The Geometry of The Golden Ratio: A Vision for a Golden Future
Greg Bushell CI 499
The Golden Ratio is nothing new to the world around us. For centuries, mathematics, art, architecture and science have been intertwined into a greater idea that is sometimes hard to express in words. The Golden Ratio is a concept that was never talked about in my classes as a high school student, but I was introduced to the golden concept in one of my undergraduate computer science courses. Although I haven’t even formally introduced the concept of the Golden Ratio, your mind should bring with excitement about a mathematical concept so intriguing that mathematicians are still arguing over the concept today. I will go into the history of the Golden Ratio and all it entails later in this paper, but first I feel its important to point our my interest in the topic and why I chose to explain a topic that I was only vaguely familiar with before I began my investigations. As I mentioned earlier, I was introduced to the Golden Ratio in one of my computer science combinatorics class. The Fibonnacci numbers can be given by the recurrence Fn = Fn-1 + Fn-2 with F0 = F1 = 1 (Erickson, 1999). Solving this recurrence with a mathematical concept we covered in my combinatorics class yields a closed for solution for the nth fibonnacci. After solving the equations1, we get: Fn = (1/5)1/2((1+51/2)/2) - (1/5)1/2((1-51/2)/2) Although it looks complicated, this equation is quite remarkable. Sitting in class as a sophomore I had no idea you could come up with 1000th Fibonnacci number without painstakingly calculating the previous 999 numbers. In terms of computer this closed form equation is invaluable since recursive equations such as Fibonnaci defined his sequence utilize much more memory than closed form equations. In the footnotes of Erickson’s notes, it proclaims the number, ((1+51/2)/2, is known in the mathematical
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The complete proof is available at Jeff Erickson’s CS 373 Website. URL: http://compgeom.cs.uiuc.edu/~jeffe/teaching/373/
world as the Golden Ratio and from that day forward, I have been interested in the topic although opportunity to further investigate the phenomenon did not occur until I needed an interesting mathematic concept to present to a class full of math educators. I think the name itself is something that interested me as well: why is this ratio golden anyway? I saw the opportunity to use the lesson to not only teach others but to also further my curiosity of the Golden Ratio, and all it entails. My combinatorics teacher went on the day I saw the magical, closed form Fibonacci numbers to explain that the Golden Ratio appears in many mathematical concepts, and I felt an urge to find out exactly where. Before beginning my research, all I knew was the decimal expansion of the Golden Ratio2 and that “it appears a lot of places.” I wanted to know more so I did a quick search of Google. To my delight, the search reveals over one million hits so I knew I would have plenty of material to learn and present. The mathematical journey of the Golden Ratio was about to begin. The first results of my search were a web page of Dr. Ron Knott, a former mathematics and computer science lecturer of 19 years at the University of Surrey in England. His web page is the one of the most thorough web sites devoted entirely to Fibonnacci numbers and their relation to the golden ratio. The first thing I learned from viewing this web site was that I didn’t really know much of anything! The amount of material available on Knott’s web sit is staggering and challenging to sift through, but it was a great place to begin my investigation. I started at the base of the Golden Ratio with its history and relation to Fibonacci. The history of the Golden Ratio dates back to Euclid and his book, Elements (Knotts, 1996). In Book 6 proposition 1, Euclid explains how to divide a line “in mean and
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1.61803398875…
extreme ratio,” which to us will involve finding the Golden Ratio (Simson, 1787). Below is the diagram provided by Euclid in Elements used to derive what we now call, the Golden Ratio.
<-------- 1 ---------> A G B
g
1–g
Solving the following ratio (1-g)/g = g/1 for g yields the golden ratio just as we had seen before in our search for a closed form of Fibonacci. This second derivation of the Golden Ratio was the first interesting aspect I found from looking at Elements and Knott’s web page. Although Euclid is generally thought of discovering the Golden Ratio, there is much debate of whether he was in fact the first one to use discover the Golden Ratio. It is interesting to note there are many who feel Pythagoras explored the concept almost 200 years before Euclid. Nevertheless, the ratio value 1.618033… was being discussed during the time of the ancient Greeks. I also came to understand that the origins of the actual term, “Golden Ratio” isn’t completely understood although artist Leonardo De Vinci did describe certain aspects of his paintings as section aurea or the golden section (Knotts, 1996). The opening activity I had the class conduct generated a little bit of buzz and controversy in the classroom. I asked the students to choose their “favorite” rectangle out of three choices. One was a square, another was a long width and short length rectangle, and another rectangle had a width to length ratio equal to the golden ratio also known as the golden rectangle. From my investigations, I learned that most people visually prefer the Golden Rectangle to the others (Nairan, 2002) and during class the Golden Rectangle received more votes than the other rectangles. Some people were very adamant that their
rectangle looked the best, which opened the door for me to explain that the Golden Ratio concept is controversial, and not all mathematicians believe that it is such an important ratio and that we simply invented it. From there, I learned more specifically about the mathematics behind the golden ratio. There are many applications and facts related to the Golden Ratio so I chose a few I felt were most interesting to teach to the rest of the class and to learn for myself. The first concept I thought about was trying to connect the Golden Ratio with the Fibonacci numbers. It is clear that the closed form of the Fibonacci numbers contains what we now call the golden ratio, but I felt there was a better way to show the connection. During the class presentation, I wrote the first 10 Fibonacci numbers on the board and I asked the class to determine the ratios of the nth Fibonacci number to the nth – 1 Fibonacci number. After a few calculations, the group seemed to grasp the concept that the ratio of successive Fibonacci numbers turns up to become the golden ratio as ninfinity. If I had to do this over again, I would show a graph like one listed on Knott’s site showing the progression of the Fibonacci ratios and also point out how if you continue to count Fibonacci numbers less than 0, you arrive at the other solution (1-51/2)/2. I think some of the people in class were confused on where that number went so explaining this concept would have been very helpful. From there, I went on to learn the connection between the Golden Ratio and nature. The connection was fascinating. Probably the greatest and most exciting fact I learned was the Optimal Packing Theorem. The optimal packing theorem states that on a seed head, placing a seed every 1.618033… turns yields optimal packing and exposure to
sunlight.3 It was quite simple to see how the number of seeds per rotation led to a certain pattern seed head, but it was still very interesting to see the connection between the Golden Ratio and Nature. Unfortunately, I didn’t go into much detail regarding the optimal packing in a seed head although a few people did ask me questions about the math behind it. At the time, I pointed them to Knott’s web page to look for the theorem since I did not feel comfortable with the proof. This was a mistake and an oversight on my part and given the opportunity to teach the subject again, I would go through how optimal packing works. Besides the obvious connection to recursion and the Fibonacci numbers, there are a few more topics that I see relevant to a middle school or high school curriculum. First and foremost, I think the Golden Ratio entails an interesting geometric construction that students have never seen before. In high school, students go through the basic geometric constructions, which at least for me were boring. The Golden Rectangle’s construction is simple yet intriguing and there are a few ways to do the construction. By showing there is more than one way to construct a Golden Rectangle, I think it opens student’s minds that there is usually more than one way to approach a math problem. Many times students get into the habit of believing mathematics is black and white, but this is not the case. There are many different interpretations on many mathematical problems and having an open mind to realize there can be multiple solutions to the same problem leads to a greater understanding. For instance going back to the Golden Rectangle, I would ask the students, if we know that as we ninfinity, the ratio of the nth/nth-1 Fibonacci numbers equals the Golden Ratio, how could we construct a Golden Rectangle using
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An animation of the optimal packing is available at: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html
successive Fibonacci numbers? In class, people grasped the idea how you can start with a 1x1 rectangle, add a unit length to one side to get a 1x2 rectangle then add 2 to the other side to get a 3x2 rectangle and so on and so forth. You construct a larger golden rectangle from smaller golden rectangle whose dimensions are Fibonacci numbers. Due to time constraints, the last Golden tidbit I explained was the Golden Spiral. The Golden Spiral is a spiral based on the construction of the golden rectangle. Below is an illustration of the Golden Rectangle complete with the Golden Spiral (Hu, 2003).
The Golden Spiral to me was fascinating to construct and when I learned it also appeared in nature, I really started to believe in the Golden Ratio. After having everyone construct and complete their golden rectangle complete with spiral, I showed the class how Golden Spirals are prominent in nature in seashells, pinecones, cauliflower, and various other plants. I think having the students look at online pictures where Golden Sprials have been clearly indicated gave students an easy opportunity to see the spirals in nature without too much difficulty. Looking through a few mathematics textbooks, I was surprised to find a few mentions of the Golden Ratio. Specifically, Adam and Wesley’s 6th and 7th grade mathematics books mention the golden ratio (CITATION). If I were going to teach the Golden Ratio in the classroom specifically in high school, I would do everything I mentioned above, but I would add and tweak a few things. I think the first thing I would change if given enough time would be to formally introduce recurrence relations. The
Golden Ratio and Fibonacci are intertwined and the just how exactly we arrive at the closed form of the Fibonacci numbers would provide an excellent and simply example of how to solve recurrence relations. Since 1989, NCTM standards have called for teachers to teach discrete mathematics for 9-12th grade students. This involves the discussion of iteration and recurrence relations (NCTM Principles and Standards, 2000). Deriving the closed form solution of the Fibonacci numbers although relatively simply given experience with recursion could possibly be intimidating for some students in high school, but I feel going through and showing the relations between the nth and nth-1 Fibonacci numbers is a great introduction to the topic. An extension of the Golden Rectangle construction would be to use the mirras we used in class. Showing another way to construct geometric figures is interesting, and at least from my perspective was much more fun than using a compass and straight edge. Furthermore, I think the construction of the Golden Rectangle offers up an excellent technological opportunity. CI 499 introduced me to Geometer’s Sketchpad, and I love it and hope to use it in the classroom when I am a math teacher. A mathematics course web site4 at Colorado State University offers up an excellent set of directions for constructing a Golden Rectangle using Geometer’s Sketchpad. Using sketchpad would be excellent for numerous reasons. First and foremost, it is fun to use. Students like doing hands-on activities and using technology – Sketchpad is both. Furthermore, you can animate constructions in Sketchpad, which I think students would find helpful especially when constructing the Golden Rectangle through successive Fibonacci numbers. In terms of a high school curriculum, the mathematics behind the optimal seed packing on a seed head is an excellent opportunity to illustrate ratios and rotation.
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http://www.math.csusb.edu/courses/m129/golden/golden_ratio.html
Simply put, I was confused and challenged myself with the optimal seed packing material I found on the Internet, but I think the challenge is appropriate and interesting enough for a high school level curriculum. There are a few animations available on the web that illustrates how as you change the number of turns per seed the spacing differs. During class, I didn’t go into the math behind how spacing works, and I feel I lost a valuable opportunity to explain one of the most interesting aspects of the Golden Ratio so in a classroom, I would definitely take the time to introduce the idea and show the animations found on the Internet. An addition that could be made in presenting the Golden Ratio would be to delve into some artistic aspects after all, De Vinci as stated before is most likely to have coined the term “Golden Ratio.” De Vinci was enamored with making aspects of his paintings proportional to the Golden Ratio. In his most famous painting, The Mona Lisa, the woman’s face is perfectly contained in a Golden Rectangle and also appears in a few more aspects of the painting. In a classroom, I would have the students try and discover exactly where the Golden Rectangle “fits” onto the Mona Lisa. Further more, although there is no direct evidence that the Greeks knew they were using the Golden Ratio in their buildings, the ratio appears in many buildings. I think for middle school students, the aspects of art and the Golden Ratio would be easy to understand. Students could spend some time possibly creating their own artwork containing the Golden Ratio or even possibly a model of a building utilizing the Golden Ratio. This activity is across contexts and illustrates that mathematics is more than just solving a formula; it is art, architecture and all things in between.
There are many instances where students seem to be afraid of mathematics. Even in my early field observations and during my tutoring sessions, there are many kids who are afraid or hate mathematics because teachers relay the information in lecture based forums where students have little hands on experience with the material except for homework. The Golden Ratio provides students with a topic where they can experience hands on some real world applications of mathematics. For students interested in art and architecture, the Golden Ratio provides a bridge to mathematics where hopefully they understand mathematics doesn’t have to be boring and only for certain people. I see mathematics as available to everyone, but the key is finding a way to bring the material to students by helping them create a personal connection to the material. There is another interesting and important aspect related to teaching the Golden Ratio in high school or middle school that I haven’t mentioned, but its importance cannot be overlooked. Although I have never been a teacher in a school, from the discussions during our class period, there seems to be a consensus that many administrators and or department heads are not comfortable with “non-traditional” teaching methods or lessons. Although the Golden Ratio is listed in some mathematics textbooks, most textbooks do not contain any information on the subject. It’s important as a math teacher to bring material into the classroom that you can relate mathematics even if it isn’t in the textbook. There are many mathematical concepts covered in high school that can be applied to the Golden Ratio so its legitimacy isn’t the question. There are many mathematical learning standards set by NCTM and the Illinois Board of Education that can be applied to the Golden Ratio. I have listed most of them above and no doubt there are some more I haven’t covered, but the point is, explaining to your superiors why you
are studying the Golden Ratio as opposed to basic recursion for example is important. When you know that the administration isn’t hoarding you over teaching something not in the textbook, you would become a better teacher more willing to take teaching risks. The Golden Ratio is an in depth mathematical concept, which I found interesting and challenging to learn and present. From my first days being introduced to the subject, I wanted to investigate further and my chance to teach a lesson in C&I 499 was a great opportunity to learn more about the “divine proportion.” From constructing Golden Rectangles to the Optimal Space Theorem, the possibilities in a classroom are endless. The Golden Ratio highlights an interesting aspect not all mathematics students realize in that not all math problems are black and white. There is endless interpretation in much of mathematics, which makes it every bit of an art as painting or music. The Golden Ratio in fact is tightly connected with art and architecture, illustrating that mathematics can break out of the textbook and into ones mind and imagination. Studying the Golden Ratio should be hands on and using technology adds to the excitement and ease of learning. In today’s growing technological society, any instance where we can use good technology to teach our students is invaluable. Although some may feel that teaching the Golden Ratio in our schools isn’t worth our time, letting our students minds imagine and debate about mathematical concepts is what drives learning and hopefully an improved grasp of just what mathematics is.
Euclid. (1787) The Elements (Robert Simson, Trans.). London: Nourse. (Original work published 300 B.C.)
Nairen. December 2002. Teaching Guide for The Golden Ratio. Retrieved Sepetember 22, 2004 from The University of Chicago: http://cuip.uchicago.edu/~dlnarain/golden/teaching_guide.htm Hu, Robin. April 17 2003. Golden Ratio. Retrieved [September 22, 2004] From http://hep.50g.com/golden.htm. Knott, Ron Dr. 30 November 2003. Fibonacci Numbers and the Golden Section. Retrieved September 22, 2004 from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Erickson, Jeff. (July 1999). Notes on Solving Reccurance Relations. Retrieved September 22, 2004 from http://compgeom.cs.uiuc.edu/~jeffe/teaching/373/notes/recurrences.pdf