Section 11C Proportion and the Golden Ratio
A. The Golden Ratio or Golden Section 1. The most visually pleasing division of a line segment was 2 portions of unequal width so the ratio of the longer portion to the shorter portion is the same as the ratio of
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2. Later in the 20th century, the Greek letter So the Golden Ratio = =
“fee” was chosen to represent this ratio.
3. Let’s do a little math on the proportion:
Cross product:
Solve for a using the quadratic formula:
Substitute this result in the original ratio
:
So what good is this ratio? It results in a ratio of a line segment that is aesthetically pleasing to the eye. It has had many uses and turns up in one form or another in art, architecture, and in natural objects…you’ll see in material yet to come . Ex 1) Suppose a rectangle has a length of 5 cm and we want the width so the ratio of the length to the width, . What is the approximate width of the rectangle so that it is a golden rectangle (using the golden ratio)?
Artists have scaled sculptures using the Golden Ratio and placed objects in artwork so they fit within a golden rectangle. The front of the Greek Parthenon is close to a golden rectangle:
Height Columns of the Parthenon here
Width = height
B. The Golden Ratio in nature 1. See figure 11.46 on page 677 – logarithmic spiral and chambered nautilus shell 2. This logarithmic spiral can also be seen in comet’s tails, the curve of ram’s horns, and other aspects of nature. 3. Consider the following Fibonacci sequence (or Fibonacci numbers). F1 1, F2 1, F3 2, F4 3, F5 5, F6 8, F7 13, F8 21, F9 34, F10 55, F11 F12 …
4. Complete the following ratios (round decimals to 3 decimal places): First ratio:
Second ratio:
Third ratio:
Forth ratio:
Fifth ratio:
Sixth ratio:
Seventh ratio:
Eighth ratio:
Ninth ratio: What do you observe about the number value results of these ratios? 5. The Fibonacci numbers appear in nature too – the number of petals on many flowers is a Fibonacci number. For example: iris has 3 petals; wild rose has 5 petals; deliphinium has 8 petals; astor has 21 petals Tree leaves grow at fractions of a full circle around a branch and the fraction includes Fibonacci numbers. For example: 2/5 (poplar, elm, apple trees); 3/5 (weeping willow, pear, locust trees); 5/13 (pussy willow, almond trees) Section 11C Assignment: page 679 # 1 – 10, 21 – 24, 26, 30, 31, 32.