The Golden Ratio is an interesting concept in mathematics. The ratio is found in a number of places throughout nature. For example, the ratio has been found in see shells and pineapples. It is common in architecture. But the Golden Ratio is rooted in mathematics, where it can be derived a number of ways including from a Golden Triangle. Someplace I had forgotten that it occurs is in the Fibonacci Sequence, which actually appears to be one of the most fundamental sources of the ratio. In case you had forgotten about the Fibonacci Sequence or Numbers, they start like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

The pattern works like this: start at 1, and then for each number take the current number + the number before it to create a new number. So in the Fibonacci Sequence, every number is the sum of the two previous brother numbers. Pretty straight forward, huh?

Where it gets interesting is when you divide each number by its previous number. The results look like this for the first few numbers:

1 : 1/1 = 1
2 : 2/1 = 2
3 : 3/2 = 1.5
5 : 5/3 = 1.6*
8 : 8/5 = 1.6
13 : 13/8 = 1.625
21 : 21/13 = 1.61538461538462
34 : 34/ 21 = 1.61904761904762

You can view a longer version of this at a Google Doc Spreadsheet I created. You will need a Google account to view it.

See that number starting 1.618? That is the golden ratio (or an approximation). The Fibonacci Sequence division scheme I have presented here converges on the ratio. So how do we ever discover the “real value”? Well, there are several ways. One way is to use a limit. But I found that the lesson plan on this showed a very interesting, purely algebraic way of doing it.

The Golden Ratio value for each number, lets call it r for “ratio”, is based on the division of any number via its previous number. If we write this formula out it is , if n is the number and f is the function that creates a Fibonacci number. Another way of writing this is . We can take any given n-1, put it into the function, then multiple it by r, and we will get the result of f(n). This part was pretty easy for me, too. The Fibonacci Sequence tells us that every number is the sum of its two previous brothers, so we can also state . So far we have two identities:

#1:
#2:

The goal here is to solve for r. This is the tough part. This stumped me, but I got some help from Reddit Math. The key here is to make an educated guess that we can do some substitution here that will get us the quadratic equation (which we can easily solve). Looking at the equation, we can see that there might be a path to this type of equation, and if we have a quadratic equation then we can solve for r. So how do we get here from there?

The first thing to note is that is the same value as it’s previous brother times r. In other words, is the same as or . OK, now we are closer to a quadratic equation, since we have a power of two. The first equation can now be written as . Now we are rocking! We can now see in our soup of equations that we have all the parts of a quadratic equation, except one part. We still have an “n-1″ floating around and everything else is an “n-2″, and we need the same variable across everything to do a quadratic equation. But we already know is the same as ! Here are those two identities again, with both transformed:

#1:
#1a:
#2:
#2a:

We can now set 1a and 2a as equal, since they both equal f(n):

or or

And that is a quadratic equation. Given a quadratic equation , then one may solve it with:

When we plug that in it looks like:

Which equals 1.6180339887498948482045868343656 (approx).

Very cool.

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