GABRIELLE NAHAS

IBDP MATH STUDIES

THURSDAY, FEBRUARY 23rd 2012

WORD COUNT: 2,839

INTRODUCTION:

The Golden Ratio, also known as The Divine Proportion, The Golden Mean, or Phi, is a constant that can be seen all throughout the mathematical world. This irrational number, Phi (Φ) is equal to 1.618 when rounded. It is described as "dividing a line in the extreme and mean ratio". This means that when you divide segments of a line that always have a same quotient. When lines like these are divided, Phi is the quotient:

When the black line is 1.618 (Phi) times larger than the blue line and the blue line is 1.618 times larger than the red line, you are able to find Phi.

What makes Phi such a mathematical phenomenon is how often it can be found in many different places and situations all over the world. It is seen in architecture, nature, Fibonacci numbers, and even more amazingly,the human body.

Fibonacci Numbers have proven to be closely related to the Golden Ratio. They are a series of numbers discovered by Leonardo Fibonacci in 1175AD. In the Fibonacci Series, every number is the sum of the two before it. The term number is known as ‘n’. The first term is ‘Un’ so, in order to find the next term in the sequence, the last two Un and Un+1 are added. (Knott).

Formula: Un + Un+1 = Un+2

Example: The second term (U2) is 1; the third term (U3) is 2. The fourth term is going to be 1+2, making U3 equal 3.

Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

When each term in the Fibonacci Series is divided by the term before it, the quotient is Phi, with the exception of the first 9 terms, which are still very close to equaling Phi. Term (n)| First Term Un| Second Term Un+1| Second Term/First Term (Un+1 /Un)| 1| 0| 1| n/a|

2| 1| 1| 1|

3| 1| 2| 2|

4| 2| 3| 1.5|

5| 3| 5| 1.667|

6| 5| 8| 1.6|

7| 8| 13| 1.625|

8| 13| 21| 1.615|

9| 21| 34| 1.619|

10| 34| 55| 1.618|

11| 55| 89| 1.618|

12| 89| 144| 1.618|

Lines that follow the Fibonacci Series are found all over the world and are lines that can be divided to find Phi. One interesting place they are found is in the human body. Many examples of Phi can be seen in the hands, face and body. For example, when the length of a person’s forearm is divided by the length of that person’s hand, the quotient is Phi. The distance from a person’s head to their fingertips divided by the distance from that person’s head to their elbows equals Phi. (Jovanovic). Because Phi is found in so many natural places, it is called the Divine ratio. It can be tested in a number of ways, and has been by various scientists and mathematicians.

I have chosen to investigate the Phi constant and its appearance in the human body, to find the ratio in different sized people and see if my results match what is expected. The aim of this investigation is to find examples of the number 1.618 in different people and investigate other places where Phi is found. Three ratios will be compared. The ratios investigated are the ratio of head to toe and head to fingertips, the ratio of the lowest section of the index finger to the middle section of the index finger, and the ratio of forearm to hand.

FIGURE 1 FIGURE 2

FIGURE 3

The first ratio is the white line in the to the light blue line in FIGURE 1 The second ratio is the ratio of the black line to the blue line in FIGURE 2 The third ratio is the ratio of the light blue line to the dark blue line in FIGURE 3

METHOD:

DESIGN: Specific body parts of people of different ages and genders were measured in centimeters. Five people were measured and each participant had these parts measured: *...