  The Golden Ratio and the Fibonacci SequenceDo you know what a Fibonacci Sequence is? a Fibonacci Sequence is a sequence of numbers in which every element of the sequence after the first two is the sum of the two preceding elements. The general example of a Fibonacci sequence starts with the number one as the first two elements. So it looks like this:1 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21 13 + 21 = 34 21 + 34 = 55 34 + 55 = 89 And so on... Now, if you've been reading from the beginning, some of those numbers should look very familiar. 3, 8, 5, 13, 34, 55, and 89 were all numbers that showed up in the geometry of pine cones, pineapples, and sunflowers! Can you guess where this is going? The image below shows a spreadsheet I created of successive ratios between Fibonacci Numbers. The first column is the Fibonacci Numbers, and the second column is the ratio of the numbers. As you can see, this ratio is getting closer and closer to The Golden Ratio.  "But wait a minute!" you say. "Do we have to start with ones as the first two numbers in the sequence?" An excellent question! Let's find out. Instead of starting with ones, let's pick some random numbers. Let's make the first element of the sequence 51, and the second will be 19. (Notice, just to be completely different, I made the second number smaller than the first one - let's see what happens!)  As you can see, it takes a bit longer for the ratio to stablize around The Golden Ratio, but it still does zero in on our special friendly number! I'm not quite finished with Fibonacci - visit the Next page to read one more thing about Fibonacci Sequences.
Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.
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