Fibonacci



How does this sequence of numbers continue: **0, 1, 1, 2, 3, 5, 8, 13**

**Hint:** -> Add two consecutive numbers from the sequence to get the next one following them.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

This sequence is called Fibonacci numbers. And it isn't just any ole sequence of any ole numbers... it has some amazing properties, plus it's found in nature in many places. For example, Fibonacci numbers are found in:

Tree branches rotate around the trunk in a pattern based on Fibonacci numbers. A 13-year old student has built a "tree model" with solar panels and proven that it collects MORE sunlight than a flat array of solar panels!
 * [|Petals on flowers]
 * [|Seed heads]
 * [|Pine cones]
 * [|Leaf arrangements]
 * [|Vegetables and Fruit]

How to find Phi with Fibonacci numbers
Here's another amazing thing about these numbers:

Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ... So what's so great about that? you ask...

It is not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818... Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time. The ratios keep approaching the number (√ 5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi. So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√ 5 + 1)/2.

The Golden Section
Phi is also called **the golden section** number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio). This is how we get this golden section or golden cut:

Take a line and divide it into two parts, L (Long part) and S (short part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line. S:L = L:W From this can be solved that L = (√ 5 + 1)/2 × S or L ≈ 1.618 × S. This number (√ 5 + 1)/2 is Phi.

So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut). And the golden ratio is the ratio Phi:1.

Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
 * Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think. ||

Check also this nice animated illustration of the golden rectangle and a spiral inside it.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
 * [|The Golden section in architecture]
 * [|Dividing the line in golden section using compass and ruler]
 * [|Phi in pentagons and pentagrams]
 * [|Where Fibonacci numbers are found in nature]
 * [|Where golden section is found in nature]

P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: [|The Fibonacci Sequence: Relationship to the Human Hand].

Check out how the golden rectangle was used to create the Apple logo...