Fibonacci Sequence

The Fibonacci Sequence is a pattern of number that is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ... Each two previous numbers added together equal the next number. For example: 3+5=8, 5+8=13, 8+13=21 and so on. This seemingly simple pattern appears in many natural things like phi (the golden ratio).

Now, if I were to take two numbers next to each other on the pattern and divide them by each other, I would get an answer approximate to phi. The further the two number are on the pattern, the closer they are to it. For example: 21÷13≈1.6153 while 144÷89≈1.6179 (phi≈1.61803).

A very bizarre way that the Fibonacci Sequence can be shown is in honeybees. Each female honeybee has a mother and a father (you might say, "duh"), but male honeybees strangely only have a mother ("what?") since males are produced from an unfertilized egg. Below is a family tree of a male bee: (The circle with the arrow means male while the plus means female.)

So to start out with, the bee here has one parent, a mother.
He has two grandparents.
He has three great grandparents.
He has five great great grandparents.
Is this pattern starting to look familiar? Yes, it is the Fibonacci Sequence.

The sneezewort (Achillea ptarmica) is a plant that also demonstrates the Fibonacci numbers in its branch numbers and leaves.


On a lot of flowing plants (not all), the flowers' petals tend to be a Fibonacci number (or the average of the number of petals is a Fibonacci number). In my yard, I had a hibiscus and an african violet both with five petals. Clovers would be a Fibonacci flower even though they can have four leaves because the average amount of clovers have three leaves. Other Fibonacci flowers include buttercup (5 petals), delphiniums (8 petals), ragwort (13 petals), black-eyed susan (21 petals), and plantain (34 petals).


— Alec

Phee, Phi, Pho, Phum

Have you ever been playing with your calculator, typing in and stumble upon the fact that no matter what number you put for "x", you end up with the same number? (Ok, I seriously doubt you have, but if you did that, you would always end up with a number called phi.)

Phi is a very interesting irrational number usually represented by the Greek symbol . Like pi, phi continues on to infinity with its first several digits as follows: 1.618033988749. This peculiar number can be found in the makeup of people, animals, nature, and geometry. By many Christians, phi is also know as the divine proportion since it seems to be the building block of nature used by God. Phi is most commonly called the golden ratio, but it is also known as the golden mean, mean of Phidias...


On a person, if you take the length of their shoulder to their fingertip divided by the length of the elbow to the fingertip, you get phi. For example, I took a measure of myself: the length of my arm was 27.5 inches and the length of my elbow to my fingertip was 17 inches. Then if I divided the lengths, I got approximetly 1.6176 inches which is very close to phi. Other parts of the hand, face, and body can be divided by each other to equal phi also. This is a fun thing you can try out too, but try to measure as exact as you can.


An example of phi in geometry can appear with a regular pentagram:
Red ÷ green, green ÷ blue, and blue ÷ pink all equal phi.

Mathematicly, phi is also the only number that its square equals one plus itself:
This is very different from any other number. At first glance, this might seem impossible, but on closer inspection, with the help of a handy-dandy calculator, this equation works.

Phi can also be used in these equations, which no other number can:
and .

This one looks really cool, but sadly, I have no earthly idea of what it means:
.

Yet another way to show phi is by looking at this simple pattern: 1, 1, 2 , 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584...
Can you see the pattern? Each previous two numbers in this pattern added together equal the next number. Now for phi to come in. Divide two numbers next to each other. The result is a number approximate to phi. The further you go down the pattern and divide the two numbers, the closer the quotient is to phi. This addition pattern is known as the Fibonacci Sequence.

The pictures following are examples of creatures who exhibit phi: .

In our world, phi isn't just a random number made up by some guy, rather is a number that God chose to build His world with.


"Measure what is measurable, and make measurable what is not so."
--Galileo Galilei

For more information, go to http://goldennumber.net/.

~ Alec

I hope this was interesting, if you have any questions, please feel free to comment.