Fibonacci sequence

A mathematical sequence consists of a series of numbers, usually starting at 0 or 1, all of which are derived from preceeding ones according to formal rules. In other words, no matter where you are in the sequence you ought to be able to deduce what the following number should be.

One of the most well-known sequences was one derived by the Italian medieval mathematician known today as Fibonacci. In his Liber Abaci, published in 1202, he explained a simple sequence, where every number is the sum of the two numbers preceeding it. So the result is:

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

The fascinating thing about it is that this sequence is actually found in nature with amazing frequency. Fibonacci himself used the cute (pre-Australian cute) example of the multiplication of rabbits in his book. Since then, it's also been found to apply to such varied things as shell spirals, branching plants, flower petals, seeds, pine cones, leaves... It seems almost everywhere you look there is lurking a Fibonacci sequence.

So what does all this have to do with fiber? There are easy ways to use the Fibonacci sequence in designs, both for color and texture. And the results somehow always manage to look pleasing to the human eye, which is probably because the sequence looks so... natural! See for instance this improvised sock - doesn't it look like someone racked her brain to come up with something so sophisticated? And you don't need to use a whole lot of the sequence for the effect to take hold visually - instead of using 3 uniform thin stripes for instance, make the 3rd twice as wide, and see the result...

So let's think of some obvious ways to use the sequence:

You have a very plain stockinette sweater which needs a bit of livening up. Instead of doing a few stripes, use a Fibonacci sequence of garter ridges as a pattern. If you want color, the width of the stripes can follow the sequence. This can work both when one color does the pattern:
```
a...a aa bbbbb aa bbb aa bb aa b aa b a...a
```
or when both colors work together: bbbbbbbb aaaaaaaa bbbbb aaaaa bbb aaa bb aa b a b a
You run out of yarn as you near the end of the sleeves. No problem, you can incorporate a few stripes in a Fibonacci sequence as you change colors, and instead of strange contrasting cuffs your sweater will look perfectly integrated. Add a couple thin Fibonacci contrasting stripes (1 1 2 or even just 1 1) in the collar and it'll look like you did it all on purpose.
You can use one color/texture in a Fibonacci sequence of varying width, separated by fixed contrasting bands:
```
aa bbbbb aa bbb aa bb aa b aa b
```
Better yet, you can have two sequences going in both directions at once:
```
aaaaaaaa b aaaaa b aaa bb aa bbb a bbbbb a bbbbbbbb
```
You can integrate the sequence into the design of Fair-Isle patterns, with progressively larger designs. Or into cables of more and more complex design.

Here is a Fibonacci rib - a 4x4 ordinary rib is livened up by changing the pattern every Fibonacci number of rows.

aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa

And this is a Fibonacci doodle - switch rows every Fibonacci number of rows.

abbbbbbbbbbbbbbbbbbbbbbbb
babbbbbbbbbbbbbbbbbbbbbbb
bbaabbbbbbbbbbbbbbbbbbbbb
bbbbaaabbbbbbbbbbbbbbbbbb
bbbbbbbaaaaaabbbbbbbbbbbb
bbbbbbbbbbbbbaaaaaaaabbbb

There is nothing original about this, it's all been very well known for a long time, and once your eye gets tuned to it you'll notice uses of the sequence in many designs. But it all works so well to give pleasing designs that there's no reason not to make use of it liberally...

References:

Complete Fibonacci reference
The short version
Fibonacci's bio