Examples of the Golden Ratio can be found everywhere in classic architecture, artwork, nature, and even music. This expression of proportion, also known as the Divine Ratio, offers harmonious composition through the application of an irrational number (1.618) in design, both natural and human-made. Most likely, you're unaware of the math behind the beauty of the Golden Ratio because it produces objects and music that you will naturally find pleasing without needing any explanation.
Golden Ratio in Architecture
The Golden Ratio creates nearly perfect beauty in nature and art. When you start looking for Golden Ratio examples in everyday life, you may be surprised by the many instances it has been used to create many monumental buildings and structures. When the Golden Ratio is used in architecture, the building is said to be created by using "sacred architecture." By applying the calculation of the Golden Ratio to a square, architects and designers are able to create the Golden Rectangle, which is believed to have the most aesthetically pleasing proportions to the human eye.
Golden Rectangle Example: The 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. This sacred structure is a beautiful example of the Golden Ratio in architecture.
Golden Triangle Example: 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 1.618 ratio, the Golden Ratio.
Other Architectural Examples
You can find many examples of ancient to modern sacred architecture and Golden Ratio buildings.
- 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 Examples
You can find many examples by master painters who 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.
Golden Ratio in Art
Golden ratio art examples include:
- Leonardo Di Vinci - Mona Lisa, Vitruvian Man
- Botticelli - Birth of Venus
- Michelangelo - Holy Family, 'David'
- Raphael - Crucifixion
- Rembrandt - Self-Portrait
- Salvador Dali - The Sacrament of the Last Supper, The Persistence of Memory
Using Golden Ratio in Art Composition
Within a Golden Rectangle are certain areas that are found to be 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 focal point of the painting is then drawn or painted within these points of interests (ratios).
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 in 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.
Composers Who Used the Golden Ratio
Some of the most known classical composers used the Golden Ratio and Fibonacci Sequencing in their music pieces, including Bach, Beethoven, Chopin, and Mozart. Some modern composers like Casey Mongoven have explored the Golden Ratio in their music.
Golden Ratio Examples in Nature
Where is the Golden Ratio found in nature? It's in the Golden or Fibonacci Spiral, which can be created by using the Golden Ratio. This is a phenomenon that's found extensively in the natural world. 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.
- Spiral cacti
- Spiral galaxies
Flowers With the Fibonacci Sequence
Some flowers have flower petals that follow the Fibonacci Sequence:
- Three petals: Iris, lily, orchids, trillium
- Five petals: Buttercups, geraniums, hibiscus, morning glory, nasturtium
- Eight petals: Delphiniums
- 13 petals: Certain varieties of daisies, ragwort, marigold
Fibonacci Spiral in Pinecones
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.
Golden Ratio in Humans
This ratio is also important to not only how humans view each other but also in how their bodies work, and in their DNA.
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.
The Math Behind the Golden Ratio
Finding the Golden Ratio in real life is simple, as it appears all around you. It is a mathematical truism that's used to define what's commonly known as the perfect number found in nature that 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.
Golden Ratio in the Real World
The Golden Ratio is often referred to as the Divine Ratio because of its prominence in nature, and in human bodies. The discovery that the Golden Ratio is present in so many living things fostered a reverence for this magical proportion, and it continues to be an inspiration for artists and creators today.