Well, I’m going to teach an art class again — thought I was finished with that, but guess it’s in my “blood.” Beginning September 15 (Thursday) from 1:30 – 3:30, I will be teaching a class on how to compose a work of art at the Maumelle Senior Wellness Center in Maumelle. This is a seven week class, and will include examples, critiques, information, exercises, and perhaps occasional homework. Students will use their own materials, as well as materials provided by the instructor. I’ve had many years of experience teaching this subject, both in high school art classes, children’s classes, and adult classes. A lot of the lessons will be based on the blogs I’ve shared on this site. Cost is $45, and there is a maximum of eight students so call MSWC as soon as possible, if you want to register (501- 851-4344). I’m looking forward to seeing you and sharing my understanding of composition and design principles.
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.