Fibonacci Rectangles and Shell Spirals

Image Reference: http://www.huybers.net/fit/fib33.gif
Here is an image of the Fibonacci Rectangle. It is a rectangle that is made up of squares that have a side length of a Fibonacci number, in order, and each new square has a side which is as long as the sum of the latest two square sides. This is just like the equation I mentioned in my last post.

Image Reference: http://upload.wikimedia.org/wikipedia/en/1/18/Fibonacci_Spiral.png
A spiral can be drawn in the squares, a quarter of a circle in each square. Though this is not a true mathematical spiral, it is a good approximation of the kind of spiral that often appears in nature.
A 'spiral' is only an approximation as it is made up of separate and distinct quarter-circles, and the (true) spiral increases by a factor of Phi every quarter turn so it is more correct to call it a Phi(to the power of 4) spiral.

Petals on Flowers:
On many plants the number of petals is a Fibonacci number.
-buttercups have 5
-lilies have 3
-marigolds have 13
-daisies can have 34, 55 or even 89

Seed Heads:

Image Reference: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
This is a Coneflower. The orange 'petals' seem to form spirals curving both to the left and to the right. At the edge of the picture if you count those spiraling to the right as you go outwards, there are 55 spirals. A little more towards the centre there are 34 spirals. This pair of numbers are neighbours in the Fibonacci series.

The same thing happens in many seed and flower heads in nature. This arrangement seems to form an optimal packaging of seeds so that no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges.

Image Reference: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
The spirals are patterns that the eye sees, "curvier" spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we look. So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head we see more spirals further out than we do in the centre. The number of spirals we see in each direction are (almost always) neighbouring Fibonacci numbers.

References:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
http://plus.maths.org/content/os/issue3/fibonacci/index