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.