Golden Ratio In Nature

Golden Ratio in Nature

Introduction

The Golden ratio is represented with the Greek letter phi ( or ) and has a value of approximately 1.618. The Golden ratio creates a special fascination that has caught the attention of mathematical minds for 2,400 years. The Golden ratio can be found in many areas such as art, architecture and nature, making it a very special value that I have found interest in exploring.

Mathematics of the Golden ratio

The Golden ratio is represented as a mathematical ratio with special properties. In this section we will be discussing about the mathematical calculations behind the Golden ratio. To begin with, we will be looking at the algebraic form of the Golden ratio that is derived from the geometric relationship

The Golden ratio is a special value where it is able to define in terms of itself. This could be shown as:

The equation above is derived from the algebraic form of the Golden ratio: .

Where could be expressed in . If then would equal to . With this in mind, would equal to , creating the above equation.

Next we multiply both sides of the equation by :

Then we move everything on the left side to the right side to make it a quadratic equation:

Now, to apply the above equation to the quadratic formula, we would need to apply their coefficient to and . By looking at the equation above, would equal to 1, that is the coefficient of the first term , would equal to -1 and c would equal to -1. After listing out the coefficient of and , we are now able to insert it into the quadratic formula:

(quadratic formula)

(coefficients)

Sunflower:

In order for a sunflower to produce as many seeds as possible to help populate its future generations, it needs to use its limited space effectively in a sunflower. So how much of a turn would be the most space efficient for each seed to grow in a sunflower? A simulation allows us to input the value of rotation for each seed in a sunflower. Below are a few examples of what would happen with different values

So why is the Golden ratio the best rotation value for sunflower seeds? Well first off, it is not a rational number (able to be expressed as a fraction) which would create a pattern of lines stacking up, thus creating gaps. Secondly, the Golden ratio is an irrational number (unable to be expressed as a fraction), irrational numbers would not create straight lines that stack up. and both are irrational numbers and creates spiral lines that do not stack. But both the values of and are too close to rational numbers, being very close to and being very close to , these are called rational approximations. However, the value of the Golden ratio does not settle down to a rational approximation for very long. This is a mathematical theory called continued fractions. This is because the Golden ratio can be defined in terms of