Chapter 35: Phi (ϕ)

Phi, the magic number was discovered by the Greeks, by Pythagoras or his followers, as best as we can tell. Originally, it was described as a ratio of line segments described by the following situation.

Screen Shot 2014-03-03 at 9.28.24 PM

For a line segment composed of two components, a and b, the ratio of the length of a/b is the same as the ratio of the combined line segment to a, that is:

Screen Shot 2014-03-03 at 9.28.35 PM

It seems simple and innocent enough, like most fundamental ideas. It was most likely investigated because there are many natural and geometrical things that are formed with this proportion. Human bodies, for example, approximately display this ratio between their upper bodies, their legs and their overall height. Pentagons can be built from isosceles triangles made with bases of length ‘a’ and matching sides of ‘b’ length.

Screen Shot 2014-03-03 at 9.30.25 PM

The Parthenon is said to have been designed emphasizing this ratio. It is evidenced in great art, like Da Vinci’s illustrations in De Divina Proportione (On the Divine Proportion), where he shows how the ratio applies to the parts of the human body. It is called the Golden Ratio because, especially when used to describe the sides of a rectangle, the size and its affect are thought to be pleasing to the eye. Some say that the pyramids were built to this ratio. There are hundreds of books written on these aspects of Phi alone, and they contain much more information then could ever be presented here. Many great artists, mathematicians, architects and musicians have spent years upon years studying the aspects of this number and its relation to the world around us. You could probably spend the rest of your life just trying to read and comprehend all that is out there, but that is not the purpose here. Go. Read if you want to. This book will wait….

Phi also has some very mysterious mathematical properties. Like other many other useful constants  (Pi, the natural log e and the square root of 2, for example), the ratio that expresses Phi, is a never ending, non repeating decimal whose value is both exactly and approximately:

Screen Shot 2014-03-03 at 9.31.43 PM And it has the following equalities:

Screen Shot 2014-03-03 at 9.31.50 PM

So, a complicated equation in Phi can be broken down to a simple addition and subtraction:

Screen Shot 2014-03-03 at 9.34.18 PM

Screen Shot 2014-03-03 at 9.34.27 PM


Sqrt ϕ ~ 4/π

Oddly enough, it is also the average and, in the limit (that is, considering all possible Fibonacci numbers), the ratio between any two adjacent values in a Fibonacci sequence. The Fibonacci sequence is a series where each subsequent value is the sum of the previous two, given 0 and 1 for starting, as shown below.

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

But, this chapter is really not about this kind of stuff either.

The general form for the expression of a vortex, a Spiral velocity is;

V = rθ/t where r, θ, & t are all variables.

These variables can be moved around to be proportional to each other, or multiplicative (excepting time, or course) depending on the form that is to be chosen. And what a universe of them that there is!

The orbit of a planet around a star, a tornado, a flushing toilet bowl, a dust devil, the airflow created by a rotating fan, and of course,

galaxies and hurricanes.

But of the many and various types of vortices found in nature, there is one type that turns up in just about every level and at every size.




And yes,


And that is the logarithmic spiral. But the pictures shown above (with the possible exception of the nautilus shell) are all of a particular type of logarithmic spiral that is based on the relationship, you guessed it, Phi. The actual equation looks like this:

r = aebθ

 Where: a is any real constant, and

b is a rotational measurement constant such that ebθ = ϕ when θ = 900 (or π/2 radians).( Pretty convenient that Sqrt ϕ ~ 4/π, huh?)

This locks the rotational change and the linear change into a constant ratio such that the overall velocity, and hence the acceleration created by it, are both constant. This special relationship has been given a moniker, the Golden Spiral, since it is based on the golden ratio, ϕ.

Screen Shot 2014-03-03 at 9.47.07 PM


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