1, 1, 2, 3, 5, 8, 13, ...

The ratio of consecutive terms in the Fibonacci series converges on the Golden Ratio, which has a value of approximately 1.618. It also shows up in a geometric analysis of the pentacle. These properties are discussed in mathematical detail on the following Web pages:

the Golden Ratio

pentacle geometry

I wanted to see this from a computational point of view, so I wrote a quick Perl program to compute the Fibonacci numbers and ratios out to around 100 terms. The program prints out deviations from the Golden Ratio, and the output shows a near-zero deviation after less than 40 terms.

#!/usr/bin/perl -w

use strict;

use diagnostics;

my $NUM_ITERATIONS = 100;

my $GOLDEN_RATIO = 0.5 * (1.0 + sqrt(5));

my ($previous_term, $current_term) = (1, 1);

my $iteration_number = 1;

while ( $iteration_number <= $NUM_ITERATIONS ) {

my $new_term = $previous_term + $current_term;

my $ratio = $new_term / $current_term;

my $deviation = abs($GOLDEN_RATIO-$ratio) / $GOLDEN_RATIO;

printf "% 3d: %e\n", $iteration_number, $deviation;

$previous_term = $current_term;

$current_term = $new_term;

$iteration_number++;

}