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# The Secret of Fibonacci Numbers

## The Secret of Fibonacci Numbers

Have you ever heard of the Fibonacci number sequence? They are really spooky, or rather one of those natural mysteries.

OK, the Fibonacci sequences looks like this:  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 how they come about?

It’s quite simple really. You take the number 1 and add it to itself which gives you 2. You then add the 2 to the 1 which =3. then 3+2 = 5. 5+3= 8 and so on like this. 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, …

A bit freaky.

But it gets freakier.

The Fibonacci numbers are Nature’s numbering system. They appear everywhere in Nature, from the leaf

Fibonacci in nature

arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

Plants do not know about this sequence – they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Fibonacci in plants

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.

In the seeming randomness of the natural world, we can find many instances of mathematical order involving the Fibonacci numbers themselves and the closely related “Golden” elements.

## Fibonacci in Plants

Phyllotaxis is the study of the ordered position of leaves on a stem. The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Click on the picture to see a more detailed illustration of leaf arrangements.

By dividing a circle into Golden proportions, where the ratio of the arc length are equal to the Golden Ratio, we find the angle of the arcs to be 137.5 degrees. In fact, this is the angle at which adjacent leaves are positioned around the stem. This phenomenon is observed in many types of plants.

Fibonacci in trees

The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory’s 21, the daisy’s 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

The head of a flower is also subject to Fibonaccian processes. Typically, seeds are produced at the center, and then migrate towards the outside to fill all the space. Sunflowers provide a great example of the

Sun flower seeds

Here are just a few plants that have their leaves and settles arranged according to Fibonacci numbers

Fibonacci Petals

3 petals lily, iris
5 petals buttercup, wild rose, larkspur, columbine
8 petals delphiniums
13 petals ragwort, corn marigold, cineraria
21 petals aster, black-eyed susan, chicory
34 petals plantain, pytethrum
55, 89 petals michelmas daisies, the asteraceae family

## Fibonacci in the planets and outer space?

Fibonacci in space

Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees. As an interesting aside, spiral galaxies appear to defy Newtonian physics. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate. Subsequently, after a few rotations, spiral arms should start to wind around a galaxy. But they don’t — hence the so-called winding problem. The stars on the outside, it would seem, move at a velocity higher than expected — a unique trait of the cosmos that helps preserve its shape.

## US …..

Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).

It’s worth noting that every person’s body is different, but that averages across populations tend towards phi. It has also been said that the more closely our proportions adhere to phi, the more “attractive” those traits are perceived. As an example, the most “beautiful” smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It’s quite possible that, from an evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.

Looking at the length of our fingers, each section — from the tip of the base to the wrist — is larger than the preceding one by roughly the ratio of phi.

Even our bodies exhibit proportions that are consistent with Fibonacci numbers. For example, the measurement from the navel to the floor and the top of the head to the navel is the golden ratio. Animals

## Animals

Animal bodies exhibit similar tendencies, including dolphins (the eye, fins and tail all fall at Golden Sections), starfish, sand dollars, sea urchins, ants, and honey bees.

Honey Bee

Honey bees follow Fibonacci in other interesting ways. The most profound example is by dividing the number of females in a colony by the number of males (females always outnumber males). The answer is typically something very close to 1.618. In addition, the family tree of honey bees also follows the familiar pattern. Males have one parent (a female), whereas females have two (a female and male). Thus, when it comes to the family tree, males have 2, 3, 5, and 8 grandparents, great-grandparents, gr-gr-grandparents, and gr-gr-gr-grandparents respectively. Following the same pattern, females have 2, 3, 5, 8, 13, and so on. And as noted, bee physiology also follows along the Golden Curve rather nicely.

## The Golden Rectangle

The Golden rectangle has been known since antiquity as one having a pleasing shape, and is frequently found in art and architecture as a rectangular shape that seems ‘right’ to the eye. It is mentioned in Euclid’s Elements and was known to artists and philosophers such as Leonardo da Vinci.

One of the interesting properties of the golden rectangle is that if you cut off a square section whose side is equal to the shortest side, the piece that remains is also a golden rectangle.

In the figure below, the yellow rectangle is in the same proportion as the original larger rectangle after the gray square is cut off.Both the rectangles ABCD and PBCQ are golden rectangles.

Golden Rectangle