Chapter 5.5, Problem 17HP

(a)

To determine

To find:. The three different objects close or similar to golden ratio by making a table to display ratio and dimensions of each object.

Answer to Problem 17HP

The three different objects close or similar to golden ratio are Divyank ratio, Fibonacci series and DNA molecules.

Explanation of Solution

Given information:

The ratio of length and width of golden rectangle is approximately equals to $1.618$ to $1$ .

Calculation:

Divyank Ratio in which Divyank is the divine constant reveals exact values of creation, maturation and development of some perfect objects. Its most precise decimal value of Golden ratio is $1:1.618034$ .

There are total $10,34419$ strands, width of double helix in angstroms is $21$ and width of each spiral in angstrom is $22$ . Thus, the divyankis calculated as:

  $\frac{22}{21}\times \left(10.34419\right)=1.618034$

Fibonacci sequence is the series of number in which each number is the sum of two numbers that precede its. The sequence is $0,1,1,2,3,5,8,1,21,34,55,89,\mathrm{144...}$ and so on.

Consider the last two values to determine Fibonacci sequence in ratio.

  $\frac{144}{89}=1.61797753$

The width of each full cycle of DNA molecule double helix spiral in angstroms is $21$ and length in angstrom is $34$ .Thus, DNA molecules in ratio is calculated as:

  $\frac{32}{21}=1.6190476$

The table showing three different objects having golden ratiois shown below:

Different Objects	Divyank Ratio	Fibonacci number	DNA molecules
Ratios	$\frac{1.618034}{1}$	$\frac{1.6179775}{1}$	$\frac{1.6190476}{1}$

(b)

To determine

To find: Comparison of each ratio with golden ratio.

Explanation of Solution

Given information:

The ratio of length and width of golden rectangle is approximately equals to $1.618$ to $1$ .

Calculation:

The expression for golden rectangle in ratio is:

  $\frac{1.618}{1}$

The expression for Divyank ratio in ratio is:

  $\frac{1.618034}{1}$

The expression for Fibonacci sequence in ratio is:

  $\frac{1.6179775}{1}$

The expression for DNA molecule in ratio is:

  $\frac{1.6190476}{1}$

On comparing the above two expressions of golden ratio,Divyank ratio, Fibonacci sequence and DNA molecules, it can be concluded that they all the above ratios are approximately equal to each other.

(c)

To determine

To find: The usage of golden rectangle in architecture in three different ways.

Explanation of Solution

Given information:The ratio of length and width of golden rectangle is approximately equals to $1.618$ to $1$ .

1. One of the Ancient Greek architecture to determine relation between width, height, size and position of the supporting architecture used the golden ratio.
2. The Notre dame in Paris, France was also found to used the golden rectangle in their structure.
3. The Great Pyramid of Giza may also be based on golden ratio revealed from the position and relative size of the pyramid.