Golden Triangles and Rectangles
Contents
Introduction
The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most artistically pleasing rectangular shapes. The Golden Rectangle is a rectangle that is based upon the Golden Mean, which is a number that is represented by the Greek Letter phi (F) or represented decimally 1.6180339887499 etc. The Golden Rectangle is said to be one of the most visually satisfying of all geometric forms; for years experts have been finding examples in everything from the edifices of ancient Greece to art masterpieces. In recent times the validity of its link with beauty has been widely debated. These rectangles, (Golden Rectangles) are used by humans in both art and architecture and even appear in nature. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. Applications of the Golden Ratio are: ratio, similarity, sequences, constructions, and other concepts of algebra and geometry.
Task
You will investigate the golden ratio,
golden triangles, and golden rectangles;
design a poster, overhead
transparency, or power point presentation to explain basic principles to the
class;
and include a demonstration
that involves a (serious)
volunteer.
Process
Complete the investigation:
Click here to see the golden ratio.
Click here to investigate the golden triangle.
Activity I.
1.
Print the regular pentagon.
- Draw segments AC and AD.
- Use a protractor to find the measures of the angles of triangle ADC. Classify triangle ADC.
- Find the ratio of the length of AD to DC (divide). How does this ratio compare to the golden ratio?
Activity II
Use your pentagon and triangle to construct another golden triangle. Follow the steps below:
1. Using a protractor, bisect angle ADC. Label the point where the angle bisector intersects segment AC as point F.
2. What are the measures of the angles in triangle DCF? Classify triangle DCF.
3. Find the ratio of the length of DC to FC. How does this ratio compare to the golden ratio?
4. What are the conclusions you draw from this activity?
The golden rectangle is found in some art,
especially 20th Century art. But, it would seem that ancient Greek architects
did not consciously use it. The Parthenon is the most famous example of the use
of the golden rectangle. People find the golden rectangle in the Mona
Lisa, and other Renaissance
art works as well.
The
golden rectangle and the golden ratio sometimes pop up in nature. Below, we see
a spiral which comes from the golden rectangle. We are told that this is very
close to the shape of the shell of a chambered nautilus. This figure is
self-similar, each part is similar to smaller parts and larger parts. This
makes it a rudimentary fractal.
Activity
III
1.
Draw a golden
rectangle.
2. The Golden
Spiral: If you continue constructing golden
rectangles inside other golden rectangles you end up with a spiral:
You can follow the squares easily enough, the biggest one is on the left, the second biggest one is in the bottom right, the third biggest one in the top left.
Inside every square you draw a quarter circle, thus constructing a continuous curve. In the above image I've stopped after drawing 5 squares, but of course you could go on forever.
If the spiral, drawn without the scaffolding, ...
... looks familiar, it is because it often occurs in natural forms, like shells and packed seeds in flowers.
The inner place where the spiral "ends" (it never ends, of course) is sometimes called "The Eye of God".
3. Teach the class some of the basics of what you have learned and design a quick problem for a volunteer to practice (Don’t pick a goof ball). DO NOT READ BORING PARAGRAPHS TO THE CLASS! Discuss it like you know what you are talking about! Use several visual aids to help the class understand the topic
Resources
More information on Everything
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Evaluation
Web Quest Evaluation Rubric
Conclusion
Congratulations! Now that you have finished the web quest on
golden triangles and rectangles, you should have a greater understanding of the
mathematics that can be applied in artistic rules. If you enjoyed this
quest, do further research that could lead you into a career.