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Graphic: Ji Sun An
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There's Pi Day and Mole Day, so it seems only right that the Golden Mean should have its own day. The Golden Mean, or Golden Ratio, is rounded off to eight digits, or approximately 1.618. That makes August 13--the day that is 61.8 percent of way through the year--a fitting day to proclaim Golden Mean Day. (Of course, if you prefer to celebrate during the traditional school year, you could have students figure out what 61.8 percent of your school year would be, or 61.8 percent of a specific month.)
The Golden Mean represents an aesthetically pleasing proportion of height to width: 1 to 1.618 . . . . Like pi, it's an irrational number. It's often represented by the Greek letter Phi.
There's no clear evidence pointing to who gave it its 24-karat name. Some people think it was Johannes Kepler in the 1600s, and others think it was the ancient Greeks hundreds of years earlier. Certainly, the ancient Greeks recognized the existence of the Golden Mean and used the ratio in the design of one of the world's most beautiful structures--the Parthenon. The Golden Mean can also be found in paintings by artists as diverse as Leonardo da Vinci and Piet Mondrian.
The Golden Mean can be heard as well as seen! From Mozart and Bach to Faith Hill and Britney Spears, at 61.8 percent of the way through a piece, something special almost always happens. These "special somethings" can be a sudden or brief change in key, a guitar or drum solo, the introduction of bridge music (a musical transition between themes), the final recapitulation (restatement) of a theme, or even silence. And to segue from music to musical instruments, violins often contain examples of the Golden Mean. For instance, on many violins the length of the fingerboard compared with the length of the total instrument is equal to about 1:1.618.
The Golden Mean also occurs in nature. A nautilus shell with its sequentially larger chambers is one of the most striking examples. In the human body, the ratio of the "lub" to the "dub" of a heartbeat is equal to the Golden Mean--the dub is 1.618 times longer than the lub.
You might introduce the Golden Mean by drawing a large rectangle whose sides are 55 by 89 centimeters (about 1:1.618). Divide the rectangle into a 55-cm square and a rectangle. Ask the class to observe the similarity between the larger rectangle and the smaller one. Have the students predict what would occur if you divided the smaller rectangle into a square and a rectangle. Then have a student do it. The hope is that students will recognize that the proportions remain similar.
Connecting the Golden Mean to the Fibonacci series is a natural extension since the Golden Mean and Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, . . .) are inextricably related. Have students calculate the ratios of neighboring numbers: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, . . . . These ratios steadily approach the Golden Mean as a limit.
In addition, you can extend the Golden Mean across the curriculum by studying Greek architecture or the works of artists such as da Vinci or composers such as Bela Bartok.
For another connection to music, have students test the hypothesis that something often "happens" in a musical composition about 61.8 percent through it. Ask students to choose a favorite song, multiply the number of seconds the song takes to play by 61.8 percent, and then listen to find out if any changes occur in the song around that time.
Another suggestion: Have students measure their total heights and then measure their height just to their navels. Ask them to divide their total heights by their "navel" heights. The result should be a number close to the Golden Mean.
Finally, every new holiday needs some good PR work. Have students create Golden Mean Day posters and cards--or even songs and poems--to spread the word. And you might want to top off the celebration with a PC--proportionally correct--Golden Mean cake!
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Nature's Golden Ratio
A brief discussion of the Golden Ratio and its pleasing proportions, with an emphasis on the chambered nautilus.
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Illumin: Phi's Presence in Many Facets
A discussion of the Golden Ratio, or Phi, that includes painting, music, and architecture.
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Fibonacci Numbers and the Golden Section
(An ENC Digital Dozen selection, ENC#011674) Two especially relevant parts of this site are "The Golden Section in Nature" and "The Golden Section--the Number and Its Geometry."
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The Golden Ratio
An illustrated explanation of the Golden Ratio, the Golden Rectangle, and the Fibonacci sequence.
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Fibonacci Numbers and the Golden Section in Art, Architecture and Music
Read about the Golden Mean and the Parthenon, about the Golden Mean and buildings designed by Le Corbusier, about Fibonacci numbers on the chimney of a Finnish power station, about the Golden Mean and Leonardo da Vinci's paintings, about the Golden Mean and music, and more.
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Science News Online: Stepping Beyond Fibonacci Numbers
This interesting article describes what happens when you begin a "Fibonacci"-type sequence with 2 and 1, instead of 1 and 1. The sequence, by the way, is called the Lucas sequence, named after Edouard Lucas, who was the person who named the Fibonacci sequence--how cool is that!
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Science News Online: A Fibonacci Fountain
Read about the Fibonacci Fountain--a jewel of a fountain in the center of Lake Fibonacci, in Bowie, Maryland. The artist-mathematician Helaman Ferguson used the golden ratio in creating the design. The article describes the math behind the design and includes photos of the fountain during the day and all lit up at night.
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I Am Golden
Grades 6-8. Well-thought-out lesson plan, based on a Math Vantage videotape, for teaching the Golden Mean and the Fibonacci sequence. As a bonus, it includes suggestions for extensions to literature, art, music, computer work, social studies, multicultural activities, and science.
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The Golden Ratio
(An ENC Digital Dozen selection, ENC#016300) Grades 5-12. Three students developed this excellent site as a final class project. It contains, in logical progression, a number of activities that students can work through to gain an understanding of the Golden Ratio and the Fibonacci sequence and their applications in the real world.
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Golden Ratio
(Part of an ENC Digital Dozen, ENC#020682) Grades 7-11. This is a nifty little activity in which you estimate the ratio of length to width of three rectangles and then go on to build your own approximation of a golden rectangle.
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The golden ratio: the story of phi, the world's most astonishing number
 Date: 2002 Grade(s): 11 - Post-Sec. Cost: $24.95 ENC#: 027748
This book is a thoughtful investigation of the Golden Ratio that explores its history and applications.
(For more details see ENC Record)
Find more materials by searching Curriculum Resources on ENC Online using these terms: Fibonacci sequences or Golden ratio.
Realizing that there are many ways to use this Classroom Calendar entry, we chose these standards because they relate to the entry, in full or in part. If you want to explore the standards further, please use the link(s) provided.
Connections to NCTM Standards
Algebra: Understand patterns, relations, and functions
Algebra: Analyze change in various contexts
Communication: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
Connections: Recognize and apply mathematics in contexts outside of mathematics
Geometry: Use visualization, spatial reasoning, and geometric modeling to solve problems
Measurement: Apply appropriate techniques, tools, and formulas to determine measurements
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Scott Bartley
Instructional Resources
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First posted: 3/19/2002
Last modified: 8/8/2003 |
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