The Fibonacci Sequence and More...

A hobby of Jamie

     Φ      The Fibonacci Sequence      Φ      The Musical Octave      Φ      Pascal's Triangle      Φ      Find Phi      Φ      The Golden Ratio and Rectangle      Φ      Pentagon & Pentagram      Φ      Special Properties of Phi      Φ      For Fun      Φ      For Real      Φ      Videos

The Fibonacci Sequence

Starts with 1, then 1, then you add the prior 2 together and keep on going in that sequence:

According to James Nickel, author of Mathematics: Is God Silent?, the Fibonacci Sequence is related to the petal arrangement of flowers; is found in the spiral arrangements of petals, pine cones, and pineapple; in the leaf positioning of the Phyllotaxis, in the mathematics of the quantum matrix, of rabbit populations, and of the genealogy of male bees.

The Musical Octave

The Fibonacci Sequence shows up in the musical octave as shown below:


There are
2 black keys, then
3 black keys, totalling
5 total black keys.
8 white keys make the octave.
13 keys total.

2, 3, 5, 8, 13 is a subset of the Fibonacci Sequence.

Pascal's Triangle

Pascal's Triangle, which is used to solve binomials and probability is related to the Fibonacci Sequence as shown below:

Pascal's Triangle

Find Phi

If you look at the Fibonacci Sequence and divide, you will find a trend. If you divide one number by the number before it continuously, the result approaches Phi (Phi) which is about equal to 1.618033989. See below:

In fact, you can choose any 2 numbers and this will work. Say I choose 8 and 26 and form a Fibonacci-like Sequence as such:

Now say I seek Phi using the same dividing method mentioned above. I get the following:

The result approaches Phi. Amazing!

The Golden Ratio and Rectangle

The Golden Rectangle, whose ratio of the longer side to the shorter side is the value of Phi (also called the Golden Ratio) also contains a spiral that is seen in the cochlea of the human ear, the chambered nautilus, the spiral galaxy, and in hurricane storm clouds according to James Nickel, author of Mathematics: Is God Silent?. Further, the proportions of the Golden Rectangle have been used in art and architecture. More information about the Golden Rectangle.

Golden Rectangle

The shorter side is 1 and the longer side is Phi (Phi).


James Nickel explains in his book Mathematics: Is God Silent? how Phi relates to the Pentagram/Pentagon of 5-petaled flowers, starfish, and sand dollars.


In the figure above AC/AB = Phi.

Special Properties of Phi

1/ = Phi - 1

Solving for Phi, I get the following:

and this equals 1.618033989... OR -0.618033989...


Phi Division

Phi cubed

Phi and its Square Roots
and if you add it to 1, you get Phi squared, which in turn equals 2.6180339887...

For Fun

Click to Zoom

Click to Zoom

Card Trick

Cool Math - lessons, games and more!

Imperfect Math

The Beauty of Math

Sator Square:


The Sator Square may be read top-to-bottom, bottom-to-top, left-to-right, and right-to-left: SATOR AREPO TENET OPERA ROTAS. Translation from Latin: The sower Arepo has work of wheels; that is, the farmer uses his plough for his work.



For Real

Algebra at The Free Information Society (web archive)

Archimedes at Wikipedia

Art of Problem Solving

Calculus at The Free Information Society (web archive)

Circle - Radians, Degrees and Coordinates

Curves and Equations Reference

Doodling in Math Class: Spirals, Fibonacci, and Being a Plant - Part 1

Doodling in Math Class: Spirals, Fibonacci, and Being a Plant - Part 2

Doodling in Math Class: Spirals, Fibonacci, and Being a Plant - Part 3

Glimmers of Light from the Eye of a Giant (web archive)

The Golden Ratio - a sacred number that links the past to the present | Ancient Origins

Golden Ratio at Wikipedia

Golden Spiral on squareCircleZ

The Greatest Mathematicians of All Time

Hero of Alexandria at Wikipedia

Interactive Mathematics

Math Articles at The Free Information Society (web archive)

The Mathematical Association of America

The Math Behind Beauty

The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics

Math Help

Mathematically Correct

Meton of Athens at Wikipedia


What pi sounds like

Where's the Math? (web archive)



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