What is the signifiance of the number "phi"?

Question

I need some serious answers here, about phi, the golden ration, and what the importance of them is. Help me out here everybody!!!

Answer 1

Phi is often called the golden ratio... The mathmatics of it have been well explained in the above answers. So much was made of the ratio because it had certain geometrical properties that were used in buildings. The parthenon and the Pyramids at Giza have structures that aproximate the golden ratio. Leonardo Davinci helped illustrate a book dedicated to the golden ratio and he noted that many parts of the anatomy seem to aproximate the Golden ratio. The total length of a leg vs the length of the femur and tibia/fibia portion. The total lenghth of an arm vs the lenght of the humures and radius/ulna protion. The height of an individual vs the upper body and lower body portions. And various facial features. Also the number was porported to be seen in nature with certian trees branching at a rate based on the golden ratio. Anatomical features of animals realting to phi. Number of worker bees vs the total hive population, the angle of seed pods in the center of a sunflower, etc... etc...

The natural observations vary pretty widely due to any number of circumstances and do not always represent phi. The only consistant observations have been the curve of the shell of the nautalis and the over all length and length between knuckles on fingers or toes

Answer 2

I actually did a project in a History of Mathematics course on phi, or the golden ratio.

The golden ratio - or "phi" (which I'll abbreviate as "p" here) describes when you have 2 quantities...let's say 2 numbers....one bigger than the other....when the ratio of the sum of these 2 numbers to the larger number equals the ratio of the larger number to the smaller number - then this ratio is p.

An easier way to visualize it is to consider the golden section.

Imagine a line segment split into 2 parts - call the larger part "a" and the smaller part "b".

So - the golden ratio is when the length of the total line segment (a+b) is to the longer segment (a) as the longer segment (a) is to the smaller segment (b).

This ratio is, in fact, a constant. Using algebra - you have:

(a+b)/a = a/b = p

Therefore a=bp - using substitution for a in the first part of that equation:

(bp + b)/bp = bp/b

which leads to: p^2 - p - 1 = 0.

This is a quadratic with 1 positive solution:

p = (1 + sqroot(5)) / 2 -- which is an irrational number approximately equal to 1.618033989...

This ratio is considered aesthetically beautiful - and whether by knowledge or coincidence -- it can be found in works of art & architecture.

For example, the Parthenon is filled with golden rectangles (a rectangle whose sides are described via the golden ratio).

Answer 3

It is defined as (sqrt(5)+1)/2. You can use it to calculate the nth Fibonacci number without actually calculating the sum up to n.
Some properties I remember:

sqrt(1+sqrt(1+sqrt(1+sqrt(1+....))) = phi.

F[n+1]/F[n]=phi as n tends to infinity, where F[n] is the nth Fibonacci number

1+1/(1+1/(1+1/(1+1/(....))))=phi
Basically solve the quadratic equation x^2-x-1=0 and take the positive root in order to get phi. This simplifies the above equations.
Phi is not transcendental because it is the root of a quadratic polynomial. But phi is irrational because it can be expressed as an irrational number + a rational number.

Answer 4

Phi: That Golden Number
by Mark Freitag
Most people are familiar with the number Pi, since it is one of the most ubiquitous irrational numbers known to man. But, there is another irrational number that has the same propensity for popping up and is not as well known as Pi. This wonderful number is Phi, and it has a tendency to turn up in a great number of places, a few of which will be discussed in this essay.


One way to find Phi is to consider the solutions to the equation


When solving this equation we find that the roots are


x = ~ 1.618... or x=~ -.618...

Answer 5

Golden ratio,usually denoted "phi" expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller, and is the irrational number 1,618033989....

Answer 6

I'm not sure about phi? Do you me pi?