Phi & Fibonacci Anomalies

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Foundations of the Fibonacci Series

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, . . .

It's just a simple series of numbers discovered by Leonardo Fibonacci that defines the ratio of the Golden Section and Divine Proportion.

But, the Fibonacci number (1.61804...), often called phi or , was described by Johannes Kepler (1571-1630) as one of the "two great treasures of geometry."   (The other is the Theorem of Pythagoras.)

The ratio of each successive pair of numbers in the series quickly converges on 1.61804. . . , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.

But that is just the beginning of its mathematical curiosities and mysteries. 

Unusual Relationships of Phi

Square phi and you get a number exactly 1 greater than phi, 2.61804...:

2 = +1

Divide phi into 1 and you get a number exactly 1 less than phi, 0.61804...:

1 / = - 1

Take the square root of 5, add 1 and then divide by 2, and you also get phi.

( 51/2 + 1 ) / 2 =

Which can also be expressed all in fives as:

5 ^ .5 * .5 + .5 =

Fibonacci, Phi and Spirals in Nature

If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number.  This property results in the Fibonacci spiral seen in everything from sea shells to galaxies:

12 + 12 + 22 + 32 + 52 = 5 x 8
12 + 12 + . . . + F(n)2 = F(n) x F(n+1)

Fibonacci series and spiral Fibonacci spiral in a nautilus shell Fibonacci spiral in a galaxy
Note:   The Fibonacci series spiral on the left is slightly different from the perfect spiral generated by Phi (1.61804...) because of the approximations early in the series leading to Phi.
(1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625)


The DNA molecule, the program for all life, is based on the golden section.  It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.

Construction of the golden section DNA and the golden section

34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.

The Solar System

The average of the mean orbital distances of each successive planet expressed in relation to the one before it approximates phi.

Solar system


Plants illustrate the Fibonacci series in the numbers and arrangements of petals, leaves, sections and seeds.

Fibonacci series in plant growth
Here a plant illustrates that each successive level of branches is often based on a progression through the Fibonacci series.
Fibonacci series in the petals of a flower
Many plants produce new petals in quantities that are based on Fibonacci numbers.

The Fibonacci Series and Music

Piano keyboard - even music is based on the Fibonacci series

The Fibonacci series appears in the foundation of many other aspects of art, beauty and life. Even music has a foundation in the series, as 13 notes make the octave of 8 notes in a scale, of which the 1st, 3rd, and 5th notes create the basic foundation of all chords, and the whole tone is 2 steps from the root tone.

Phi and the Mind of God

Any sign that would prove the existence of another intelligence must have a pattern or sequence that could not be the result of natural processes.

Phi's unique mathematical qualities are inherent, but what is the likelihood that creation could be based on this number by chance?

The Fibonacci number, phi, is more than a mathematical curiosity. It is one of the most amazing and pervasive numbers known to man.  This, above all numbers, would be a perfect sign to tell us that creation is not just the result of random processes.

Might this number be intriguing and beautiful to a mind of advanced intellect, to the mind of God?

It is a perfect way to not only demonstrate that there is design in the universe, but to assure its unity.

Could it be that this is why it is found throughout creation in the design of plants, animals, galaxies ... and even you,

Intrigued? Click at
right to learn more about the golden section, Fibonacci
series and phi.


as a sign from,
and the signature of,
the Creator . . .



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