Golden Ratio

Introduction

Many ideas can be connected to one another by a single mathematical concept. Consider the following ideas: Pascal's triangle, Newton's binomial formula, the Fibonacci sequence, probability, the golden mean, the golden rectangle, the equiangular spiral, the golden triangle, pentagram, limits, series, platonic solids, regular decagon... Also, consider the fields of psychology, art, architecture, physiology, statistics, and nature. There are many branches of mathematics and fields of study in our world that seem independent of one another, but if one were to look closely a connection could be found. It is exciting to learn or discover these connections.

Guiding Question

What is the "golden ratio" and what are its connections to mathematics, nature, and other disciplines?

Questions

1. What are the different expressions for its value (approximate and exact)?
2. How have artists used the ratio in their work?
3. What is gnomonic growth (pronounced "nawmic") and how is it related to the ratio?
4. How was the golden ratio discovered and who discovered it?
5. How does the ratio relate to probability and statistics?
6.What are the geometric applications for the ratio?
7. What objects in nature display the ratio and how?
8. What is the connection between the ratio and the Fibonacci sequence?
9. What is the "golden mean?"
10. How is the ratio related to psychology?
11. How is the golden ratio used in architecture, construction, or just the production of our daily products?
12. What are the applications of the ratio in human physiology?
13. How can the "golden rectangle" be constructed?

Internet Resources

The Golden Section Ratio: Phi
THE GOLDEN RATIO IN THE ARTS
Gnomonic Growth
History of the Golden Ratio
THE GOLDEN RATIO IN NATURE
Nature's Golden Ratio
The Golden Ratio and Aesthetics
The Fibonacci Sequence and the Golden Ratio
Golden Rectangles
The Golden Section in Biology

Activities
1. Research and describe the connections between the ratio and the discipline(s) you discovered.
2. Create a concept map describing the connections between the disciplines and the ratio.
3. Write a mathematical lesson for inclusion in a geometry textbook that would teach the ideas of the golden ratio to another student.
4. Create a children's book about the golden ratio.
5. Write a report on the evolution of the discoveries associated with the golden ratio.
6. With a group, prepare a PowerPoint presentation or presentation board discussing  the connections between the golden ratio and the above named disciplines.

Jason Murphy-Thomas, Teacher; and Yezhou Shi, AP student , collaborated to develop this web hunt.

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