Fibonacci vs. Kilometers

This experiment was based on this Reddit posting: http://www.reddit.com/r/fifthworldproblems/comments/s0217/i_can_only_speak_in_a_recurring_meter_the/c4a4c18

Here’s a brief excerpt that sums up the point of this experiment:

Cool fact:

The Fibonacci sequence fairly accurately converts miles to kilometers. For each number in the sequence that represents miles, the next number is the same distance in kilometers

Wanting to test this, I threw together a quick Perl script to plot this out. The results were, surprisingly, “fairly accurate.”

I’ve included here a few quick plots, the numbers, and the raw source code should anyone wish to tinker further with this. Granted these are unrefined results from a presentation stand-point, but you get the point…

The raw results:

Miles, Calculated from Fibonacci, True
1,1,1.609
1,2,1.609
2,3,3.218
3,5,4.827
5,8,8.045
8,13,12.872
13,21,20.917
21,34,33.789
34,55,54.706
55,89,88.495

Miles, Percent Difference
1,-37.8495960223741
1,24.3008079552517
2,-6.77439403356122
3,3.58400662937643
5,-0.559353635798632
8,0.994406463642015
13,0.396806425395604
21,0.624463582822808
34,0.537418199100641
55,0.570653709249105
89,0.557956997507005
144,0.562806436019613
233,0.560954075386042
377,0.561661608360142
610,0.561391353961833
987,0.561494581832304
1597,0.561455152276278
2584,0.561470213023878
4181,0.561464460329799
6765,0.561466657663354
10946,0.561465818356607
17711,0.561466138943263
28657,0.561466016490064
46368,0.561466063263029
75025,0.561466045397334
121393,0.561466052221422
196418,0.561466049614862
317811,0.561466050610479
514229,0.56146605023018
832040,0.561466050375433

The source code:

 Perl | 
 
 copy code |
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#!/usr/bin/perl
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use GD::Graph::lines;
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##################
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# Computed values
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$last = 1;
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$cur = 1;
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my @a;
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print "Miles, Calculated from Fibonacci, True\n";
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for ($i = 0; $i < 10; $i++) {
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        $a[0][$i] = $last; #miles
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        $a[1][$i] = $cur;  # Fibonacci Calculated value
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        $a[2][$i] = $last * 1.609;  # computed value
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        print $a[0][$i] . "," . $a[1][$i] . "," . $a[2][$i] . "\n";
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        # increment the Fibonacci sequence
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        $x = $cur; $cur += $last; $last = $x;
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}
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my $mygraph = GD::Graph::lines->new(600, 400);
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$mygraph->set(
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    x_label     => 'Miles',
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    y_label     => 'Kilometers',
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    title       => 'Miles to Kilometers',
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) or warn $mygraph->error;
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my $myimage = $mygraph->plot(\@a) or die $mygraph->error;
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open OUTPUT, ">Fib_vs_KM.png";
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print OUTPUT $myimage->png;
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close(OUTPUT);
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#################
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# % Error 1 - 55
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print "\nMiles, Percent Difference\n";
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$last = 1;
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$cur = 1;
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my @a;
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for ($i = 0; $i < 10; $i++) {
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    $a[0][$i] = $last; #miles
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    $a[1][$i] = ($cur - ($last * 1.609)) / ($last * 1.609) * 100;  # percent difference
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        print $a[0][$i] . "," . $a[1][$i] . "\n";
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    # increment the Fibonacci sequence
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    $x = $cur; $cur += $last; $last = $x;
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}
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my $mygraph = GD::Graph::lines->new(600, 400);
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$mygraph->set(
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    x_label     => 'Miles',
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    y_label     => '% Difference',
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    title       => 'Percent Difference from Real (1 - 55 miles)',
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) or warn $mygraph->error;
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my $myimage = $mygraph->plot(\@a) or die $mygraph->error;
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open OUTPUT, ">Fib_vs_KM_diff1.png";
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print OUTPUT $myimage->png;
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close(OUTPUT);
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############
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# 89 - 6765
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for ($i = 0; $i < 10; $i++) {
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    $a[0][$i] = $last; #miles
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    $a[1][$i] = ($cur - ($last * 1.609)) / ($last * 1.609) * 100;  # percent difference
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        print $a[0][$i] . "," . $a[1][$i] . "\n";
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    # increment the Fibonacci sequence
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    $x = $cur; $cur += $last; $last = $x;
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}
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my $mygraph = GD::Graph::lines->new(600, 400);
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$mygraph->set(
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    x_label     => 'Miles',
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    y_label     => '% Difference',
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    title       => 'Percent Difference from Real (89 - 6765 miles)',
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) or warn $mygraph->error;
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my $myimage = $mygraph->plot(\@a) or die $mygraph->error;
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open OUTPUT, ">Fib_vs_KM_diff2.png";
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print OUTPUT $myimage->png;
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close(OUTPUT);
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#################
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# 10946 - 832040
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for ($i = 0; $i < 10; $i++) {
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    $a[0][$i] = $last; #miles
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    $a[1][$i] = ($cur - ($last * 1.609)) / ($last * 1.609) * 100;  # percent difference
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        print $a[0][$i] . "," . $a[1][$i] . "\n";
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    # increment the Fibonacci sequence
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    $x = $cur; $cur += $last; $last = $x;
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}
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my $mygraph = GD::Graph::lines->new(600, 400);
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$mygraph->set(
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    x_label     => 'Miles',
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    y_label     => '% Difference',
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    title       => 'Percent Difference from Real (10946 - 832040 miles)',
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) or warn $mygraph->error;
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my $myimage = $mygraph->plot(\@a) or die $mygraph->error;
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open OUTPUT, ">Fib_vs_KM_diff3.png";
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print OUTPUT $myimage->png;
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close(OUTPUT);
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