Examples of Golden Ratio

Sally Painter
Taj Mahal, India

Examples of the Golden Ratio can be found in nature and in manmade artifacts, as well as buildings and even music. The Golden Ratio is also called the Divine Ratio because it's an infinite number that can't be used as a whole number or fraction. The number is written as 1.62, an abbreviation of 1.618033989. This numeric value is known as Phi. You can find the Divine Ratio in all of nature. Mathematicians, musicians and artists also use the Golden Ratio. Due to its unique properties, many believe the Golden Ratio, Golden Rectangle (also known as the Golden Proportions), and the Golden Triangle to be divine.

The Math Behind the Golden Ratio

The Golden Ratio is a mathematical truism that's used to define what's commonly known as the perfect number that's found in nature and has been duplicated and imitated by humans for centuries. The simplistic beauty of this number disguises its complexity in execution. To understand the theory behind the Golden Ratio, you must first explore the Fibonacci Sequencing of the ratio.

Fibonacci Sequence and the Golden Ratio

The Fibonacci Sequence or Series has a relationship to the Golden Ratio. The Fibonacci Series shows up in the number of leaves on a plant and the number of petals on a flower. The Fibonacci Spiral, which is found in nature, is always part of a Golden Rectangle with a Golden Ratio.

The Fibonacci Series math is simple:

  • The sequence begins with 0 and 1.
  • Just add the last two numbers together to get the next number in the series.
  • 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, and so on.
  • This Fibonacci Series example becomes: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so forth.

The Fibonacci's relationship to the Golden Ratio is realized when it's added forward, further, and further. The more you add the series, the closer you get to the Golden Ratio.

Creating a Golden Rectangle and Triangle

To create a Golden Rectangle with the Fibonacci Sequence, you start with a square. You will begin to build a rectangle by adding another square to the original square. Remember to use the formula: 0+1=1 is the first square, 1+1=2 - you'll add another square. 1+2=3 you will add three squares and next, 2+3=5, you'll add five squares. You'll continue to add squares and eventually form a Golden Rectangle.

A Golden Triangle can be created by bisecting a Golden Rectangle from one corner to the opposite corner. This creates a triangle where its three sides or angles have a 2:2:1 proportion, meaning the two long sides are equal in length and the short angle is exactly one-half the length of the two longer ones.

Architecture Examples of Golden Ratio

Once you understand the math of how a Golden Ratio is created, you gain a better appreciation of the near perfect beauty it creates in nature and art. When you start looking for examples of the Golden Ratio, you may be surprised by the many instances it has been used by mankind to create some monumental buildings and structures. When used in architecture, the building is said to be created by using "sacred architecture."

The Parthenon

parthenon

Phidias, the Greek sculptor, used the Golden Ratio in his work, especially when he began to work with the bands he sculpted just above the Parthenon columns. It's also important to note that the numeric value assigned to the Golden Ratio, Phi, was named in his honor.

If you measure the dimensions of the Parthenon's exterior, you'll discover that it not only forms a Golden Rectangle, but that there are also many Golden Rectangles between the columns. The use of the Golden Ratio accounts for the genius and beauty of this example of sacred architecture.

Great Pyramid of Giza

The Golden Ratio, Golden Rectangle, and Golden Triangle can all be found in the perfection of one of the Seven Wonders of the World, the Great Pyramid of Giza. To find the Golden Ratio, you'll need to halve the square base of the pyramid and draw a vertical line up the center of the pyramid. When this is connected to an angled side of the pyramid, you can easily see how it forms Golden Triangle with a 1.62 ratio, the Golden Ratio.

Other Architectural Examples

You can find many examples of ancient to modern sacred architecture that have the Golden Ratio in them:

  • Chartres Cathedral - Centre, France
  • Notre Dame - Paris, France
  • Porch of Maidens - Acropolis, Athens
  • Taj Mahal - Agra, India
  • United Nations Building - New York City, New York

Golden Ratio in Art

You can find many examples by master painters who understood and used the Golden Ratio. These works of perfection were created by using the ratio of Golden Rectangles and Golden Triangles. Art created based on the Golden Rectangle proves to be more pleasing to the human eye. It's one of the mysteries that surrounds this perfect rectangle and Golden Ratio.

It is known that within a Golden Rectangle are certain areas that are more visually appealing than other areas. These points are discovered by drawing a line from the bottom corner of the rectangle to the opposite corner and repeating it with the other bottom corner. These lines will intersect at the exact center of the Golden Rectangle. Next, measure the midway along each line starting from the center point. These four points are called the eyes of the rectangle (Golden Ratio). The main focus of the painting is then drawn or painted within these points of interests (ratios).

Examples of artwork featuring the Golden Ratio include:

  • Botticelli - Birth of Venus
  • Leonardo Di Vinci - Mona Lisa, Vitruvian Man
  • Michelangelo - Holy Family', David''
  • Raphael - Crucifixion
  • Rembrandt - Self-Portrait
  • Salvador Dali - The Sacrament of the Last Supper, The Persistence of Memory

Golden Ratio in Music

Music is composed of numeric value and when the Golden Ratio is used to create a musical piece, it becomes a living example of math.The Fibonacci Sequence is also prevalent in music:

  • There are eight notes to a scale
  • The third and fifth notes are the basis of chords.
  • The length, or octave, of any note is 13 notes

The sequencing continues throughout a piece of music and becomes more complex as it reaches the Golden Ratio.

A few of the classical composers used the Golden Ratio and Fibonacci Sequencing in music pieces including Bach, Beethoven, Chopin, and Mozart. Some modern composers like Casey Mongoven have explored these age-old truisms in their music.

Nature and the Golden Ratio

Fibonacci sequence in seashells

A Fibonacci Spiral can be created by using the Golden Ratio. This is a phenomenon that's found in nature. A plant's leaves grow so as many as possible can spiral up the stem. A new leaf only forms after the one proceeding it has formed.

  • Nautilus Seashell
  • Spiral Cacti
  • Spiral Galaxies
  • Sunflowers

Some flowers that have flower petals that follow the Fibonacci Sequence:

  • Three petals: Iris, lily, orchids, trillium
  • Five petal: Buttercups, geraniums, hibiscus, morning glory, nasturtium
  • Eight petals: Delphiniums
  • 13 petals: Certain varieties of daisies, ragwort, marigold

Depending on the tree species, you can also see the Golden Ratio at work within a Fibonacci number series in pinecones. You can find a series of eight spirals on one side of the pinecone with 13 spirals on the other. Another pinecone pattern has five spirals on one side with eight on the other.

The unique pattern of a pineapple is composed of diagonal shapes with eight moving in one direction and 13 in the opposite direction.

Ratio in Human Beings

This ratio is also important to not only how humans view one another, but also in how their bodies work.

Humans and Concept of Beauty

The human body and facial construction are considered beautiful the closer the features and bone structures are to the Golden Ratio. The number five and phi have been found to be the basis of the human body.

DNA Reveals the Golden Ratio

One of the most amazing examples of Golden Ratio is found within the human DNA structure. This can be seen in a single DNA cross section that reveals the DNA double helix forms a decagon shape. This is a combination of two pentagons, rotated 36 degrees from each other forms the DNA double helix The double helix spiral itself forms a pentagon. Even a single DNA molecule reveals a basis of the Golden Section or Divine Proportion.

Golden Ratio Is Divine

The Golden Ratio is often referred to as the Divine Ratio. It's easy to understand why this mathematical phenomena is considered divine. The complexity and consistent presence of the Golden Ratio in all of nature astounds and leaves the world in awe.

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Examples of Golden Ratio