TYPE WORKSHOP - PHIL

Fibonacci Sequence - Not a mathematical equation it is a ratio.

The Fibonacci Sequence is the series of numbers:

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

The next number is found by adding up the two numbers before it.

• The 2 is found by adding the two numbers before it (1+1)
• Similarly, the 3 is found by adding the two numbers before it (1+2),
• And the 5 is (2+3),
• and so on!

Example: the next number in the sequence above would be 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...

Can you figure out the next few numbers?

Makes A Spiral

When you make squares with those widths, you get a nice spiral:

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

TYPE GUIDE

A 50 point title should be complimented with 

34 point body copy by applying Fibonacci's sequence.

The Rule

The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n =	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	...
xn =	0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	...

So term number 6 is called x₆ (which equals 8).

Example: the 8th term is the 7th term plus the 6th term: x8 = x7 + x6

So we can write the rule:

The Rule is x_n = x_n-1 + x_n-2

where:

• x_n is term number "n"
• x_n-1 is the previous term (n-1)
• x_n-2 is the term before that (n-2)

Example: term 9 would be calculated like this:

x₉ = x_9-1 + x_9-2 = x₈ + x₇ = 21 + 13 = 34

Golden Ratio

And here is a surprise. If you take any two successive (one after the other)Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034... In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A	B		B / A
2	3		1.5
3	5		1.666666666...
5	8		1.6
8	13		1.625
...	...		...
144	233		1.618055556...
233	377		1.618025751...
...	...		...

Note: this also works if you pick two random whole numbers to begin the sequence, such as 192 and 16 (you would get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

A	B		B / A
192	16		0.08333333...
16	208		13
208	224		1.07692308...
224	432		1.92857143...
...	...		...
7408	11984		1.61771058...
11984	19392		1.61815754...
...	...		...

It takes longer to get good values, but it shows you that it is not just the Fibonacci Sequence that can do this!