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Aim: What is the Golden Section?

Aim: What is the Golden Section?. Do Now: What do you know and/ or what do you want to know about the Golden Section and Fibonacci Numbers?. The Golden Section is often represented by the Greek letter phi. The Golden Section precisely equals: (1+ √ 5) 2

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Aim: What is the Golden Section?

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  1. Aim: What is the Golden Section? Do Now: What do you know and/ or what do you want to know about the Golden Section and Fibonacci Numbers?

  2. The Golden Section is often represented by the Greek letter phi. • The Golden Section precisely equals: (1+√5) 2 • The Golden Section is also known as: • Golden Ratio • Golden Spiral • Golden Mean • Divine Proportion • Golden Number • Golden Rectangle • Phi

  3. The golden ratio is an irrational number. It is usually rounded to 1.618. There is a way to find it more accurately though. If you use Fibonacci numbers you will find a pattern in dividing them.

  4. Fibonacci Numbers Start with any two numbers and add them together. For example: 0 and 1. So you have 0, 1, 1… Now add the next two numbers together and keep adding. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…

  5. Finding the Golden Section 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… 1/1=1 21/13=1.615 2/1=2 34/21=1.619 3/2=1.5 55/34=1.618 5/3=1.666 89/55=1.618 8/5=1.6 144/89=1.618 13/8=1.625 233/144=1.618 Notice a pattern? As the numbers get higher, they become closer to the golden ratio.

  6. The Golden Ratio on a line The Golden Section is when the ratio of the smaller section of the line to the larger section is congruent to the ratio of the larger section of the line to the whole line. Confused? Let’s take a look at it: B A M According to the Golden Ratio, AM/MB ≅ MB/AB

  7. The Golden Rectangle The Golden Rectangle is a rectangle whose sides are in proportion to the Golden Ratio. The sides are in the ratio of 1:(1+√5)/2 or 1:1.618

  8. Notice that if you take away the square portion from this rectangle the rectangle can still be divided to receive the Golden Rectangle once again. The removal of squares can be performed infinitely.

  9. Architecture Egypt: Used in the Great Pyramids. The geometry shows an angle of 51.83 degrees which is the cosine of Phi Greece: “Phi” was named for the Greek sculptor Phidias. The exterior Parthenon dimensions form a golden rectangle.

  10. Art • Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings, for example in the famous "Mona Lisa". Mona Lisa’s face is composed of golden rectangles. • In the “Vitruvian Man”, Leonardo Da Vinci shows that the human body is proportioned according to the Golden Section.

  11. Let’s Practice What We Know • List the first 20 numbers in the Fibonacci Sequence. (start with 1) • A golden rectangle has a shorter side of 100 ft. How long is the longer side? Round your answer to the nearest thousandth. • The Golden Ratio can sometimes be approximated to 5/8. This is the ratio of the two sides of a golden rectangle. If the length of a rectangular frame is 2 ft. longer than its width, what would be the length and width of the rectangle according to the golden ratio? • If a rectangular billboard is to have a height of 18 ft., how long should it be if it is to form a golden rectangle? Round to the nearest tenth.

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