Imperfect Beauty

CREATER/S/. Young Ae Kim

CATEGORIES. Experimental Design, Visual Illusion, Golden Ratio

KEYWORDS. Systems Design, Symmetry in Mathematics, Learning Style, Language & Culture, Symmetry in Physics, Cognitive Psychology, Memory & Perception, Symmetry and Chaos, Gestalt principles, Universal Design, Perfection, Illusions

ABSTRACT.

Perfect beauty has been defined in many different ways by many different academics such as physicists, psychologists and mathematicians. They analyze divergent ideas of beauty to try to find out why people perceive things to be beautiful or not. In the process they develop theories that manipulate our thinking about ideas of beauty and our viewpoints about beauty.

Using the principles of symmetry found in the study of physics (including entropy), mathematics and Gestalt theory, I will investigate human perception to see how it determines the idea of beauty. I will investigate various measurement systems that were developed to find perfect proportions, and/or, perfect beauty. Despite varying definitions of beauty, these theorists thought that they could devise a kind of formula for creating absolute beauty; a universal definition. This thesis not only focuses on reconsidering what we believe beauty is and understanding how beauty is perceived and defined, but also understanding visual perceptions.

Inspiration

In a world where people see, process and remember information differently, questions about beauty arise: What is real beauty? How do we determine what true beauty is? Can the disciplines of mathematics or physics, for example, help us arrive at a definition, and can we believe their answers?

Many thinkers have tried to find absolute definitions. For example, they have devised ways to relate the human form and other natural forms to the mathematical ideals of proportional “beauty.” Alberti, a philosopher, believed that “beauty is a kind of harmony and concord of all the parts to form a whole which is constructed according to a fixed number, and a certain relation and order, as symmetry, the highest and most perfect law of nature, demands.” We have since come to recognize that this kind of effort is a rather futile attempt at codifying our understanding of visual perception within the physical realm, yet it consoles us. Many thinkers and artists have been defining proportions based on mathematical principles, such as the Golden Section, the Fibonacci series, etc. It is not difficult to figure out that these numbers are not absolute numbers. “The three dots after the numbers indicate that these numbers are ‘irrational,’ so called because they can only be approximated, never expressed fully.” (Doczi,5)

Highly educated people have tried for centuries to define beauty, but they have not been able to find an absolute definition even after spending their whole lives analyzing it. Is it possible to perceive anything, including beauty, without preconceptions? In other words, is it possible to explain beauty in absolute terms? It is a challenge to define beauty, as we shall see. We perceive things based on our own individual past experiences, education, knowledge and judgments taught to us by teachers and society. Even philosophers have tried to find ideal beauty. Aristotle, for example, stated that beauty can be achieved in the form of a well-made artifact or artwork, whether it is a candelabra or the creation of the universe by God. However, how do we define something that is well-made? Even the idea of “well-made” is relative. It is a justification given by humanity and human longing.

The focus of this thesis study will be to examine beauty through the process of exploring imperfect beauty and attempt to define our understanding of what true beauty is, how it is perceived, and how it is defined. This analysis will encourage readers to reconsider their thoughts and ideas about beauty.

Golden Section.

The golden section — otherwise known as phi, the golden mean, or the golden ratio — is one of the most elegant and beautiful ratios in the universe. Defined as a line segment divided into two unequal parts, such that the ratio of the shorter portion to the longer portion is the same as the longer portion to the whole, it occurs throughout nature — in water, DNA, the proportions of fish and butterflies, and the number of teeth we possess — as well as in art and architecture, music, philosophy, science, and mathematics. Beautifully illustrated, The Golden Section tells the story of this remarkable construct and its wide-ranging influence on civilization and the natural world, this is the most accessible and appealing guide to the subject ever assembled. The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years. In particular, Pythagoras believed that beauty was associated with the ratio of small integers and tried to prove it. “Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.” (Zeising, 27) However, haven’t we believed the Golden Ratio as a perfect proportion without any question? Just because we have used it for century that does not mean that it is the answer of the perfect proportion. It is merely a place where preconception starts.

Ancient mathematic tradition centers on a study of numbers, harmony, geometry and cosmology, and stretches back through the mystification of time into the Egyptian, Babylonian, Indian and Chinese cultures. It is evident in the layout and relationships of the stone circle and underground chambers of ancient Europe, as well as in Neolithic stones discovered in Britain. There are further clues in Mayan and other Mesoamerican artifacts and buildings and across the ocean to the Gothic masons who embedded them in their cathedral designs. The great philosopher, Plato, in his writings and oral teaching, said that there was a golden key unifying these mysteries. Despite its use in ancient Egypt and the Pythagorean tradition, the first definition comes from Euclid [325~265 BCE], who defines it as the division of a line in extreme and mean ratio. The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter φ (phi). It illustrates the geometric relationship. The ratio is important in the geometry of regular pentagrams and pentagons. Actually, the name “extreme and mean ratio” was the principal term used from the 3rd century BC until about the 18th century. Since the twentieth century, the golden ratio has been represented by the Greek letter φ. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden cut, and mean of Phidias.

The golden ratio is approximately 1.6180339887, which is from the quadratic formula. This number can be directly linked to the Fibonacci sequence. Fibonacci numbers indicates neighboring old and new stages of growth, in which each number is the sum of the two previous one — 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. Any number in this series divided by the following one approximates 0.618… and any number divided by the previous one approximates 1.618…, these being the characteristic proportional rates between minor and major parts of the golden section. Still one may ask, what is so special about these numbers? This is where the infinite numbers repeat, and the symmetry starts. For example, the sunflower’s spirals keep growing based on the Fibonacci series of numbers.

These numbers give us the idea of perfection, such as perfect measurement, perfect numbers, and perfect symmetry. They have a tendency to be close to being a perfect numerical system. However, “The three dots after the numbers indicate that these numbers are “irrational,” so called because they can only be approximated, never expressed fully.” (Doczi, 5) It seems unreasonable to believe these numeric standards. Since the Renaissance, many artists and architects have proportioned their works based on the golden ratio and believed this as a perfect proportion. However, this ratio (1.6180339887…) is approximate and cannot be definable with a rational number.

Leonardo da Vinci, especially, studied the proportions of the human body with the golden ratio and Fibonacci numbers in order to find the secret of beauty. His illustrations in De Divina Proportion and his view that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own painting. And many architects reflect this idea onto their design and system theory. Mondrian used the golden section extensively in his geometrical painting (Bouleau, 247). Interestingly, a statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in this size of their canvases. The study concluded that the average ratio of the two sides of the painting studied is 1.34, with averages for individual artists ranging from 1.04(Goya) to 1.46(Bellini) (Livio, 125).

Some scholars deny that the Greeks had any aesthetic association with the golden ratio. For example, Midhat J. Gazalé says, ‘It was not until Euclid, however, that the golden ratio’s mathematical properties were studied. In the Elements (308 B.C.) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties.’ And Keith Devlin says, ‘Certainly, the often repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value.’ (Keith, 62) Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

As illustrated above, even over an extended time span artists used measurements that are not absolute. Their obsessive trust in rules and regulations predominated their way of perceiving and creating. If we were to see da Vinci’s Mona Lisa today, there is no doubt that we would immediately identify it as a symbol of beauty, not based upon our own distinctive predispositions, but on those placed upon us by others. With this said, it is next to impossible to have an absolute guide to achieve perfect beauty through these irrational means. So why then, shall we believe something to be considered beautiful because of standards imposed upon us by popular demand? Or is it possible to see from our individual receptors?