THE GOLDEN RATIO (WITHOUT GOD)

The Golden Ratio

The golden ratio, mathematically represented as the Greek letter phi (φ) is 1 : 1.618, or more precisely

Leonardo Fibonacci (1175) is commonly cited as having discovered this ratio, although it has existed from the beginning of time, and has been discovered and rediscovered throughout the centuries.  Mathematically, two quantities have the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Expressed algebraically:

This proportion has been given many different names, the Golden Section, the Golden Ratio, the Golden Mean, the Golden Cut, and the Divine Proportion. And it has inspired many applications, including the Golden Rectangle, the Fibonacci Spiral, the Golden Angle, and the Fibonacci Gauge.
The Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, 21, 34 55, 89, 144 ...) is derived from the Golden Ratio, and represents a series of ratios which approximate φ with increasing precision as the sequence progresses. For instance, 2:3 equals 1.5, 5:8 equals 1.6, and 89:144 equals 1.6179. The next number in the sequence always equals the sum of the previous two numbers.

So what's so important about this ratio? Only that it keeps popping up in nature, science, and art, and has been identified as the ratio that is the most visually pleasing.

Shells - A Fibonacci Spiral is created by drawing arcs connecting the opposite corners of squares, whose relative sizes follow the Fibonacci Sequence. Many shells follow the shape of the Fibonacci Spiral.

Sunflowers and Pinecones - The individual florets of the sunflower (and of the daisy as well) grow in two spirals extending out from the center in opposite directions. The first spiral has 21 arms, while the other has 34. These are Fibonacci numbers, and have the Golden Ratio. Similarly, pinecones have 5 and 8 arms, or 8 and 13 arms depending on their size. This arrangement has been identified as the most efficient way of filling the space on the pinecone with seeds.
 
Daisies - Most daisies have 21, 34, 55, or 89 petals - all Fibonacci numbers.

Spiral Growth - The Golden Angle, also derived from the Golden Ratio, approximates to 137.51°. This is often the angle found between successive florets or leaves, in spiral growth.


Moths & Butterflies - The proportions and placement of colorings on a moth's wings follow the Golden Ratio.
Human body - probably the most amazing example of the Golden Ratio is the human body. Obviously everyone is different, but if you take the average proportions for several people, you will start to find a pattern. Just in the human face, the following Golden Ratios are found:

• Center of pupil : Bottom of teeth : Bottom of chin = φ
• Outer & inner edge of eye: Center of nose = φ
• Outer edges of lips : Upper ridges of lips = φ
• Width of center tooth : Width of second tooth= φ
• Width of eye : Width of iris = φ

Examples from the entire body:

• The human head forms a golden rectangle (width : height = φ)
• Whole body height : head to fingertips = φ
• Top of head to fingertips : head to navel and elbows = φ
• Top of head to navel and elbows : head to pectorals and inside top of arms = φ
• Top of head to navel and elbows : width of shoulders = φ
• Top of head to navel and elbows : length of forearm = φ
• Top of head to navel and elbows : length of shinbone = φ
• Top of head to pectorals : top of head to base of skull = φ
• Top of head to pectorals : width of abdomen = φ
• Length of Forearm : length of hand = φ 

Music - How many notes are there in an octave (black and white)? How many white notes? How many black notes? That's right - all of those numbers (5, 8, 13) are Fibonacci numbers!


Architecture - Nobody knows quite why, but things that use the Divine Ratio just look good - plain and simple. Ancient ruins show the use of the Golden Ratio, and designers and architects today still refer to it as the Golden Rule.


Woodworking - The Golden Ratio makes for the most aesthetically pleasing furniture. Want to learnhow to use a Fibonacci Gauge in woodworking?


And this is just the tip of the iceberg! Other examples where the Golden Ratio has been observed in creation include the size of DNA molecules, ants, dolphins, pineapples, cactus, romanesque cauliflower, fruit seeds, the size of Saturn's ring, and the orbital periods, mean distances, and orbital velocities of the planets in the solar system and galaxies themselves.