Now, how can you use the Fibonacci numbers to make a composition that conforms to the Golden Mean or Golden Section? If you use the correct size for the overall composition, it would be about 10 x 16″ – (1-1.618) – a larger format would be 13 x 21″. In order to break this apart, and position your center of interest in the correct spot, you will need to plot the numbers on a sheet of graph paper the correct size. For instance, using a format 13 x 21″, I measured off a 13 x 13″ square on one side, which left a rectangle of 8 x 13″. From this, I formed an 8 x 8″ square leaving a 5 x 8″ rectangle. Then I measured inside this rectangle a square that is 5 x 5″ leaving a rectangle of 3 x 5″. Inside this rectangle, I formed a 3 x 3″ square, leaving a rectangle of 2 x 3″. Inside this rectangle I measured a 2 x 2″ square, leaving a rectangle of 1 x 2″. This rectangle was divided into two squares 1 x 1″ each. So you see, the sequence from inside out is 1,1,2,3,5,8,13,21 — the Fibonacci number sequence. Here’s my result–
If you draw a spiral connecting the corners of each square, it looks like the kind of spirals you see in shells, pine cones, flowers, etc. How about that!
The center of interest should be placed in the smallest squares — the largest section should conform to the principle of using the same value with different hues to keep it integrated. Here is a collage I made using this type of composition. It is a poem collage which reads from the inside out: The earth turns round. Faces the sun; A new day is born. Shall I change another’s life today? Shall I reach a goal or realize my life’s mission? Or shall I still live one day at a time, in the hope that the path I follow is His?
I invite you to try a composition in this format and see how you like it!
The Golden mean and Fibonacci numbers have been used since the time of Ancient Greece, especially in the design of the Parthenon. This system might have even been used by the Egyptians in building the pyramids. It has been used by artists such as Leonardo da Vinci, Michelangelo, Picasso, Seurat, Signac, Hopper, and Mondrian. Even musicians have used it in their works — Mozart, Beethoven (his 5th Symphony), Bach, Schubert, Bartok, Satie, and DeBussy have all been thought to use the divisions. An article in The American Scientist of March/April 1996 points out that many of Mozart’s sonatas can be divided into two parts exactly at the golden section point in almost all cases. The Mathematics Teaching magazine in 1978 points out that Beethoven used the system. It is even thought that Virgil structured the Aeneid in this way.
In architecture, the Golden Mean is a standard proportion for width in relation to height, in first story to second story buildings, in the sizes of windows. Look at any three-story bank building for instance to see the proportion in use. The College of Engineering at the California Polytechnic State University built the new engineering plaza based on the Fibonacci numbers. Plaza designer Jeffry Gordon Smith said, “As a guiding element, we selected the Fibonacci series spiral, or golden mean as the representation of engineering knowledge. ” The United Nations Building in New York is supposedly built on a golden rectangle.
What is most interesting is the way Leonardo Da Vinci’s The Last Supper was composed. The scene itself is based on two squares, with Christ in the center. All converging lines lead to the vanishing point on the horizon line, his face. The top of the windows lies at a golden section as do the outer edges of the side windows. Christ’s hands are at the golden section of half the height of the composition. The figures are grouped in threes, in a series of four shapes, with Christ forming the fifth. Application of the Fibonacci numbers includes: 1 table, 1 central figure, 2 side walls, 3 windows and figures grouped in 3’s, 5 groups of figures, 8 wall panels and 8 trestle legs, 13 individual figures.
Realizing how often the Golden Mean and Fibonacci numbers have been used in all forms of art, I tried it myself in writing a poem. I admit the structure is a little different, but here’s what I came up with based on the number of syllables in each line:
NOW YOU TRY IT!
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!
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.
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:
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:
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.
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.
Sometimes we find ourselves at a hiatus. We’ve run out of ideas and have not been creating for a spell. This is the time to explore fresh experiences in art– use new and different materials, use new and different subject matter, and look at our subjects from different angles. What subjects have you never tried before? Interior scenes, fish, old people, old family photos, gnarled hands, rock patterns, fabrics, toys, masks, crowds, death and dying, roots, nightmares, trees that look like people, mothers and children, Bible stories. The list could be endless. Pick one thing and draw different versions in your sketchbook every day until you find something that excites you.
What different angles and formats have you not tried before? Looking down, looking up, upside down or from the top– something from the back, a vignette, a silhouette, a panoramic view, or a detailed closeup.
What fresh materials could you try? Encaustic painting seems to be popular right now. I’m interested in trying silverpoint drawing. I’ve spent a lot of time with colored pencil and pen and ink lately. At one time, I made several collages. Maybe I’d like to do some printmaking one of these days, and ceramics has always interested me! “So much to do and so little time!”
One other thing you could try is to place new subjects in unrelated settings. Take four subjects and four settings, and cross-match them. For instance, apples and oranges on a theater stage, peppers on a beach, animals at a card game, books scattered all over the lawn! So try some of these ideas when you have “artist’s block.” You’ll get all creative and excited about doing art again.
To continue the discussion of design principles, we turn to RHYTHM. We can understand the use of rhythm in dance and music,but it is also important in the visual arts. And it is common to human nature. All we have to do is look around us to discover the RHYTHM in nature. The cycles of the seasons — the growth, production, death, and rebirth of the land is familiar to each of us. Even the simplest one-celled organism has a rhythmic pattern that relates it to the complex world outside. Man’s own internal system demonstrates the rhythm of life. Music and the dance began with the simple rhythms of primitive man, as he beat patterns on animal skin drums and stamped out the rhythm with his feet.
These five basic principles of design (UNITY, VARIETY, DOMINANCE, BALANCE, AND RHYTHM) work together to form aesthetic wholes in any form of art: dance, drama, literature, music, or the visual arts. The major difference between art forms is the matter of timing. The musician and the writer can manipulate an audience over a period of minutes or hours, attracting our attention, building up suspense to a climax, and unfolding the denouement to our enthralled eyes or ears. The visual artist, however, places his entire composition before the eyes of the viewer all at one, and it is the knowledge and experience of the viewer that determines how much he gleans from it. no one would leave a play in progress, but many walk by a painting with just a cursory glance. A work of visual art deserves the time and study necessary to discover the artist’s design – his plan of arrangement to achieve his total effect.
More on design principles later.
In any field of art, the first thing for the artist is his idea, or subject matter. After this comes the composition of his ideas to best achieve the effect wanted. To do his planning, the artist must be aware of certain principles or rules to be followed. No matter if the field is visual art, dance, music, literature, or drama; we still see the same principles at work. These are not considered to be rules arbitrarily made up by a teacher – they are basic to the human condition. An understanding of these principles is inherent in good art, whether you are an observer or a doer.
For example, it is a psychological truth in human nature that all men feel a “rage for order” – the need to control his situation and bring unity to his existence. We all strive for order – we organize into families, into clubs, companies, societies, and nations so that we can be stronger. “United we stand, divided we fall.” This is called UNITY.
However, UNITY can become boring at times. We need some VARIETY to avoid monotony. This often leads to CONFLICT. Biological psychological, and emotional needs trigger competition between individuals, and between parts of an individual. Life is full of conflict, and it must be resolved, or it leads to the breakdown of the individual or the society. “A man cannot serve two masters.” But CONFLICT can be constructive — it leads to growth and maturity. Every story or play must have a conflict that leads to a solution; otherwise we lose interest in it.
DOMINANCE resolves the CONFLICT. One of the opposing forces becomes stronger than the other and takes over the situation; or a decision is made that leads to a solution. DOMINANCE, or EMPHASIS, restores UNITY until the cycle is again broken. In a play or story, the solution is often called the denouement.
Although these are the most important principles of design, there are two others that are also basic to nature: BALANCE and RHYTHM. BALANCE, or stability, is indispensable to human life. For every breath we inhale, we must exhale as well. Our days of work must be balanced by nights of rest. Disease is an upset of balance, either by germs or the action of our environment. Proper medication or living style will restore the balance. In nature the rough bark of a tree is balanced by the smoothness of its leaves, and sometimes the smallest flowers have the strongest fragrance, while bright, showy flowers have no noticeable scent. This too is BALANCE.
TO BE CONTINUED!
Make a format about 7 x 9″ in your sketchbook. Start drawing a face you’ve seen in a newspaper or magazine article somewhere in the center of your sketchbook page. Then start to add faces on top of faces — all sizes, shapes, genders, ages, ethnic groups, and poses superimposing and overlapping each other. These should be purely imaginative — don’t worry if they don’t look accurate–that’s not the point. Keep going until you’ve used up most of the paper. Fill in negative shapes with dark values. Use either pen and ink, charcoal, or pencil.
Design a frame within a frame – make a 3-5 inch margin completely around your rectangular format. Extend your composition all the way to the edge, but reverse or modify the colors withing the margin.
Try a different format – a square, a circle, an elongated vertical or horizontal. or even a triangular format. I’ve seen stretched canvases in different shapes if you work with oil or acrylic.
Paint a panoramic view — looking down from a mountain. Arkansas has a lot of these — some may be painted from memory or from putting several scenes together as the one above — called Looking Southward.