Thursday, March 15, 2012

Fibonacci ...

Well, here's a change!!  My daughter had to calculate the ratio of the (n+1)th and nth elements in the Fibonacci series.  Not that bad really.

The Fibonacci series goes like this;
1, 1, 2, 3, 5, 8, 13, 21, 34,  and so on.

You can see how to calculate the next number in the series ... you just add together the previous two.  So, bear with me a little now.  We need to know the ratio of adjacent entries in the series as we go long way down the list.
I can represent the ratio and the value to which the ratio will "tend" as we go further down the list with the the equations you see to the left.

Solving the final equation will give me the solution required.

But why am I showing you this?  Well, in nature, you will notice many things in a proportion known as the Golden Ratio, or Golden Mean.  This ratio is just what the Fibonacci Series tends towards as you go further down the list.

Now lets look at something else.

Often when people are taking picture they talk about the "Rule of Thirds".  This is where the image is split into three equal parts and instead of placing important parts of the image in the middle, they'd be placed on a third.  For example, you'll often see the horizon is placed on a horizontal third.
Splitting the frame into thirds for "better" composition.
This "rule"of thirds is very common amongst photographers, but isn't really a "rule" at all.  It's just an easy way to get a safer composition than placing objects of interest smack in the middle of the frame.  What may be a better thing to consider is the Golden Ratio.
Lets split our frame vertically in some ratio, say a:b. The Golden Ratio says the ratio of the smaller part to the bigger part is the same as the ratio of the bigger part to the whole frame (little plus big).

OK, so we can do a bit more math now and see what pops out.  Take a little look to your left and you'll see I've solved the equation (look familiar to the Fibonacci series equation earlier?).  Now you'll see that the ratio of small to big isn't 1:2 and is the case with the rule of thirds, but 1:1.62, or very crudely 3:5.

The thing that is particularly fascinating is how often this Golden Ratio appears in nature, and also how much more powerful your photographs will be if you use the Golden Ratio instead of the rule of thirds.  Take a look at some really good photographs or paintings and check just where the objects of interest are placed in the image.  Are they on thirds, in the centre, or perhaps somewhere closer to the Golden Ratio.

Oh well, just a bit of fun with some math thanks to a question from my daughter.  Hopefully I didn't bore you witless ;-)

No comments:

Post a Comment