Visualizing $\sqrt{2}$, $\pi$, and $e$ Using Random Walk 〜1 Million Digits〜 [C, gnuplot]

YouTube

Simulation [gnuplot]

Case 1 : $\sqrt{2}$

Make DAT file [C]

Get "r2_1M.dat" (line:21) → The Square Root of Two to 1 Million Digits (APOD)
▼ RW-r2.c

#include <stdio.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>
#define LIMIT 1000000  // Digits
#define LENGTH 80      // The number of characters per line
#define NUM0 4         // Number of leading zeros
  //  Link(APOD)  :  1.414213562373095048...
  // -> r2_1M.dat:0000414213562373095048...
#define PI 3.1415

int main(void){
    FILE *fp, *fq;
    char str[256];
    double x = 0;    // Position.x
    double y = 0;    // Position.y
    int i, j, k;     // For Loop counter
    int digit;       // Current digit
    int num;         // Number
    
    fp = fopen("r2_1M.dat", "r");          // Make this file by yourself
    fq = fopen("randomwalk-r2.dat", "w");

    if(fp==NULL){
        printf("File not found. Program terminated.\n");
        exit(1);
    }

    fprintf(fq, "%.3lf\t%.3lf\n",x, y);     // Initiate value
    
    // i: row, j: column, k: 0 - 9
    for(i=1; fgets(str, sizeof(str), fp)!=NULL; i++){
        for (j=0; j<LENGTH; j++){
            // Calculate current digit
            digit = (LENGTH*(i-1)+j+1) - NUM0;

            // Ignore leading zero NUM0 times
            if(digit<=0) continue;
            
            // From (NUM0+1)th digit on
            for(k=0; k<10; k++){             // Check matching between k and num
                num = str[j] - '0';

                if(num ==k){                 // Calculate x and y
                    x += cos(2*PI*k/10);
                    y += sin(2*PI*k/10);
                    fprintf(fq, "%.3lf\t%.3lf\t%d\t%7d\n",x, y, k, digit-NUM0); 
                    break;
                } 
            }
            if(digit==LIMIT) break;
        }
        if(digit==LIMIT) break;
        
    }
    fclose(fq);
    fclose(fp);
    return 0;
}

→make "randomwalk-r2.dat"

Draw $\sqrt{2}$ random walk [gnuplot]

▼ RW-r2.plt

# Setting --------------------
reset
set terminal gif animate delay 6 size 1280, 720
set output "randomwalk-pi.gif"
set size ratio 1
set nokey

set xr[-2000:200]
set yr[-600:1600]
set tics font 'Times, 20'
unset xl
unset yl

N = 1000000                      # Digits
dec = 200                        # Decimation

#system "mkdir png"    # Make folder for storing PNG files

DIGIT(i) = sprintf("Digits: %d\n", i)    # Label displaying digits

# Plot --------------------
do for[i=0:N]{
    # Digits
    set label 1 left DIGIT(i) font 'Times, 22' at graph 1.04, 0.03
    
    # Decimate and draw (the number of files: N/200+1)
    if(i%dec==0){        
        # Draw initiate value
        if(i==0){
            plot 1/0
            continue
        }
        
        # Update
        plot 'randomwalk-r2.dat' every ::0::i-1 using 1:2 with lines lc rgb "red"
    }
}

set out

▼ randomwalk-r2.gif (dec=5000)

Case 2 : $\pi$

Make DAT file [C]

Get "pi_1M.dat" (line:23) → 1 Million Digits of Pi (Pi Land by Eve)
▼ RW-pi.c

#define LENGTH 50      // The number of characters per line
#define NUM0 0         // Number of leading zeros
  // Link(Pi Land) :141592653589793238462643...
  // ->  pi_1M.dat :141592653589793238462643...

    fp = fopen("pi_1M.dat", "r");
    fq = fopen("randomwalk-pi.dat", "w");

→make "randomwalk-pi.dat"

Draw $\pi$ random walk [gnuplot]

▼ RW-pi.plt

        plot 'randomwalk-pi.dat' every ::0::i-1 using 1:2 with lines lc rgb "blue"

▼ randomwalk-pi.gif (dec=5000)

Case 3 : $e$

Make DAT file [C]

Get "e_1M.dat" (line:23) → The Number e to 1 Million Digits (APOD)
▼ RW-pi.c

#define LENGTH 80      // The number of characters per line
#define NUM0 4         // Number of leading zeros
  //  Link(APOD) :  2.718281828459045235360...
  // -> e_1M.dat :0000718281828459045235360...

    fp = fopen("e_1M.dat", "r");
    fq = fopen("randomwalk-e.dat", "w");

→make "randomwalk-e.dat"

Draw $e$ random walk [gnuplot]

▼ RW-e.plt

        plot 'randomwalk-e.dat' every ::0::i-1 using 1:2 with lines lc rgb "green"

▼ randomwalk-e.gif (dec=5000)

Extra

Draw random walks of $\sqrt{2}$, $\pi$, and $e$ simultaneously

Read DAT files created already and plot them

▼ 3RWs.plt

# Setting --------------------
reset
set terminal png enhanced font "Times" 20 size 720, 720
set xr[-2000:200]
set yr[-800:1400]
set size ratio 1
set tics font 'Times, 20'
set xtics 400
set ytics 400
unset key

# Axes
set arrow 1 nohead from 0, -800 to 0, 1400 lc rgb "black" lw 2
set arrow 2 nohead from -2000, 0 to 200, 0 lc rgb "black" lw 2

# Digits
N = 1000000

# Arrays
array num[3]    = ["r2", "pi", "e"]             # Number
array color[3]  = ["red", "blue", "green"]      # Color

# Functions --------------------
PNG(i) = sprintf("img_%04d.png", i)             # Output img_000i.png
DAT(i) = sprintf("randomwalk-%s.dat", num[i])   # num[i] -> name of DAT files

# Plot DAT files --------------------
do for[i=1:3]{
    set output PNG(i)
    str = "plot "              # Command (need space)

    # Make the command using string concatenation ("A"."B" -> "AB")
    do for [j=1:i] {
        if(j!=1){
            str = str.", "     # Need ", " for plotting DAT files together
        }
        str = str.sprintf("'%s' every ::0::N with lines lc rgb '%s' lw 0.5", DAT(j), color[j])
    }

    eval str                   # Execute this command
    # i=1: plot 'randomwalk-r2.dat' every ::0::N with lines lc rgb 'red' lw 0.5
    # i=2: plot 'randomwalk-r2.dat' every ::0::N with lines lc rgb 'red' lw 0.5, 
    #           'randomwalk-pi.dat' every ::0::N with lines lc rgb 'blue' lw 0.5
    # i=3: plot 'randomwalk-r2.dat' every ::0::N with lines lc rgb 'red' lw 0.5, 
    #           'randomwalk-pi.dat' every ::0::N with lines lc rgb 'blue' lw 0.5,
    #           'randomwalk-e.dat' every ::0::N with lines lc rgb 'green' lw 0.5
}

set out

Output PNG files

img_0001.png ($\sqrt{2}$)	img_0002.png ($\sqrt{2}, \pi$)	img_0003.png ($\sqrt{2}, \pi, e$)