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7.4 Fibonacci & Other Patterns

7.4 Fibonacci & Other Patterns
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  • 7. Golden Rectangle

    Determine whether the golden ratio applies to the object. (See Example 4.)

    • Worked-Out Solution

      The Golden Rectangle has a length and width whose ratio is about 1.62. The dimensions of the credit card are about 26 units by 16 units. The ratio of the length to the width is

      So, credit cards have dimensions that are close to the Golden Rectangle.

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      _____    __   __    _  __   _    _     _  __  
     /  ___||  \ \\/ //  | |/ // | || | ||  | |/ // 
    | // __     \ ` //   | ' //  | || | ||  | ' //  
    | \\_\ ||    | ||    | . \\  | \\_/ ||  | . \\  
     \____//     |_||    |_|\_\\  \____//   |_|\_\\ 
      `---`      `-`'    `-` --`   `---`    `-` --` 
                                                    
    
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  • 8. Golden Rectangle

    Determine whether the golden ratio applies to the object. (See Example 4.)

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    __    __     ___    __    __    _____    __   __  
    \ \\ / //   / _ \\  \ \\ / //  |  ___||  \ \\/ // 
     \ \/ //   | / \ ||  \ \/ //   | ||__     \   //  
      \  //    | \_/ ||   \  //    | ||__     / . \\  
       \//      \___//     \//     |_____||  /_//\_\\ 
        `       `---`       `      `-----`   `-`  --` 
                                                      
    
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  • 9. Golden Rectangle

    Determne whether the golden ratio applies to the object. (See Example 4.)

    • Worked-Out Solution

      The Golden Rectangle has a length and width whose ratio is about 1.62. The dimensions of the mirror are about 26 units by 16 units. The ratio of the length to the width is

      So, the mirror's dimensions are close to the Golden Rectangle.

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      _____    __   __    _  _      ___     __   _   
     /  ___||  \ \\/ //  | \| ||   / _ \\  | || | || 
    | // __     \ ` //   |  ' ||  / //\ \\ | '--' || 
    | \\_\ ||    | ||    | .  || |  ___  ||| .--. || 
     \____//     |_||    |_|\_|| |_||  |_|||_|| |_|| 
      `---`      `-`'    `-` -`  `-`   `-` `-`  `-`  
                                                     
    
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  • 10. Golden Rectangle

    Determine whether the golden ratio applies to the object. (See Example 4.)

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      _  __    _____    __   __   __   __    ______  
     | |/ //  |  ___||  \ \\/ //  \ \\/ //  /_   _// 
     | ' //   | ||__     \   //    \ ` //     | ||   
     | . \\   | ||__     / . \\     | ||     _| ||   
     |_|\_\\  |_____||  /_//\_\\    |_||    /__//    
     `-` --`  `-----`   `-`  --`    `-`'    `--`     
                                                     
    
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  • 11. Beauty

    Since ancient Greece, symmetry and the golden ratio often have been thought to embody ideal beauty. Dr. Stephen Marquardt has studied human beauty for years across both genders and for all races, cultures, and eras. Through his research, he developed and patented the RF Mask, which he claims is the most beautiful shape for a human face.

    Use the Internet to describe why Dr. Marquardt believes that the RF Mask models the perfect human face. (See Example 4.)

    • Worked-Out Solution

      Dr. Marquardt believes that the golden ratio of about 1.62 is the most pleasing ratio to humans. His RF Mask identifies several prominent dimensions on a human face. He believes that if these dimensions are close to the golden ratio, then the person's face will be perceived by other humans to be beautiful.

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      ______     ___       _____   _    _    __   _   
     /_   _//   / _ \\    / ___// | || | || | || | || 
       | ||    / //\ \\   \___ \\ | || | || | '--' || 
      _| ||   |  ___  ||  /    // | \\_/ || | .--. || 
     /__//    |_||  |_|| /____//   \____//  |_|| |_|| 
     `--`     `-`   `-` `-----`     `---`   `-`  `-`  
                                                      
    
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  • 12. Beauty

    Since ancient Greece, symmetry and the golden ratio often have been thought to embody ideal beauty. Dr. Stephen Marquardt has studied human beauty for years across both genders and for all races, cultures, and eras. Through his research, he developed and patented the RF Mask, which he claims is the most beautiful shape for a human face.

    Compare the distance between the pupils to the length of the nose on the RF Mask. What do you notice? (See Example 4.)

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      ____      ______   _    _      ___      _____   
     |  _ \\   /_   _// | |  | ||   / _ \\   / ____|| 
     | |_| ||   -| ||-  | |/\| ||  / //\ \\ / //---`' 
     | .  //    _| ||_  |  /\  || |  ___  ||\ \\___   
     |_|\_\\   /_____// |_// \_|| |_||  |_|| \_____|| 
     `-` --`   `-----`  `-`   `-` `-`   `-`   `----`  
                                                      
    
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  • 13. Beauty

    Since ancient Greece, symmetry and the golden ratio often have been thought to embody ideal beauty. Dr. Stephen Marquardt has studied human beauty for years across both genders and for all races, cultures, and eras. Through his research, he developed and patented the RF Mask, which he claims is the most beautiful shape for a human face.

    The RF Mask is applied to the face of the Egyptian queen Nefertiti (14th century B.C.). According to the RF Mask, is Nefertiti beautiful? Explain. (See Example 4.)

    • Worked-Out Solution

      When the RF Mask is superimposed over the face of Nefertiti, her features align perfectly with the mask. So, according to the RF Mask, Nefertiti is beautiful.

      Scientists have done fascinating studies about what humans perceive as beautiful in the faces of other humans. What they have discovered is that humans who have "average" features tend to be considered the most handsome or the most beautiful.

      You can check this out for yourself at faceresearch.org.

      Reset the image. Select 4 or more male faces or 4 or more female faces. Then, click "Average." Does the composite face appear more "beautiful" to you? For most people, it does.

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      ____     _    _     _  _    __   __   _    _   
     |  _ \\  | || | ||  | \| ||  \ \\/ // | || | || 
     | |_| || | || | ||  |  ' ||   \ ` //  | || | || 
     | .  //  | \\_/ ||  | .  ||    | ||   | \\_/ || 
     |_|\_\\   \____//   |_|\_||    |_||    \____//  
     `-` --`    `---`    `-` -`     `-`'     `---`   
                                                     
    
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  • 14. Golden Ratio

    It is common for humans to find or see patterns in everything they do. Do you think the golden ratio is a law of nature or just a coincidental pattern detected by humans? Explain your reasoning. (See Example 4.)

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      _____      ___      ______    ______    ______  
     /  ___||   / _ \\   /_   _//  /_   _//  /_   _// 
    | // __    | / \ ||  `-| |,-    -| ||-     | ||   
    | \\_\ ||  | \_/ ||    | ||     _| ||_    _| ||   
     \____//    \___//     |_||    /_____//  /__//    
      `---`     `---`      `-`'    `-----`   `--`     
                                                      
    
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