2.5.12. Fibonacci (Optional study)

Mathematics in nature - magical, aesthetic, and somewhat of a mystery (the math is simple, the magic is everywhere). 

There are two mathematical patterns that appear throughout the universe in both natural and built environments. There are some practical applications that are known, such as stability and efficiency of space, but the reasons nature uses this pattern so deeply and why it is found throughout the physical universe as well from galaxies to molecular structure is not completely understood scientifically. 

Fibonacci sequence

The Fibonacci sequence was discovered by, yes, Fibonacci, an Italian mathematician of the 13th century, who noted there were mathematical patterns in nature that repeated themselves constantly.  This pattern could be expressed by the following formula:

1+1=2
1+2=3
2+3=5
3+5=8
5+8=13

Etc., continuing to add up the last two numbers in the formula.

This sequence then becomes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 etc.

You can see this sequence in things like the ratio in a human hand.

Flowers often have petal numbers that conform to the Fibonacci sequence.

Tree branches tend to branch in Fibonacci numbers too. Some branches will stop branching while others continue, ending up with Fibonacci totals. 

Honey Bees reproduce in the Fibonacci sequence, and DNA molecules twist in that sequence as well.

The Golden Ratio

The Golden Ratio, Golden Mean or Golden Rectangle is another pattern that is found throughout nature.

The Golden ratio is found by combining a circle with a perfect square, thus:

This could be done with sticks, by laying out a perfect square (one size stick), dividing it in half, and then tracing a circle from the dot in the middle of one of the sides.

Now, you can do a lot with this approximately ⅓ to ⅔ Golden Rectangle. One can see this ratio in the human body:

If you add a spiral into it, you get a “Golden Spiral”.

Notice, that this now becomes the Fibonacci sequence as well.

And this Golden Spiral and Fibonacci sequence in this form are found in numerous places!

This pattern is found at macro and micro levels.

Hair flung back with water in it, regardless of length, will form a Golden Spiral.

It can even be found in land masses.

And in numerous plants.

Many plants have multiple golden spirals in them, like these Spiral Geraniums, or aloe.

Sunflowers reflect Fibonacci numbers and spirals in several ways. Seeds are pushed out from the center in a spiral to maximize numbers and efficiency and they usually end up with Fibonacci totals.

This Fibonacci curve (Golden Spiral) can expand without changing its shape. This may be the secret to its ubiquitousness in the natural world. Virtually all growth spirals will reflect the Fibonacci sequence.

And cats! (this is apparently a thing on the internet; Fibonacci cat pictures).

We also see the Golden Ratio pattern reflected in numerous buildings throughout the world. 

And in primitive and modern art from all cultures.

Why does this pattern exist so ubiquitously?

The respected naturalist Sir D'Arcy W. Thompson pointed out that "the shell, like the creature within it, grows in size but does not change its shape. The existence of this constant relativity of growth, or constant similarity of form, is of the essence." In other words, the very shape of the logarithmic spiral enables growth to occur without any change in form. 

Growth from the inside out is associated with living organisms and allows for greater movement, while crystals grow from outside, with growth causing no motion. Ian Stewart, a British mathematician and writer, called for the development of a new kind of mathematics that can explain the patterns in nature, complaining that our mathematical schemes are "too inflexible, geared to the constraints of pencil and paper."