = intpl(B0,B360,a,0,360);
        double const Rt = intpl(Rs,1,i,0,m+1);
        double const Gt = intpl(Gs,1,i,0,m+1);
        double const Bt = intpl(Bs,1,i,0,m+1);
        const char * const align = angle2align(a);
        printf("\\Spiral");
        printf("{%7.3f}",a);
        printf("{%7.3f}",a+aSpoke[i]/2);
        printf("{%7.3Lf}",r);
        printf("{%7.3Lf}",r+h/2);
        printf("{%7.3f,%7.3Lf}",h,r*sin(aSpoke[i]*2*M_PI/360));
        printf("{%4.2f,%4.2f,%4.2f}",Rs,Gs,Bs);
        printf("{%4.2f,%4.2f,%4.2f}",Rt,Gt,Bt);
        printf("{%s}",align);
        if (i < m)
            printf("{$%s$}",lb[i]);
        else
            printf("{}");
        printf("%%\n");
        h *= qh;
        a += aSpoke[i];
        sda += aSpoke[i];
    }
    long double const r = expl(lnr0 + lnqr * sda / 360.0);
    *r0 = r;
}

static inline void matchstickSpiral(
    double r0,          /* spiral start radius */
    double qr)          /* spiral radius quotient per 360 deg */
{
    double da;
    double a0 = 0.0;
    double h0;
    double aSpoke0[220];
    double aSpoke1[25][220];
    int i;

    printf("%% !}" Path " -spiral %5.3f %5.3f\n",r0,qr);

    /* ------------------------------------------------------------- */
    printf("%% 0 ... omega\n");
    char * lb0[] = { "0", "1", "2", "3", "4",
                     "5", "6", "7", "8", "9"};
    mSeg(&r0,qr,
         &a0,70,360.000,
         8,0.90,
         10,
         lb0,
         0.0,0.0,0.0, 0.8,0.0,0.0,
         40,aSpoke0);

    /* ------------------------------------------------------------- */
    printf("%% omega ... omega^2\n");
    char * lb11[] = { "\\omega",
                      "\\omega\\!\\!+\\!\\!1",
                      "\\omega\\!\\!+\\!\\!2",
                      "\\omega\\!\\!+\\!\\!3",
                      "\\omega\\!\\!+\\!\\!4"};
    char * lb14[] = { "\\omega\\!\\cdot\\! 4",
                      "\\omega\\!\\cdot\\! 4\\!\\!+\\!\\!1",
                      "\\omega\\!\\cdot\\! 4\\!\\!+\\!\\!2",
                      "\\omega\\!\\cdot\\! 4\\!\\!+\\!\\!3",
                      "\\omega\\!\\cdot\\! 4\\!\\!+\\!\\!4"};
    char lb1buf[64];
    char * lb1[] = { lb1buf };
    h0 = 6.0;
    a0 = 0;
    da = 15;
    for (i=0; i<25; ++i) {
        sprintf(lb1buf,"\\omega\\!\\cdot\\! %d",i+1);
        mSeg(&r0,qr,
             &a0,da,aSpoke0[i],
             h0,0.90,
             (i==0 ?5    :i==3 ?2    :i<7 ?1   :0),
             (i==0 ?lb11 :i==3 ?lb14 :i<7 ?lb1 :NULL),
             0.8,0.0,0.0, 0.5,0.5,0.0,
             25-i/2,aSpoke1[i]);
        da *= 0.80;
        h0 *= 0.90;
    }
    mSeg(&r0,qr,
         &a0,da,360.0-a0,
         h0,0.98,
         0,
         NULL,
         0.8,0.0,0.0, 0.5,0.5,0.0,
         30,NULL);

    /* ------------------------------------------------------------- */
    printf("%% omega^2 ... omega^3\n");
    char * lb21[] = { "\\scriptstyle\\omega^2",
                     "\\scriptstyle\\omega^2\\!+\\!\\omega",
                     "\\scriptstyle\\omega^2\\!+\\!\\omega\\cdot 2",
                     "\\scriptstyle\\omega^2\\!+\\!\\omega\\cdot 3" };
    char lb2buf[64];
    char * lb2[] = { lb2buf };
    a0 = 0;
    h0 = 4.0;
    for (i=0; i<25; ++i) {
        sprintf(lb2buf,"\\scriptstyle\\omega^2\\!\\cdot %d",i+1);
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             (i==0 ?4    :i<7 ?1   :0),
             (i==0 ?lb21 :i<7 ?lb2 :NULL),
             0.5,0.5,0.0, 0.0,0.8,0.0,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.80;
    }

    /* ------------------------------------------------------------- */
    printf("%% omega^3 ... omega^4\n");
    char * lb31[] = { "\\scriptstyle\\omega^3" };
    char lb3buf[64];
    char * lb3[] = { lb3buf };
    a0 = 0;
    h0 = 3.0;
    for (i=0; i<25; ++i) {
        sprintf(lb3buf,"\\scriptscriptstyle\\omega^3\\!\\cdot %d\\!",i+1);
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             (i==0 ?1    :i<7 ?1   :0),
             (i==0 ?lb31 :i<7 ?lb3 :NULL),
             0.0,0.8,0.0, 0.0,0.5,0.5,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.80;
    }

    /* ------------------------------------------------------------- */
    printf("%% omega^4 ... omega^5\n");
    char * lb41[] = { "\\scriptscriptstyle\\!\\omega^4" };
    a0 = 0;
    h0 = 2.0;
    for (i=0; i<25; ++i) {
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             (i==0 ?1    :0),
             (i==0 ?lb41 :NULL),
             0.0,0.5,0.5, 0.0,0.0,0.8,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.90;
    }

    /* ------------------------------------------------------------- */
    printf("%% omega^5 ... omega^6\n");
    a0 = 0;
    h0 = 1.5;
    for (i=0; i<25; ++i) {
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             0,
             NULL,
             0.0,0.0,0.8, 0.5,0.0,0.5,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.90;
    }

    /* ------------------------------------------------------------- */
    printf("%% omega^6 ... omega^7\n");
    a0 = 0;
    h0 = 1.0;
    for (i=0; i<25; ++i) {
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             0,
             NULL,
             0.5,0.0,0.5, 0.0,0.0,0.0,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.90;
    }

    /* ------------------------------------------------------------- */
    printf("%% omega^7 ... omega^8\n");
    a0 = 0;
    h0 = 0.5;
    for (i=0; i<25; ++i) {
        tSeg(&r0,qr,
             a0,
             h0,0.90,
             0,
             NULL,
             0.0,0.0,0.0, 1.0,1.0,1.0,
             20,aSpoke1[i]);
        a0 += aSpoke0[i];
        h0 *= 0.90;
    }

    /* ------------------------------------------------------------- */
    printf("%% done\n");
}

/* ***************************************************************** */
/* ***** main ***************************************************@@* */
/* ***************************************************************** */

int main(
    int argc,
    const char * const argv[argc])
{
    int i;

    for (i=1; i<argc; ++i) {
        if (strcmp(argv[i],"-spiral") == 0) {
            double r0;
            double qr;
            sscanf(argv[i+1]," %lf",&r0);
            sscanf(argv[i+2]," %lf",&qr);
            i += 2;
            matchstickSpiral(r0,qr);
        } else {
            fprintf(stderr,"illegal argument '%s'",argv[i]);
            return 1;
        }
    }
    return 0;
}

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

to share – to copy, distribute and transmit the work
to remix – to adapt the work

Under the following conditions:

attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	11:55, 9 February 2023		647 × 560 (95 KB)	Jochen Burghardt (talk \| contribs)	fixed mult opnds; darker colors
	18:03, 8 February 2023		647 × 560 (93 KB)	Jochen Burghardt (talk \| contribs)	Uploaded own work with UploadWizard

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Short title	Omega-exp-omega-normal.dvi
File change date and time	13:52, 9 February 2023
Date and time of digitizing	13:52, 9 February 2023
Software used	dvips(k) 2021.1 (TeX Live 2022/dev) Copyright 2021 Radical Eye Software
Conversion program	GPL Ghostscript 9.55.0
Encrypted	no
Page size	311.81 x 269.29 pts
Version of PDF format	1.7

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