Ron Larson
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7.4 Fibonacci & Other Patterns

7.4 Fibonacci & Other Patterns
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  • 1. Daisies

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Daisies

    • Worked-Out Solution

      Ordinary field daisies have 34 petals. It is common for other members of the daisy family to have 13, 21, 55, or even 89 petals. All of these numbers are in the Fibonacci sequence.

      In the arrangement of tiny florets in the core of a daisy, you can see 21 counterclockwise spirals and 34 clockwise spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence.

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     _    _    __   __   __   __     ___      _  _   
| \  / ||  \ \\/ //  \ \\/ //   / _ \\   | \| || 
|  \/  ||   \ ` //    \   //   / //\ \\  |  ' || 
| .  . ||    | ||     / . \\  |  ___  || | .  || 
|_|\/|_||    |_||    /_//\_\\ |_||  |_|| |_|\_|| 
`-`  `-`     `-`'    `-`  --` `-`   `-`  `-` -`
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  • 2. Phyllotaxis

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Phyllotaxis

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      _____     _____    _    _      ___      _____   
 /  ___||  |  ___|| | || | ||   / _ \\   / ____|| 
| // __    | ||__   | || | ||  | / \ || / //---`' 
| \\_\ ||  | ||__   | \\_/ ||  | \_/ || \ \\___   
 \____//   |_____||  \____//    \___//   \_____|| 
  `---`    `-----`    `---`     `---`     `----`
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  • 3. Pineapple

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Pineapple

    • Worked-Out Solution

      Pineapple scales are patterned into spirals and because they are hexagonal, there are 3 sets of spirals.

      There is one set of 5 paralllel spirals.

      There is a second set of 8 parallel spirals.

      And there is a third set of 13 parallel spirals.

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               _    _    _____      ______    _____   
  ____    | || | || |  __ \\   /_   _//  / ____|| 
 |    \\  | || | || | |  \ ||   -| ||-  / //---`' 
 | [] ||  | \\_/ || | |__/ ||   _| ||_  \ \\___   
 |  __//   \____//  |_____//   /_____//  \_____|| 
 |_|`-`     `---`    -----`    `-----`    `----`  
 `-`
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  • 4. Pinecone

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Pinecone

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      _____      ___      ______     ___      _  __  
 |__  //    / _ \\   /_____//   / _ \\   | |/ // 
   / //    / //\ \\  `____ `   / //\ \\  | ' //  
  / //__  |  ___  || /___//   |  ___  || | . \\  
 /_____|| |_||  |_|| `__ `    |_||  |_|| |_|\_\\ 
 `-----`  `-`   `-`  /_//     `-`   `-`  `-` --` 
                     `-`
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  • 5. Sneezewort

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Sneezewort

    • Worked-Out Solution

      The branches and leaves on a sneezewort plant grow in a Fibonnaci pattern as shown in the diagram below.

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     __   _     ______    ______    ______   ______   
| || | ||  /_   _//  /_____//  /_   _// |      \\ 
| '--' ||   -| ||-   `____ `    -| ||-  |  --  // 
| .--. ||   _| ||_   /___//     _| ||_  |  --  \\ 
|_|| |_||  /_____//  `__ `     /_____// |______// 
`-`  `-`   `-----`   /_//      `-----`  `------`  
                     `-`
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  • 6. Human Body

    Use the Internet to describe how the Fibonacci sequence is related to the object shown. (See Example 1 and Example 2.)

    Human Body

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