The Golden RatioAdvisor: Jack Heidel

Jessica Stein

Abstract: This report will discuss the many appearances of the Golden Ratio (also represented as Phi) in nature and in the arts and its association with beauty. The Golden Ratio is not just a number because of its amazing characteristics and its amazing ability to fascinate humankind throughout the centuries because of its ubiquity. It is an entire subject in itself because it is so complex and yet it is so beautifully simple. The Fibonacci Number Sequence often occurs in nature and that fact that this series converges to the Golden Ratio is nothing but incredible. The Golden Ratio is often associated with beauty but it has been argued that the connection between the Golden Ratio and beauty is vague. An attempt will be made to connect the mathematical information and the intuitive nature of Phi.

The Golden RatioAn Introduction

The Golden Ratio is not just a number because of its amazing characteristics and its amazing ability to fascinate many civilizations throughout the centuries. It is an entire subject in itself because the Golden Ratio is so complex and yet it is so beautifully simple. Represented as Φ this number has captured my attention in the infancy of my mathematical journey and the Golden Ratio has inspired me to continue in this study. I am not a math person or a science person by any means but I feel a deep connection with this number like many others throughout history. The significance of Phi has captured the attention of many intellects over the span of time including Euclid, Leonardo Da Vinci, and Johannes Kepler. Interestingly 1.618… is associated with beauty and ironically this number seemingly appears almost everywhere in nature and in the arts.

Mathematical Properties of the Golden RatioThe Golden Ratio: The Story of Phi, the World’s most astonishing NumberMario Livio

What is the Golden Ratio? Euclid of Alexandria contributed greatly to the discovery of the Golden Ratio as he describes it in his own words. “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser” (Livio 3). Consider the line AB intersected at C. What can be concluded is that AB is to AC as AC is to CB or AC/CB = AB/AC. If the length of the line AB is equal to Phi then AC will equal one and CB will equal Phi minus one as shown below.

A__________________C___________B 1 Φ-1

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Then rearranging things a bit: if AC is called x and CB is called one and because AB is to AC as AC is to CB the proportionality becomes x/1=1/(x-1). The proportionality then yields the quadratic equation x²-x-1=0 and when the positive solution is determined it is found that x = (1+ √5)/2 which is the Golden Ratio.

A__________________C___________B x 1

The Golden Ratio can be defined as a continued fraction that will infinitely break down almost like a fractal. Mario Livio describes the nature of a fractal as, “endless sequences of motifs repeating themselves within motifs, on many scales” (Livio 215). An example of this special case of a continuing fraction let Phi equal [1+1/(1+1…)]. Then the section (1+1) will break down to [1+1/(1+1…)] and the result becomes 1+[1/(1+1/[1+1…])] and the process will continue to repeat breaking (1+1) down and the next result is (1+[1/(1+1/[1+1/(1+1…)])]). As this procedure progresses the fraction becomes Phi= 1+[1/(1+[1/(1+[1/(1+(1/[1+…)])])]. An interesting find is that when this happens the evaluated fraction will eventually become Phi when it is broken down an infinite amount of times. Each time the fraction breaks down the evaluated fraction will osculate and a more accurate form of the Golden Ratio will appear as shown.

1+1/(1+1) =1.5000000001+1/(1+1/(1+1)) = 1.6666666671+1/(1+1/(1+1/(1+1))) =1.6000000001+1/(1+1/(1+1/(1+1/(1+1)))) =1.62500000001+1/(1+1/(1+/(1+1/(1+1/(1+1))))) = 1.61538461541+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1)))))) = 1.61904761901+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+(1/(1+1)))))))) = 1.61764705881+1/(1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+(1/(1+(1/(1+1))))))))))) = 1.6181818182

The Golden Ratio as a continued fraction brings up an interesting point. How can the Golden Ratio be achieved using a similar concept? It has been discovered that the Golden Ratio and the Fibonacci Number Sequence are very much related. The Fibonacci Number Sequence is computed as Fn = F(n-1) + F(n-2) which generates the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… The Fibonacci Number Sequence is affiliated with many natural occurrences including the breeding of rabbits, the optics of light rays, and the arrangements of the growth of leaves (Livio 97-109). The limit as n approaches infinity of consecutive ratios of The Fibonacci Number Sequence converges exactly to the Golden Ratio as shown [Φ =Limn→∞ (Fn /(Fn-1))]. Notice that the evaluated ratios of consecutive Fibonacci Numbers are exactly the same as that of the continued fraction demonstrated earlier.

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(Fn /(Fn-1) 1/1 = 1.000000 2/1 = 2.000000 3/2 = 1.500000 5/3 = 1.666666 8/5 = 1.600000 13/8 = 1.62500 21/13 =1.615385 34/21 = 1.619048 55/34 = 1.617647 89/55 = 1.618182 144/89 = 1.617978233/144 = 1.618056377/233 = 1.618026610/377 = 1.618037987/610 = 1.618033 (Livio 101)

After the line AB is divided into two parts according to the Golden Ratio then a second dimension can be added which results in a Golden Rectangle. The Golden Rectangle is nothing short of impressive. To geometrically construct the Golden Rectangle let the width of the rectangle equal one and the length equal to the Golden Ratio. This rectangle can be cut according to the Golden Ratio as demonstrated earlier in the line AB. If the length of the rectangle is equal to Phi then it can be cut precisely at one. As a result two sections remain. One section is a square with sides of length one and amazingly enough the other section is in fact another Golden Rectangle.

The proportionality remains the same even though it is a smaller Golden Rectangle. If the smaller rectangle is taken and reset with the length equal to Phi then the width will be equal to one. If the process of splitting Golden Rectangles can happen once why then it might as well be that the process can happen an infinite number of times. The fact is that the Golden Rectangle has very remarkable characteristics because of its infinite nature and this can be done. As a result of splitting an infinite amount of Golden Rectangles the geometric structure will be as shown. These rectangles can be infinitely large as well as infinitely small.

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(Chu-Carroll)

Using the same geometric structure of the splitting of Golden Rectangles another concept evolves. A Golden Spiral can be derived from the Golden Rectangles. Also known as a logarithmic spiral its shape remains constant as it gets infinitely large and infinitely small just like the Golden Rectangles. Its debut in nature is completely breathtaking. “Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies,” (Livio 117). Here is how it works as described by astrophysicist Mario Livio, “If you connect the successive points where these “whirling squares” divide the sides in Golden Ratios, you obtain a logarithmic spiral that coils inward toward a pole” (119).

(Obara)

Notice that the spiral occurs in the squares only and travels from one corner to the opposite corner and enters the next square at the point of intersection at where the Golden Rectangle is cut into two parts that correspond to the Golden Ratio.

To think that the Golden Rectangle is the only geometric composition would be thinking inside the box. An isosceles triangle can be derived from the original line AB intersected at C to construct this triangle. Let the two sides of identical length equal one as AC is equal to one and let the third side equal Φ-1 as CB is equal to one. This creates a triangle with the smaller angle equal to 36 degrees and the two angles that are equal to each other then become 72 degrees.

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The Golden Triangle is associated with a symmetrical pentagram. This shape had divine meaning to the ancient Greeks. “The construction of the pentagram was the main reason for the Greek interest in the Golden Ratio” (Livio 79). If the Golden Triangle is inserted in the pentagram it would be exactly proportional. One of the five sides of the pentagram is taken and would equal the length of the base of the Golden Triangle which is Φ-1. The distance from of the points at the base of the Golden Triangle to the point that is opposite to the base is equal to one. Adding the distances of the base of the triangle and the length of one of the equal sides is Phi. If the length of the base of the triangle and the length of one of the identical sides is straightened into a line then the result is the original line AB intersected at C according to the Golden Ratio.

The Golden Ratio in Science and Nature

Most artistic and creative people see incredible beauty nature. This does not assume that the less artistic people are blind to this beauty. The fact that science and math are there amongst nature and beauty is truly fascinating. One could argue that beauty is merely subjective but some things are just universally beautiful where the opinion of beauty becomes insignificant. For centuries many intellectuals have discovered that this natural beauty is connected to the Golden Ratio. The Golden Ratio is a natural phenomena that appears in such things ranging from the structure of a little nautilus shell to giant galaxies. Because of the numerous and sometimes surprising appearances of the Golden Ratio in nature it is easy to develop a deeper appreciation for this amazing number. Many scientific journals can be found referring to the occurrences of the Golden Ratio in nature and are nothing but astonishing. Such articles include topics concerning the occurrence of the Golden Ratio in special relativity, quasicrystals, atomic and ionic bond lengths, and the list vastly continues. Surely anyone with such a background would appreciate these articles. More information on these articles can be found on the works cited page.

The question is why does the Golden Ratio appear so ubiquitously in nature? Is it all coincidence or is it something else? These are questions that one can only attempt to answer but the fact that the Golden Ratio remains present in nature is a proven fact. One could continue to ponder the laws of mathematics and its application to the Golden Ratio. Many have done so and discovered the unique mathematical qualities of Phi. That

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knowledge existed even in the ancient days and it continues to be most valuable to this day. Albert Einstein mentions the nature of mathematics itself. He said, “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” The reality is that a complete reason to the natural phenomena of the Golden Ratio might always remain a mystery but it never hurts to wonder.

The Golden Ratio in DanceEntertaining with Science, Educating with DanceJennifer Burg, Wake Forest UniversityKarola Luttringhaus, Alban Elved Dance Company

An article titled Entertaining with Science, Educating with Dance attempts to bridge the seemingly the opposite subjects of math and art. This article tells of a group of scientists from Wake Forest University and the Alban Elved Dance Company. They worked together to create something that would merge these very different subjects. The scientists developed a fancy computer system for lack of better words that could make this possible. The choreography of the performance called “Fibonacci and Phi” was specifically designed to relate to the Golden Ratio. This was done by a series of fractals which appeared as animation projected on a very large screen behind the dancers as they performed. These fractals relate directly to the Golden Ratio as seen in the article. The movements of the dancers coincided with the fractals so it is as they were performing as one.

It takes extreme endurance to put on such a performance. To dance well it takes years of practice, commitment, and a certain focus because these movements are so deeply involved. The training is very intense to say the least. Then to change it up a bit the fractal animation is added and the skill required to pull that off is incredible. It’s difficult to write about a performance that I have never seen before but one can only imagine. The idea to twist science and dance in this way and including the Golden Ratio is a very creative idea because both the scientists and artists are able to learn about each other. This is important because sometimes these fields can seem so alienated from each other.

The Golden Ratio in Art

Art is a fascinating subject because one can only speculate what the artist is trying to say especially if no description of the piece survives through out history. Through the span of time the Golden Ratio has made its appearance known in these compositions. Artists are very aware of the Golden Ratio as evidence to its numerous appearances in art whether they realize it or not.

A few famous masterworks include: Bathers by Seurat, St Jerome by Leonardo Da Vinci, The Mona Lisa by Leonardo Da Vinci, and self-portrait by Rembrandt. It has been speculated whether or not the existence of the Golden Ratio in countless works of

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art were done so with a conscious attempt by the artist. In other words was it done on purpose? Because many original documents by Da Vinci have survived about his interest in the Golden Ratio it can be easily concluded that he was aware of these proportions. With other artists the connection between the Golden Ratio and their art work is a mystery. Because of the accuracy and precise proportion of the Golden Ratio in these works, it is difficult to say one way or another. In many cases the artists probably did not use the proportions of the Golden Ratio on purpose.

How is it possible that these artists can create a piece without realizing that they are in fact using the Golden Ratio’s proportions? It is sort of a paradox in its own right. Professor Bryce Speed from the University of Nebraska-Omaha’s art department described this paradox:

Artists have an innate strong visual aptitude for design and harmony. Some artists are always striving for harmony and balance and a way to organize their compositions. These choices are successful of not successful based on their ability to sustain the viewer. While an artist may not be aware that they are using the golden section, they possess some ability to compose their canvases in a harmonious way to where most good designs can somehow take on the proportions of the golden mean. Most masterworks can be broken down according to this.

Others call it pure intuition. When something in a composition is not balanced it is obvious to the artist when others might not even notice. This intuition is a feeling or even an instinct. Any result of attempting to describe an artist’s intuition would be horribly vague. To describe intuition would result in a sacrifice to one’s academic credibility. However sometimes there is a need to attempt to describe this. The Golden Ratio has a certain balance that appeals to artists because it is in fact beautiful. In art balance equals beauty in most cases and this is how artists connect the Golden Ratio with beauty. Some could say that it is just pure coincidence that the beautiful ratios used in art are in fact the Golden Ratio even if the Golden Ratio was used on purpose or not. On the contrary the Golden Ratio appears too often to call it just pure coincidence. In both cases more research could be done to determine if the appearance of the Golden Ratio in art is just a red herring or if it does have a deeper meaning but when dealing with the intuition of an artist this task could prove very difficult.

The Golden Ratio in Music

Music itself has a very mathematical background that has been around since the ancient days and still it is implemented today.

Every string quartet and symphony orchestra today still uses Pythagoras’ discovery of whole-number relationships among the different musical tones. Furthermore, in the ancient Greek curriculum and up to medieval times, music

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was considered a part of mathematics, and musicians concentrated their efforts on the understanding of the mathematical basis of tones (Livio 183).

Pythagoras noticed that “…the musical intervals and the pitch of the note correspond to the relative lengths of vibrating strings. He observed that dividing a string by consecutive integers yields (up to a point) harmonious and pleasing intervals.” (Livio 28). Interestingly there are only certain combinations of these lengths that are pleasing to the ear. Such ratios include 1:1, 1:2, 2:3, 3:4, etc (Livio 29). Finding the harmonic progressions in the notes of a musical scale was a huge discovery in music. It could be in fact one of the greatest finds in music throughout history. Pythagoras’ discovery of the relationship of musical tones set the stage for the all of the music that was to come.

The speculation of the Golden Ratio appearing in music is quite natural because there is a strong relationship between mathematics and music (Livio 184). Livio explains in detail how this works. The major sixth and minor sixth are said to be the most aesthetically pleasing of all musical intervals (185). Each note has its own frequency that identifies a particular pitch. The note A vibrates at 440 vibrations per second and the note C vibrates 264 vibrations per second. A major sixth is produced when these notes are combined together. Interestingly the ratio of these frequencies is 440/264 with becomes 5/3. The same occurs with a minor sixth with High C that vibrates at 528 vibrations per second and E that vibrates at 330 vibrations per second. The ratio of these frequencies becomes 528/330 and this results in a ratio of Fibonacci numbers when reduced to 5/3. Although it is said that these are the most pleasing intervals it is still quite vague. Still the value is not diminished because it is a little unclear (186).

The Golden Ratio is said to appear in music but patterns in music are more subtle as explained by Roy Howat in the article “Debussy, Ravel, and Bartok: Towards Some New Concepts of Form.” “Of all traditional compositional techniques…In painting and architecture this facet of the craft is self evident to a far greater degree, working as it does in spatial dimensions; in music it is less obvious, since there it must operate in temporal terms.” Professor Sever Tipei, Manager of Computer Music Project of the University of Illinois Experimental Music Studios responded to a student who inquired about the Golden Ratio in music. Since Dr. Tipei formulated his words so intelligently here are his exact words.

The golden mean ratio can be found in many compositions mainly because it is a "natural" way of dealing with divisions of time. One can find it in a lot of works by Mozart, Beethoven, Chopin, etc., etc. It is a question if it was used in a deliberate way or just intuitively (probably intuitively). On the other hand, composers like Debussy and Bartok have made a conscious attempt to use this ratio and the Fibonacci series of numbers which produces a similar effect (adjacent members of the series give ratios getting closer and closer to the golden mean ratio). Bartok intentionally writes melodies which contain only intervals whose sizes can be expressed in Fibonacci numbers of semitones. He also divides the formal sections of some of his pieces in ratios corresponding to the golden

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mean. Without going into much detail, Debussy also does this in some of his music.

Erno Lendvai was a Hungarian musicologist that claimed that the Golden Ratio occurs in Bartok’s compositions like Music for Strings, Percussion and Celeste. From the words of Lendvai concerning the Golden Ratio in Bartok’s music, “from stylistic analyses of Bartok’s music I have been able to conclude that the chief feature of his chromatic technique is obedience to the laws of Golden Section in every movement” (Livio 188). There has been hardly any or no scholarly criticism on the books Lendvai has written on the subject at hand (Howat 286). Howat continues to mention the calculations of Lendvai, “Many of the calculations indeed are not particularly accurate. However they are near enough frequently enough to show some clear cases of musical structures according almost totally to the principle of the Golden Ratio” (Howat 286). Not only does the Golden Ratio appear in the structure of Bartok’s work but in the dynamics as well as mentioned in the same article, “Debussy, Ravel, and Bartok: Towards Some New Concepts of Form.” “A well-known aspect of the Golden Ratio, which Lendvai also mentions, is that it is usually a dynamic division, implying tension, whereas symmetrical division tends to be static and neutral. …the point of maximum tension from the tonic at G, the tritone, is situated at the Golden Mean between the beginning and the return to the opening tonality,” (Howat 289). Although some parts of the Golden Ratio occurring in music is somewhat vague. The evidence of the Golden Ratio in the works of Bartok and others is more solid.

A Connection/Conclusion

The Golden Ratio has in fact many names and is referred to as the Golden Number, the Golden Section, the Golden Mean, and it has even been called the Divine Proportion (Livio 3). The names are different but the magic is still the same. There are any names but it is all one number, 1.618... The Golden Ratio is equally incredible in all the places it makes its debut whether it is in music, in art, in mathematics, in nature, or in what ever it may be. These occurrences stirred the fascination of Phi that has never ceased. There is a rich history of the Golden Ratio that can only be described as magnificent because Phi has captured the attention and hearts of many generations throughout the centuries.

Many if not all things are described in terms of mathematics as illustrated by Albert Einstein, “Mathematics is only a means for expressing the laws that govern phenomena” (Livio 30). Mathematics governs our universe and also the reality in which we live. Someone once said, “If it's not mathematical in nature, then it cannot be.” The association between art and beauty and nature and beauty is quite obvious but mathematics is also very beautiful. Unfortunately, mathematical beauty is often unnoticed and under appreciated. That fact is that everything is mathematical in nature and this makes it beautiful.

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The nature of mathematics as a description of reality brings up an interesting point. What is reality? This question of metaphysics branches off into three categories. They are: what is being, what is the highest form of being, and what is the cosmos? The Pythagorean school of thought was, “…if musical harmony can be expressed in number, why not the entire cosmos?” (Livio 29). It has been discovered that mathematics does in fact express the cosmos as we know it. But because the complete knowledge of reality is limited by our perception of reality these are questions that cannot be answered fully. An interesting fact about our perception of reality is that the human brain processes 400 billion bits of information per second but we are only aware of about 2,000 of them. So what else is out there? Mathematics is beyond what we perceive to be reality because it is perfect. They way the laws of nature are conducted through mathematics is nothing but spectacular because these laws are virtually flawless. The truth is we will probably never have an omniscient grasp of reality but if one searches for absolute truth it can be found in mathematics.

What does this have to do with the Golden Ratio? From a mathematical perspective the fact is the unique characteristics of Phi define a part of reality. The Golden Ratio has great significance because it is so ubiquitous and perfect. The question is why? It is truly a great phenomenon that the Golden Ratio is so consistent in nature and in art. Going back to Pythagoras’ concept of numbers defining the cosmos is a good place to start to search for the answers. But to attempt to find absolute truth as to why this happens may prove to be a difficult task.

How is the Golden Ratio connected mathematically to the arts? Math and art can seem completely opposite but the truth is they are one in the same. Every curve, every shape, every color, every texture, every line can somehow be defined mathematically. It just so happens that ratios that seem to be the most harmonious in art seem to be the Golden Ratio or something very close to it. To try to explain completely why this happens would result in stepping into grey areas. To prove anything in art can be complicated because it is so subjective. Artistically speaking the Golden Ratio is universally pleasant and beautiful without reason.

The Golden Ratio is stunning and impressive. An attempt has been made to connect the seemingly opposite and different characteristics of Phi. Hopefully some grey areas were covered and some conclusion was reached even though it would not be much more than the original material. Attached to this report is a mathematical journey of the Golden Ratio by Dirk Huylebrouck. It includes a different mathematical approach to the Golden Ratio by optimization.

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Works Cited

Burg and Luttringhaus. “Entertaining With Science, Educating with Dance.” Computers in Entertainment. 4.2. Apr.-June 2006. CSA. University of Nebraska, Omaha Library. 12 Dec. 2006 <http://proquest.umi.com/pqdlink?Ver=1&Exp=012012&FMT=7&DID=1041971741&RQT=309>

Chu-Carroll, Mark C. 2006. Good Math, Bad Math 12 Mar. 2007. <http://scienceblogs.com/goodmath/2006/08/_the_golden_ratio.php>

Cooke, Todd. “Do Fibonacci Numbers Reveal the Involvement of Geometrical Imperatives or Biological Interactions in Phyllotaxis?” Botanical Journal of the Linnean Society 150.1 Jan. 2006: 3-24 Academic Search Premier. EBSCO. University of Nebraska, Omaha Library. 8 Jan. 2007 <http://web.ebscohost.com.leo.lib.unomaha.edu/ehost/detail?vid=21&hid=112&sid=11c28cf0-d976-4836-84f6-73af483569ce%40sessionmgr109>

Falcon, Sergio. “Classroom Note: A Simple Proof of an Interesting Fibonacci Generalization.” International Journal of Mathematical Education in Science & Technology 35.2 Mar.-Apr. 2004: 259-261 Academic Search Premier. EBSCO. University of Nebraska, Omaha Library. 12 Dec. 2006 <http://web.ebscohost.com.leo.lib.unomaha.edu/ehost/detail?vid=10&hid=112&sid=11c28cf0-d976-4836-84f6-73af483569ce%40sessionmgr109>

Golden Ratio and Fibonacci Numbers. Tipei, Sever. University of Illinois Experimental Music Studios. 22 Mar. 2007 <http://web.hep.uiuc.edu/home/karliner/golden.html>

Golden Ratio in Art and Architecture. Obara, Samuel. The University of Georgia Department of Mathematics Education. 22 Mar. 2007. <http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html>

The Golden Section as an Optimal Solution. Huylebrouck, Dirk. 2006. 16 Nov. 2006 <http://members.tripod.com/vismath7/proceedings/huylebrouck.htm>

Heyrovska, Raji. “The Golden Ratio, Ionic and Atomic Radii and Bond Lengths.” Molecular Physics 103.6-8 Mar. 2005: 877-882 Academic Search Premier. EBSCO. University of Nebraska, Omaha Library. 12 Dec. 2006 <http://web.ebscohost.com.leo.lib.unomaha.edu/ehost/detail?vid=9&hid=112&sid=11c28cf0-d976-4836-84f6-73af483569ce%40sessionmgr109>

Howat, Roy. Debussy, Ravel and Bartók: Towards Some New Concepts of Form

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Music & Letters. 58:3 285-293. July 1977. <http://links.jstor.org/sici?sici=0027-4224%28197707%2958%3A3%3C285%3ADRABTS%3E2.0.CO%3B2-5>

Livio, Mario. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number. New York: Broadway Books, 2002.

Livio, Mario. “Searching for the Golden Ratio.” Astronomy 31.4 Apr. 2003. Academic Search Premier. EBSCO. University of Nebraska, Omaha Library. 12 Dec. 2006 <http://web.ebscohost.com.leo.lib.unomaha.edu/ehost/detail?vid=8&hid=112&sid=11c28cf0-d976-4836-84f6-73af483569ce%40sessionmgr109>

Locke, Derek. Numerical Aspects of Bartók's String Quartets. The Musical Times, Vol. 128, No. 1732. (Jun., 1987), pp. 322-325: <http://links.jstor.org/sici?sici=0027-4666%28198706%29128%3A1732%3C322%3ANAOBSQ%3E2.0.CO%3B2-H>

Medway Crop Circle The Crop Circle Phenomenon in Kent. 2006. Text & Diagrams. 22 Mar. 2007. <http://www.medwaycropcircle.co.uk/article6.htm>

Sigalotti, Mejias. “The Golden Ratio in Special Relativity ” Chaos, Solitons & Fractals. 30:3. Nov. 2006: 521-524. Academic Search Premier. EBSCO. University of Nebraska, Omaha Library. 12 Dec 2006 <http://web.ebscohost.com/ehost/detail?vid=5&hid=114&sid=2fa10f55-d80c-4c1c-894b-46893f2dd52a%40sessionmgr102>

Speed, Bryce. “Re: question…” Email to Jessica Stein. 9 Apr. 2007.

Symmetry as a Compositional Determinant. Solomon, Larry J. 2002. 22 Mar. 2007. <http://solomonsmusic.net/diss7.htm>

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