The Golden Ones

Have you ever wondered about the arrangement of florets in a Sunflower?

Whats the secret behind this pattern?

Its Nature's best kept secret - The Golden Ratio.

"Two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887."

The golden ratio can be expressed as a mathematical constant by the Greek letter φ. Expressed algebraically-

(a + b)/a = a/b = φ

This equation has as its unique positive solution the algebraic irrational number.

φ = (1 + √5)/2 ≈ 1.6180339887

There are a several geometric entities that are based on Golden Ratio.

Golden Rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ration, 1:φ, that is, approximately 1:1.618. A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first.

Artisits have used Golden Rectangle in their art works. The dimensions of many paintings are Golden Rectangles. Architects have also used the Golden Rectangle in their designs. The Parthenon's, the UN Headquarters in NYC are examples where Golden Rectangles have been used. Golden Ratio was also used in constructing Egyptian Pyramids.

Golden Angle

The golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio that is, into two arcs such that the ratio of the length of the larger arc to the smaller is the same as the ratio of the full circumference to the larger.

Algebraically, let c be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that

c = a + b and c/a = a/b

The golden angle is then the angle formed by the smaller arc of length b. It measures approximately 137.51°.

The Plant kingdom uses Golden Angle extensively in the arrangement of leaves, branches, petals, florets. In the picture, a new leaf is formed at golden angle to the previous one.

The florets in Sunflower are arranged at golden angles forming a spiral arrangement. Golden angle results in the most optimal way of arranging the florets. In the figure below depicting arrangement of florets in Sunflower, Floret 1 is at a golden angle to floret 2. Similarly Floret 2 forms a golden angle with floret 3.

Any better mathemetician than nature?

Read More

• Fibonacci Numbers
• Fibonacci Sequence and the Golden Ratio
• Fibonacci Sequence in Nature
• Logarithmic Spiral