Wednesday, April 23, 2014

Golden Ratio=Beauty

Phi and the Golden Ratio in Art

“Without mathematics there is no art.”  Luca Pacioli

Art 101 – Laying out a painting on a canvas

As the Golden Section is found in the design and beauty of nature, it can also be used to achieve beauty and balance in the design of art.  This is only a tool though, and not a rule, for composition.
The Golden Section was used extensively by Leonardo Da Vinci.  Note how all the key dimensions of the room and the table in Da Vinci’s “The Last Supper” were based on the Golden Ratio, which was known in the Renaissance period as The Divine Proportion.
The Last Supper by Leonardo Da Vinci makes extensive use of phi, the golden ratio, known then as the Divine proportion, in its composition
A more detailed view of Da Vinci’s intricate use of the Divine proportion is available by using PhiMatrix golden ratio design and analysis software:
Da Vinci Last Supper showing golden ratio or phi proportions

In Michelangelo’s painting of “The Creation of Adam” on the ceiling of the Sistene Chapel, look at the section of the painting bounded by God and Adam.   The finger of God touches the finger of Adam precisely at the golden ratio point of the width and height of the area that contains them both.  Alternatively, you can use the horizontal borders of the width of the painting and get the same result.  Click on the photo to see a larger version of the image.
Golden ratio composition in Michelangelo's Creation of Adam

Some say that Bottocelli composed “The Birth of Venus” such that her navel is at the golden ratio of her height, as well as the height of the painting itself. Some argue this isn’t the case. Close examination shows that you can take the golden ratio point using several different logical variations, and they all come to her navel, as well as the bottom tip of her right elbow:
  • Red line – From the very top of her hair to the bottom of her lower foot.
  • Green line – From her hairline at the top of her forehead to the bottom of her upper foot.
  • Blue line – Her height, as measured from the middle of the feet to the top of her head at the back of the part in her hair.
Perhaps a coincidence in composition, but then again perhaps not.  See a more extensive analysis yet of golden ratios in The Birth of Venus.
Bottocelli's Birth of Venus and golden ratio of navel

The French impressionist painter Georges Pierre Seurat is said to have “attacked every canvas by the golden section,” as illustrated below left.
Note that successive divisions of each section of the painting by the golden section define the key elements of composition.  This principle is illustrated in the “Golden Ruler™”below:
The "Golden Ruler" - a Golden Ratio Measuring Stick based on Phi (copyright Gary Meisner - EOT 1997)
The horizon falls exactly at the golden section of the height of the painting.  The trees and people are placed at golden sections of smaller sections of the painting.
At right, Edward Burne Jones, who created “The Golden Stairs” at right (Click for enlarged view), also meticulously planned the smallest of details using the golden section.  Golden sections appear in the stairs and the ring of the trumpet carried by the fourth woman from the top.  The lengths of the gowns from the sash below the breast to the bottom hem hits the phi point at their knees.  The width of the interior door at the back of the top of the stairs is a golden section of the width of the top of the opening of the skylight.  How many more can you find?

In “The Sacrament of the Last Supper,” Salvador Dali framed his painting in a golden rectangle.  Following Da Vinci’s lead, Dali positioned the table exactly at the golden section of the height of his painting.  He positioned the two disciples at Christ‘s side at the golden sections of the width of the composition.  In addition, the windows in the background are formed by a large dodecahedron.  Dodecahedrons consist of 12 pentagons, which exhibit phi relationships in their proportions (see Geometry for details).
The Last Sacrament by Salvador Dali uses phi, the golden proportion, in its composition as did Leonardo Da Vinci in The Last Supper
Note:  Insights on the use of the Golden Section by Seurat and Dali were provided by Jill Britton.

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