Monthly Archives: June 2014

THE GOLDEN MEAN/ FIBONACCI NUMBERS

Look at this number sequence:  1,3,5,7,9,11 – what number should be next?  13 of course.  What about this sequence?  3,6,12,24?  The answer is 48. Now take a look at this one:  0, 1, 1, 2, 3, 5, 8, 13, 21 — what number comes next?  If you said 34 – you’d be right! You had to add the last two numbers to get the next – and so forth.

This last is called the Fibonacci sequence after its discoverer — Leonardo of Pisa known as Fibonacci (son of Bonacci) who wrote a book about math in 1202 in which he was trying to determine how fast rabbits could breed.   He was educated in North Africa and learned his mathematical system from the Moors. He helped Europe replace the Roman numeral system with the “algorithms” that we use today.

It has been found that this number sequence corresponds closely with the golden mean or section: if you divide each number by the number before it, your results get closer and closer to Phi (1/66, 1/62, 1/615, 1/619, 1/6176, 1/6181818 etc).  This sequence is found in nature – in the spirals of flower petals, seed heads, pine cones, vegetables, leaf arrangements, nautilus shells, even the human body and face.  The French architect LeCorbusier thought that the human body when measured from foot to stomach and then again from stomach to top of the head was very close to the Golden Mean.  Even the span of the arms and legs adhere to this proportion.  Dentists and oral surgeons use the proportion because the relative sizes of the jaws and teeth conform to the ratio.  The proportional ratio of the upper lateral incisors to the upper front incisors is 1:1.618!  Some believe that the more closely a woman’s face conforms to the ratio, the more beautiful she’ll look.  Leonardo da Vinci’s drawing of the Vitruvian Man depicts where he marked off proportions according to the phi progression.

Take a look at these images from nature.  Do you see where the spiral starts in the middle and progresses outward, enlarging proportionally until the sequence is completed?  More on this as it applies to the arts later!

 

100_0641nautilus

 

 

Check out how many examples of Fibonacci numbers you can see in nature — look at broccoli, cauliflower, a pine cone, etc.  Remember, though, that everything does not correspond.

 

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THE GOLDEN MEAN: PERFECT PROPORTION

A discussion by Steven Sheehan in the American Artist Magazine, September 2007, included this definition of the Golden Mean or Golden Section:  “Also known as the Golden Mean, the Golden Section is a canon of proportion used in painting, sculpture, and architecture thought to have special meaning because of its correspondence to the principles of the universe.”  This proportion is thought to be most pleasing to the human eye, and can be used in designing visual art compositions.

In the 1930’s, Pratt Institute in New York interviewed several hundred of its art students as to which vertical frame they liked the best and the least.  The ratio of 1:2 was the least liked, while the 1:618 ratio  was the preferred frame.  If this ratio was to be used in a compositional format, the shape of your paper or canvas should be 10 x 16″ rather than 11 x 14″ or 12 x 16″ (standard sizes).  To figure out a larger format using the golden mean start with a square.  Using a compass, place the center pin at the midpoint of the bottom edge (B).  Swing an arc out from an upper corner and extend the bottom edge of the square out to meet the arc (segment C).  Complete the rectangle with B=C as the base.  Now A (height) is in the same proportion of B+C as B+C is of A+B+C (the Golden Mean).

We all know how to find the “sweet spot” in a composition for the center of interest: divide the format into thirds both horizontally and vertically, and where any of the sections cross is a good place to put your center of interest.  This is the easy way, but not quite in the same proportions as the Golden Mean.  The Pastel Journal of December 2005 features an artist who uses the Golden Section for her compositions: Sydney McGinley.  Not only does she use the ratio as her format and for placing shapes within the composition, but to choose the right proportion of hues.

Here is an illustration of how to devise your own format in the Golden Section using the method outlined above:

Golden Section

 

 

 

THE GOLDEN MEAN

Have you ever heard of “The Golden Mean” or the “Golden Section?”   It is a method of design that has been used throughout the ages as the most natural and satisfying proportion known to man.  Occurring naturally in sea shells,  flowers, tree branching, certain vegetables , and even in the human body— it is thought to correspond with the principles of the universe.   Since the first century BC, it has been used in architecture, sculpture, painting, music, geometry, film-making,  furniture-making, and writing.   Modern architecture still uses the golden section, such as the United Nations building in New York. It has become a standard proportion for width in relation to height as used in facades of buildings, windows, second and third stories, and in paintings.

Vitruvius, an architect and engineer in the 1st century BC, was the first to write about the Golden Mean as the perfect proportion for buildings, rooms, and columns.  The Greeks and Romans used it to build the Parthenon, the Pantheon, and the buildings on the Acropolis.  Vitruvius’ theory became the standard for architecture, expressed in the ratio of the number 1 to the irrational 1.618034… or Phi.  In the Renaissance, Luca Pacioli of Venice published Divina Proportione, and explained the golden section thusly:

ratio

The line AB is divided  so that the length of the shorter portion is in the same ratio to the larger as the larger is to the whole.  In other words, the Golden Mean is the division of a given unit of length into two parts such that the ratio of the shorter to the longer equals the ratio of the longer part to the whole.

Phi is named for the Greek sculptor Phidias, who carved the entablature above the columns of the Parthenon.  Golden sections are formed by the distance between the columns in the ratio of 1:1.618 or Phi.  Here is a photo of the east facade of the Parthenon.

Parthenon

 

It will take several blogs to do any kind of justice to this topic, so watch for continuing articles.  Email me questions and comments, if you are interested.

PASTEL PAINTING PROCEDURES

For those pastel painters who like a step-by-step method to painting an artwork, this is for you.  I think I got this from Larry Blovits, my first workshop teacher, but I probably added to it over the years.  Some useful information is included.

PASTEL PAINTING PROCEDURES

1. SELECT A BALANCED COMPOSITION: Is there a center of interest? A variety of shapes? Forms? Colors? Values? Sizes? Movements? Does your eye lead you from one area of interest to another? Are there clues of depth? Is there a statement to be made?

2. DECIDE ON YOUR GOAL FOR THE PAINTING: What do you want to emphasize? The illusion of depth? The contrast of light against dark? A shape or pattern that is repeated throughout? An emotional feeling? Set a goal and stick to it.

3. DECIDE WHETHER THE FORMAT IS VERTICAL OR HORIZONTAL: Decide on the major divisions of the picture plane – where will the main objects be situated? You may want to lightly sketch in this composition, using a NuPastel stick. Or you may plan by doing a thumbnail sketch or two. If using paper, pad underneath with several other sheets.

4. DRAW THE MAJOR SHAPES OF THE COMPOSITION AS ACCURATELY AS POSSIBLE: Keeping shapes simple at first, work on creating proper proportion of the first shape you put into your picture plane and then measure every other shape’s height and width in proportion to this first shape.

5. MASS IN LOCAL COLOR VALUES: Mass in the foundation colors with hard pastel as close to the value as possible. The colors should be darker and more intense to begin with as they really are. Work from dark to light. Don’t put in any highlights at this point. Working with a darker, more intense color creates a stronger foundation of color as well as providing contrast for modeling with lighter colors without bleaching or “chalking out” the color.

Some artists put a hint of their lightest and their darkest values in at this point so that the range of values can be adjusted. Begin to solve basic color problems. Work on the whole picture. You can always dull an intense color, but not vice-versa. You can also lighten the dark, but not vice-versa.

6. ADD SHADE AND SHADOW: Establish the dark values – look for purples and blues in the shadow areas. Use black if your colors are not dark enough, but always layer a red, green, blue or purple dark over it so that it doesn’t look so dead. Squint your eyes or take off your glasses so that you can see the values. Think in terms of pattern, shape, and value at first.

7. REWORK AS NEEDED TO INCLUDE LIGHT, VOLUME, TEXTURES, AND DEPTH: You may want to use fixative spray between layers of pastel. Use hard and semi-hard pastels at this point. DO NOT WORK ON THE DETAILS AS YET; THIS IS THE LAST STEP IN DEVELOPING THE ARTISTIC STATEMENT. Take your time. Avoid blending – you can use a hard pastel to glaze or blend over colors. Glazing with a complement makes the colors vibrate.

8. REDUCE CONTRAST IN THE BACKGROUND, INCREASE CONTRAST IN THE FOREGROUND. Contrast should be strong in the foreground and weak in the background for the illusion of space (Aerial Perspective).   If the light is warm, the shadows should be cool, and vice versa. Have little contrast where you want the objects to recede. You only want one focus, so some things need to be unclear of hard to see (Refraction). Play with lost and found edges, soft and hard edges.

9. RESTATE ELEMENTS FOR FURTHER CLARITY: Add color accents where needed, and add foreground detail. Leave some calligraphic strokes, and arbitrary colors for pizzazz!

10. ELIMINATE MISTAKES AT ANY TIME: Cover over a color with its complement, and then change the color. You can also remove layers of pastel with a stiff brush.

11. ABOVE ALL – take your time. Don’t try to put in the excitement too early. Speed is not important; focus and perseverance is.