An interesting number and sequence

PhiOverviewWhat is Phi?

Phi ( = 1.618033988749895... ), most often pronounced fi like "fly," is simply an irrational number like pi ( p = 3.14159265358979... ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation.Phi is the basis for the Golden Section, Ratio or Mean

The ratio, or proportion, determined by Phi (1.618 ...) was known to the Greeks as the "dividing a line in the extreme and mean ratio" and to Renaissance artists as the "Divine Proportion" It is also called the Golden Section, Golden Ratio and the Golden Mean.Phi, like Pi, is a ratio defined by a geometric construction

Just as pi (p)is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.Divide a line so that:

the ratio of the length of the entire line (A)

to the length of larger line segment (B)is the same asthe ratio of the length of the larger line segment (B)

to the length of the smaller line segment (C).This happens only at the point where:

A is 1.618 ... times B and B is 1.618 ... times C.Alternatively, C is 0.618... of B and B is 0.618... of A.Phi with an upper case "P" is 1.618 0339 887 ..., while phi with a lower case "p" is 0.6180339887, the reciprocal of Phi and also Phi minus 1.

Phi appears throughout life and the universe. Some believe that it is the most efficient outcome, the result of natural forces. Some believe it is a universal constant of design, the signature of God.

Whatever you believe, the pervasive appearance of phi in all we see and experience creates a sense of balance, harmony and beauty in the design of all we find in nature.And Fibonacci numbers.The human body is based on Phi and 5

The human body illustrates the Golden Section or Divine Proportion. We'll use the same building blocks again:

Another interesting relationship of golden section to the design of the human body is that there are:The Divine Proportion in the Body

The white line is the body's height.

The blue line, a golden section of the white line, defines the distance from the head to the finger tips

The yellow line, a golden section of the blue line, defines the distance from the head to the navel and the elbows.

The green line, a golden section of the yellow line, defines the distance from the head to the pectorals and inside top of the arms, the width of the shoulders, the length of the forearm and the shin bone.

The magenta line, a golden section of the green line, defines the distance from the head to the base of the skull and the width of the abdomen. The sectioned portions of the magenta line determine the position of the nose and the hairline.

Although not shown, the golden section of the magenta line (also the short section of the green line) defines the width of the head and half the width of the chest and the hips.

The golden section in turn, is also based on 5, as the number phi, or 1.6180339..., is computed using 5's, as follows:

- 5 appendages to the torso, in the arms, leg and head.
- 5 appendages on each of these, in the fingers and toes and 5 openings on the face.
- 5 senses in sight, sound, touch, taste and smell.

5 ^ .5 * .5 + .5 = Phi

In this mathematical construction "5 ^ .5" means "5 raised to the 1/2 power," which is the square root of 5, which is then multiplied by .5 and to which .5 is then added.

Fibonacci Numbers, the Golden section and the Golden StringFibonacci Numbers and the Golden Section

This is theHome pagefor Dr Ron Knott'smultimediaweb site on the Fibonacci numbers, the Golden section and the Golden string hosted by the Mathematics Department of the University of Surrey, UK. TheFibonacci numbersare0, 1, 1, 2, 3, 5, 8, 13, ...(add the last two to get the next) Thegolden section numbersare±0·61803 39887... and ±1·61803 39887...

Thegolden stringis1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...

a sequence of 0s and 1s that is closely related to the Fibonacci numbers and the golden section.

There is a large amount of information here on the Fibonacci Numbers and related series and the on the Golden section, so if all you want is a quick introductionthen the first link takes you toan introductory pageon the Fibonacci numbers and where they appear in Nature.

The rest of this page is a brief introduction to all the web pages at this site on

Fibonacci NumberstheGolden Sectionand theGolden String

together with.their many applications

Have fun.