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Lacee Lambert

Math 1030

John Schweitzer

Final

Is the golden ratio/rectangle the true definition of beauty?

“Without mathematics there is no art.” (Pacioli) Through out the ages beauty has been argued by the most talented minds. Balance, symmetry, order, and proportions are all contained within the Golden Ratio. Anciently this was purported to be the ideal, many structures and art were laid out accordingly. However, is the Golden Ratio still considered to be epitome of beauty, and how does this feature into design today.

“This ratio, symbolized by ϕ appears with amazing frequency in mathematics and nature.” (Pickover, 2009)

In Ancient Greece, Pythagorean contemporaries sought to find the most balanced mathematical proportion found in nature. It was decided, “…the most visually pleasing division of line…had the property that the ratio of the length of the long piece to the length of the short piece is the same as the ratio of the length of the entire line segment to the length of the long piece:

L = L + 1 or ϕ = 1 + √5 = 1.61803… or ϕ ≈ 8

1 L 2 5

(Bennett, 2011)

Since the golden ratio appears ten times within the pentagram Pythagoreans adopted it as their symbol. This was due to their admiration of the Golden Ratio or the “divine proportion” as it was often referred to. By extension the ancient Greeks adopted the Golden Rectangle as the shape with the most aesthetic balance. “…a rectangle whose side is ϕ times as long as its short side.” (Bennett, 2011) This form was used repeatedly in ancient Greek structural design such as the Parthenon, Pantheon and the Acropolis.

With the birth of Neoclassical design during the Renaissance, artists such as Leonardo da Vinci and mathematicians such as Luca Pacioli brought back the idealism of the Golden Ratio. The golden ratio was reintroduced as a form of proportion in both architecture and art alike. With the renewal of the synchronicity of art and mathematics others such as Albrecht Dürer went on to study and later further develop forms of perspective and symmetry which affect art and design today.

In 1200 AD Leonardo Fibonacci developed a numeric sequence in answer to a biological question: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on, becomes productive?” (Smoller, 2001).

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

This numeric sequence closely follows the golden ratio or proportion. Each number is equal to the sum of the pair which precedes it. This numeric sequence is found in many places in nature such as nautilus shells and flower petals. Even the fleurettes that make up the centers of daisies and sunflowers follow a pattern of a superimposed clockwise and counter-clockwise Fibonacci sequence. “The higher you go in the Fibonacci sequence, the more closely the ratio between two successive numbers in the sequence approximates phi.” (Smoller, 2001)

Some of the most useful applications of the golden ratio are those of proportion and composition. In realistic visualization and portrayal of a natural subject such as in a human or in a plant require the use of what’s known as proportion. This is a skill that allows the artist to break the subject into measurable distances from one key point or physical landmark to another. This is often done by triangulation as well as by using the thumb and pencil to visually break your subject down into measurements and angles. The golden mean, however, allows the artist to visually break a subject down into a more sophisticated and organized geometrical measure reference. If one knows the visual shape of the golden rectangle one can visually superimpose it on the subjects face or form. This allows the artist to create a measure from which to judge proportions of facial landmarks as well as determine distances and reference landmarks one to another. For example, the measurement of a person’s face might approximate golden rectangle from chin to hairline. This in turn can be divided using one and ϕ. Say ϕ falls at the eye or brow level of the subject’s face. Within that new golden rectangle composed of the forehead and brow ridge one can find ϕ again going from either direction. This should fall in conjunction with eye placement respectively. The golden rectangle can be used as a means of finding the proper positioning of joints, muscles, bones and other landmarks used to indicate length and proportion on the human body.

Composition too may be influenced by the golden ratio. Most often arts are taught to place the central focus to the side rather than the center of a piece. This placement often alludes to either the golden third or the golden rectangle. To create balance with negative space as well as focal point, each must have its own weight and prominence within a visual piece. Subjects with identifiable faces are often placed at the line of ϕ, then the gaze of the figure is directed toward the larger negative space to encourage visual motion and interplay with the viewer.

The golden ratio and the Fibonacci sequence have influenced art, architecture and design for over more than 1,000 years. Painters and architects like Leonardo da Vinci to modern artists have included the golden ratio in their work. Famous pieces like the Mona Lisa have been broken down using the golden mean proportionally to the subject’s face and body proportion. The golden ratio is also used frequently in pastoral landscapes to add emphasis. Whether this mathematical theory is still an ideal for beauty and design today as much as was in ancient Greece is still a mystery.

“Beauty: the quality present in a thing or person that gives intense pleasure or deep satisfaction to the mind, whether arising from sensory manifestations (as shape, color, sound, etc.), a meaningful design or pattern, or something else (as a personality in which high spiritual qualities are manifest).” (, 2013)

In the 1800’s Gustav Fechner, a German psychologist, did a test to determine the truth about our preference of the golden ratio. He gave a test in which he presented subjects with 10 rectangles of different proportions and oritentation and asked the subjects to arrange them in order of preference. A little over 75% of the participants chose one of three rectangles closest to ϕ and 35% chose ϕ itself as the most pleasing. Per results available none of the participants chose ϕ as the least aesthetic rectangular selection. One of Fechner’s colleagues, Oswald Külpe was said to have stated, “We have in the pleasingness of the golden section simply the pleasingness of apparently equal differences. It represents, so to speak, a symmetry of a higher order” (Green, 1995)

From Pythagoras to contemporary design the concept of the golden ratio has transcended time and directed much of what we define as art. It has been proven to be a thing of visual beauty as well as a theory of mathematical perfection. As it has been stated in the past, “Beauty is in the eye of the beholder.” However the establishment of Phi as a building block to aesthetic composition and visual balance has shaped how our various cultures interpret the world around them.

Bibliography:

• "beauty." Unabridged. Random House, Inc. 29 Apr. 2013. .

• Bennet, Jeffrey and William Briggs. Using and Understanding Mathematics: A Quantitative Reasoning Approach. Boston: Pearson Education, Inc., 2011. Print.

•   Green, Christopher D. "All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section." Perception 24.9(1995): 937-968. Web. 20 April 2010.

• “Phi Quotations.” ϕ Phi1.68, The Golden Number. PhiPoint Solutions, LLC. 2013. Web. 29 April 20013. ‹›.

• Pickover, Clifford. The Mαth βook. New York City: Sterling Publishing Co., Inc., 2009. Print.

• Smoller, Laura. “Applications: We-based Precaluculus” ULAR College of Information Science + Systems Engineering, June 2001. Web. 29 April 2013.

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