Analysis of the Golden Proportion - Essay Example

The Golden Ratio is a mathematic expression that traces its origins to the ancient Greeks. The Golden Ratio is expressed many different ways. When described using a segment, the Golden Ratio is realized when the segment is divided in a manner where the length of the longer section in ratio to the shorter section is equal to the ratio of the longer section to the whole of the segment. When expressed numerically, the Golden Ratio is the irrational number 1.61803399… An irrational number continues indefinitely without repeating in any salient pattern. In geometric terms, the Golden Ratio is expressed using a variety of shapes. Most commonly, golden triangles and golden rectangles are used. A shape qualifies as “golden” when the ratio of longest side to shortest side equals 1.61803399… Regular pentagons and pentagrams are also closely associated with the Golden Ratio. When a regular pentagon is constructed and the interior angles are connected with straight lines, the result is a series of triangles, each one of them golden, that form the image of a pentagram. This symbol, a pentagram transcribed by a regular pentagon became significant to the Pythagoreans. This group of people, led by the Greek mathematician Pythagoras played an important role in the early history of the Golden Ratio. Throughout history, the Golden Ratio has been known by many different names. It has been called the golden section, golden mean, symbolized by the Greek letter phi (Φ), the divine proportion, and the mean of Phidias. Some of these names solidly link the origins of the Golden Ratio with the ancient Greeks. In ancient Greece, many famous mathematicians and artists are associated with the use and development of our current understanding of the Golden Ratio. These groundbreaking individuals were interested in the Golden Ratio for different reasons. Some were intrigued simply by the unique mathematical properties of this peculiar ratio, while others sought to utilize the aesthetic strengths of the ratio when developing their art and architecture. A notable mathematician and mystic that had great influence on the discovery and utilization of the Golden Ratio was Pythagoras. The personal history of Pythagoras is colorful and full of legends. None of his actual writings have survived, so it is impossible to tell which mathematical concepts he personally discovered. Pythagoras is most famous for the discovery of the Pythagorean Theorem. Pythagoras (or one of his followers) discovered that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the shorter two sides of the triangle. Pythagoras and his followers who were know as Pythagoreans also discovered rules concerning squared numbers and squared roots. This fascination with mathematics went beyond a mere interest in numbers and their relationships with each other for the Pythagoreans. The Pythagoreans mixed mystical ideas such as the transmigration of souls with mathematical concepts because the heart of their theology stated that numbers were the foundation of all in existence. They believed in something called the Harmony of the Spheres. This belief held that mathematical rules could be applied to the movement of heavenly bodies as well as to musical notes and sounds. The discovery of mathematical truths was essential for the Pythagoreans to accomplish their ultimate goal, which was the acquisition of knowledge and wisdom. The discovery of the Golden Ratio was a monumental event for the Pythagoreans and Pythagoras himself. Some sources attribute the discovery to a Pythagorean named Hippasos. The problem with this discovery was the introduction of a new concept into the theology of the Pythagoreans, namely the irrational number. Pythagoras taught that all numbers were rational, meaning they had a rational sequence and were able to be quantified in relation to one another. When Hippasos introduced his discovery of the Golden Ratio, the Pythagoreans denounced the discovery because it did not fit with their theology and current understanding of mathematics. They reacted much the same way members of other sects have reacted through the ages when dearly held beliefs have been challenged. Depending on the source, the fate of Hippasos was either banishment from association with the Pythagoreans or execution by being thrown overboard from a ship. A second Greek that chose to further explore the Golden Ratio rather than discounting its existence was Euclid. This mathematician was the first to actually write a definition of the Golden Ration. He referred to the ratio in different terms, calling the segments of the whole the extreme and mean ratio. Euclid used the Golden Ratio in some of the proofs he developed in his monumental book Elements. In these proofs, Euclid reinforced the idea that the Golden Ratio was indeed an irrational number. A final ancient Greek that utilized the Golden Ratio was the artist and architect Phidias. Phidias was involved in the creation of one of the most famous monuments of ancient Greece, namely the Parthenon atop the acropolis in Athens. Phidias was the sculptor that constructed the monumental figure of Athena that the Parthenon was constructed to house. The figure was constructed on ivory and gold. The original figure has not survived the ages, but many true to form replicas have been preserved. Studies of these replicas reveal that the sculpture of Athena located in the Parthenon was constructed with the Golden Ratio in mind. Everything from the torso of the sculpture to the dimensions of the face indicate that Phidias was aware of this ratio. This is little wonder, because the Greeks considered golden triangles and golden rectangles to be the most aesthetically pleasing shapes that could be devised. While Phidias was not the chief architect of the Parthenon, many sources describe him as having a contributing role. His use of the Golden Ratio is evident in everything from the floor plan of the Parthenon to the front elevation of the structure. In the floor plan of the Parthenon, the interior space is divided into two segments that correspond to the Golden Ratio. One segment was a hall for the general worship of Athena, the other was a room that acted as a sanctuary and treasury for the priests of the temple. The golden rectangles created by this division are found throughout the structure. The front elevation, when measured from the top of the steps, creates a golden rectangle. Phidias appears to have been aware of the aesthetically pleasing proportions of the Golden Ratio. What is not clear is whether he intentionally utilized this ratio with the understanding of this mathematical constant, or if he just had an innate sense to how to convey aesthetically pleasing proportions in his art and architecture. No records of his thoughts or plans have been preserved so anything connecting Phidias directly with a knowledge of the Golden Ration would be speculative. The ancient Greeks are not the only ancient civilization to appear to have some knowledge of the Golden Ratio. While they do get credit for the “discovery” because theirs is the first written records to allude to the Golden Ratio or to try to define it, the ancient Egyptians appear to have some knowledge of the concept as well. The greatest evidence for this is none other than the great pyramid of Khufu at Giza. Mathematicians have estimated the original height of the finished pyramid as it would have been upon completion in ancient times. The existing structure is devoid of the original limestone casing blocks that were polished to make the pyramid gleam under the desert sun. If the height of these blocks is added to the existing structure, a golden triangle can be detected when looking at a cross section of the pyramid. If an axis is drawn from the peak of the pyramid straight down to the base of the pyramid, the right triangle formed by using the sloping side of the pyramid as a hypotenuse is a golden triangle. This fact has led many amature Egyptologists to speculate that at the very least the ancient Egyptians had some knowledge of the attributes of the Golden Ratio. Others, more on the fringes of academia and good sense, have suggested that this utilization of the Golden Ratio is proof of cosmic assistance given to the Egyptian builders by extraterrestrials. Other famous works of art from the ancient and medieval world are said to have been influeced by the Golden Ratio. Most famous of these is possibly Leonardo Da Vinci’s painting of the Mona Lisa. As Dan Brown’s novel, The Da Vinci Code so dramatically asserts, the proportion of Mona Lisa’s face, the size of the frame and tne positioning of the figure in the frame all are dictated by the Golden Ratio. Sandro Boticelli’s The Birth of Venus is also proportioned according to the Golden Ratio. In addition to being painted by masters of their craft, many feel that the use of the Golden Ratio helps to elevate these painting from outstanding works of at to true masterpieces. The Golden Ratio is far from being passé. Artists and architects still utilize the aesthetically pleasing nature of this ratio in modern creations. The recent construction of the Nerman Museum of Contemporary Art near St. Louis is a good example of architecture being influenced by the Golden Ratio. One report of the new gallery states, “Galleries defined by big walls and right angles and proportioned in accordance with the so-called golden mean or golden ratio are ideal for displaying large paintings and playing the straight man to unruly sculptures and installations.” (Thorson) The Golden Ratio has had such a ubiquitous presence in art and architecture, it is perhaps inevitable that its presence would be felt in the performing arts sooner or later. That is exactly what the Barabbas theater company has attempted during the development of their recent play, Luca. The play is the result of a 10 day workshop that focused on the influence of the Golden Ration in art. What came out of the workshop was a dodecahedron inspired set that helped tell the story of a time traveling hero. The dodecahedron was one of the regular solids that had a special relationship to the Golden Ratio. It represented the element of ether to the ancient Greeks. Ether was the element that supported all of the other elements. In the play, the protagonist and his companions dwell in the dodecahedron inspired set as they travel through the ether of time. (Ahlstrom) The play is an interesting example of art informed by the Golden Ratio because the introduction of mystical and metaphysical elements closely mirrors the beliefs of the Pythagoreans and their relationship with mathematics in general. So is the Golden Ratio the end-all and be-all of art, architecture and theater? If this is the case, why would any artist ever deviate from the ratio if it guarantees to please aesthetically? Some recent investigations into the golden Ratio have actually caused some to question the relevance of this ratio in art, architecture and even mathematics. While The Da Vinci Code has piqued interest in the use of the Golden Ratio by everyone from the ancient Greeks to modern interior designers (Breckenridge) some historians and mathematicians question the veracity of the claims made in the book. According to Gordon Williams, the theory that structures such as the Parthenon are designed according to the Golden Ratio is preposterous. (Blagger) He points out the fact that the measurements used to show that the Parthenon conforms to the Golden Ratio entirely ignores the pedistal steps and the crumbled apex of the roof. Whe the Parthenon was newly constructed, both of these elements would have been intact and they would have made the structure appear very differently than it does today. In his book, "The Golden Ratio: The Story of Phi, the Worlds Most Astonishing Number,” Mario Livio contends that the claims concerning the pyramids and their connection to the Golden Ratio are questionable at best. He points out that there is no evidence the Egyptians of other ancient civilizations such as the Babylonians or Sumerians understood the mathematical concept of the Golden Ratio. (Livio) While some may feel that the importance of the Golden Ratio is overstated by some historians, there is no denying that this very special number has had an impact on the history of civilized societies and continues to intrigue mathematicians, artists and architects today. Works Cited Ahlstrom, Dick. "Bringing the golden ratio to the stage." Irish Times. Dublin, Ireland: Irish Times, 27 October 2005. Blagger, Tim. "Da Vincis golden flaw: Use of math ratio doesnt add up, experts say ." The Morning Call. Allentown, PA: Knight Ridder/Tribune Business News, 22 May 2006. Breckenridge, Beth. "Phi designe: Arrangements prove calming, inviting." Akron Beacon Journal. Akron, OH: McClatchy-Tribune Information Services, 26 March 2007. Livio, Mario. The golden ratio: The story of phi, the worlds most astonishing number. New York, NY: Random House, 2003. Thorson, Alice. "Proportions of new Nerman Museum of Contemporary Art make it a perfect place for art." Kansas City Star. Kansas City, MO: Kansas City Star-Ledger, 14 October 2007. Read More