The Golden Rectangle

      The Golden Rectangle is produce from the construction of the Golden Section where a line is divided into two parts following the ratio of the lengths according to the Golden Section. Then, the side is use as the side of the Golden Rectangle (see Figure 2.6).

Figure 2.6 The Golden Rectangle

It is also can be produce from a square (see Figure 2.7) with vertices label as A, B, C and D. 

The Figure 2.7 A Square

            Then, find the midpoint between line A and B and the midpoint is labeled as p.  Next, from point p, make a straight line and connect it to the vertices C (see Figure 2.8). 

The Figure 2.8

            By placing the compass on point p, draw an arc from vertices C downward. Then, the side AB is extend until there is an intersection between the arc and the line. The intersection point is labeled as Q (see Figure 2.9).

Figure 2.9

            After that, a parallel line to the side BC is drawn from the intersection point Q upward. Afterwards, the side DC is extended until it touches the parallel line that is drawn earlier. The intersection point is labeled as R (see Figure 2.10).  From this step the Golden Rectangle is formed.

Figure 2.10 A complete Colden Rectangle

           

            By calculation, assume AB is 2 units of length. Then, the length of PC is equal to the length of PQ which is square root of five units. From the formula, we get

            In the golden rectangle, it could form Golden Spiral by modelling a series of squares using the first ten Fibonacci numbers with each squares has side with length of Fibonacci Numbers.

Figure 2.11 The Golden Rectangle from Fibonacci Sequence

            This shows that there exist the relationship between the Fibonacci Sequence and the Golden Ratio by applying the concept of the Golden Rectangle.

Fibonacci Sequence

Leonardo Fibonacci (see Figure 1.1) (Leonardo, the son of Bonaccio) lived around 1175–1250 A.D. He was born in medieval Pisa.  He had travelled to many parts of the Mediterranean region with her father as a boy.  During this period, he lived for a while in North Africa.  He encountered Eastern and Arabic mathematics during this time and became convinced that current European mathematical practices were inferior (Gosett, 2013).

Figure 1.1 Leonardo Fibonacci

In 12th century, the Leonardo Fibonacci questioned about the population growth of the rabbits (see Figure 1.2) under ideal circumstances, such as no predators to eat them or no dearth of food and water that would affect the growth rate.

 

Figure 1.2 Fibonacci’s Rabbit Problem

The answer of the question is the Fibonacci Sequence of Numbers, also known as Fibonacci Numbers that starts from 1 and each new number of the series is simply the sum of the previous two numbers.  So the second number of the series is also 1, the sum of the previous 1 and 0 of the series. The sequence of the number looks like the series bellow:

            Fibonacci numbers are said as one of the Nature's numbering systems because of its existence not only in the population growth of rabbits, but also everywhere in nature, from the leaf arrangements in plants to the structures in outer space.  The special proportional properties of the golden ratio have a close relationship with the Fibonacci sequence. Any number of the series divided by the contiguous previous number approximates 1.618. (Amir, 2011).

What is the golden ratio?

Mario (2002) states that the Golden Ratio was become known as a formalised deductive system around 300 BC by Euclid of Alexandria, the founder of geometry.  Euclid defines a proportion derived from a simple division of a line (see Figure 2.1) into what he called as “extreme and mean ratio”.

            From the straight line in Figure 2.1, the ratio of the whole segment (AB) is the same to the larger segment as the larger segment is to the shorter segment (Rune, 2005).

In other words, line AB is absolutely longer than the segment AC but at the same time, the segment AC is longer than CB. Thus , then the line has been cut into an extreme and mean ratio which is the Golden Ratio.  The relationship between the ratios of AC to CB in Figure 1 produce the never-ending number 1.6180339887…, and such never-ending numbers have create an interest among the researchers (Mario, 2002).