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, the table and ornamental shields 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 lines showing Da Vinci’s intricate use of the Divine proportion were creating using PhiMatrix golden ratio design and analysis software:
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.
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.
The French impressionist painter Georges Pierre Seurat is said to have “attacked every canvas by the golden section.” In the examples, below the horizons falls exactly at the golden section of the height of the paintings, as are other key compositional elements of the paintings. A more detailed analysis and commentary is provided the Georges Seurat and the Golden Ratio in Art Composition page.
Below, Edward Burne Jones, who created “The Golden Stairs” (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:

Draw a horizontal line halfway between the eye line and the bottom of the chin. This is the nose line.

Draw a horizontal line onethird 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:
Reality  By the books 
The mathematical differences in the two approaches are small, but enough to make a difference:

This explains why portraits drawn “by the books” sometimes look a little “long in the face.”
Image source: www.dickblick.com
{ 22 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/whatsreal
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 selfproclaimed mathemusician and brilliant educator.
would like to know how to use phi programme for interior designing for a residential layout.
thanks!
very helpful article especially about painting faces “by the book”
I loved this website but i was wondering about why and how and what does putting golden rectangles help the artist in any way to make their paintings…o3o
Golden ratios appear in nature, so applying them in design and composition can help to give a painting an instinctive natural appeal or sense of harmony and balance. Golden rectangles though are one specific expression of a golden ratio. There are many other ways to apply golden ratios in design.
Thank you for a very good description and examples of the golden mean.
At last.
Jan
Hypothetically speaking, should we not all, if made by divine proportions, be of the divine? Basically we are all the exact same looking along the lines of a proven proportion in which we are able to show our alikeness but should that not also theoretically mean that we should therefore be the same in other aspects? Hypothetically speaking of course.
I