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.
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:
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 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).
Note: Insights on the use of the Golden Section by Seurat and Dali were provided by Jill Britton.
Art 102 – Painting faces like Da Vinci instead of Picasso … or “Why so long in the face?”
Many art teachers and books will tell you that a face can be drawn by dividing the face in halves and thirds, as follows:
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Draw a horizontal line halfway between the eye line and the bottom of the chin. This is the nose line.
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Draw a horizontal line one-third of the distance below the nose line and the bottom of the oval. This is the mouth line.
That’s a nice approximation, but if you want your faces to have both reality and beauty, use phi. More information on the pervasive appearance of phi in the human face is presented on the Face page, but look at the subtle difference this creates in the length of the nose and overall facial proportions:
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| Reality | By the books |
The mathematical differences in the two approaches are small, but enough to make a difference:
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This explains why portraits drawn “by the books” sometimes look a little “long in the face.”
Image source: www.dickblick.com
{ 17 comments… read them below or add one }
Hi there;
what about using Phi for a music album price? I actually saw an artist used it. Check out the below link.
http://silentaria.bandcamp.com/album/whats-real
cool
this is a cool site
Interesting
I had question about golden ratio, and this site solved it! I like it!
Hi there,
If it’s a universal proportion, does it applies also to concept of chess? Defense or offense when a certain piece is moved in specific square?
Thanks.
hey great question, i had trouble with it but i worked it out. Yes u r right.
Great take on phi, and how it applies to human faces. However, the “by the book” methods are not too inaccurate.
The line halfway between the eye line and the chin line is the line that defines the *very bottom* of the nose (not the tip). While this was recognized in your chart, it was not in your example picture; the one labeled “by the books” puts the tip of the nose at the line instead of the bottom of the nose. When the two pictures are put into photoshop, and a line is drawn on the tip of “by the books”, the line is close to the very bottom of the nose in “reality”. However, I will agree that it is still not completely accurate because this line seems to land in the nose filament (maybe 1/6 (1/4 is too low) of the way down from the tip of the nose).
Also, the line halfway between the bottom of the nose and the chin defines the very bottom of the lower lip, not the actual mouth. This was not mentioned in your chart, as both measurements are listed as the mouth, not the bottom of the lip. The “by the book” picture also assumes that the bottom of the lip line is actually the mouth line.
…although different people that are considered aesthetically pleasing have different lip sizes, so basing something on “the bottom of the lip” may not be the best method to decide where the mouth goes.
Nevermind the mouth comment, I seem to have mysteriously skipped over “one third”.
The “by the book” way I have seen often involved putting a line 3/4 of the way down between the eye line and the chin line, which would indicate the bottom of the lip. With this method, the mouth placement in the “by the book” picture would actually be considered correct, and the face would be stretched downward because of the odd placement of the nose.
I think you may be misinterpretting the context of “by the books.” It is meant to illustrate the deviation from natural facial proportions that occurs when the “rule of thirds” is applied to a face rather than drawing one based on phi, the golden ratio. So yes, the “by the books” illustration has many proportions that do not conform to the natural ones that actually appear on attractive faces.
how do we draw simple shapes like square, rectangle and circle using golden ratio? it is urgent please. i need HELP. WAITING 20 FEB
The golden ratio geometry page shows a variety of constructions of the golden ratio using triangles, squares, rectangles and circles. For practical applications with overlays that work on any program or image take a free trial of PhiMatrix Golden Ratio Software.
Very usefull………keep uploading more
Great site thx a lot!! Really useful and explains a gold section quite well:) do u have one for architecture rather than art?
See http://www.goldennumber.net/architecture/
Phi is popular in music.
TOOL’s album “Lateralus” was constructed in large part using the phi ratio and the fibonacci sequence. Drum beats, syllables per line, lyrical imagery, tempo, track ordering, all wonderfully incorporating this beautiful mathematical marvel.
Also find Vi Hart of YouTube. She’s a self-proclaimed mathemusician and brilliant educator.
would like to know how to use phi programme for interior designing for a residential layout.
thanks!