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Math Help - Fractals (Mandelbrot set and Koch snowflake) problems

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    Fractals (Mandelbrot set and Koch snowflake) problems

    I don't get how this question on the Mandelbrot set is possible:

    You are to develop a spreadsheet to generate the first 20 elements in the sequence for any value of z and c.
    Design your spreadsheet with column headings that include the term number, real component of each term, imaginary component of each term in the sequence, and the modulus of the complex number.

    Or this question on the Koch snowflake, where stage 0 is the equilateral triangle:

    Calculate the areas for stages 0 to 4 of a snowflake that starts with an equilateral triangle of side 1 unit.
    Show that the total area of the Koch snowflake will never exceed 8/5 of the original area.

    Can someone show me how to do these? Thanks.
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  2. #2
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    Quote Originally Posted by Ruscour View Post
    I don't get how this question on the Mandelbrot set is possible:

    You are to develop a spreadsheet to generate the first 20 elements in the sequence for any value of z and c.
    Design your spreadsheet with column headings that include the term number, real component of each term, imaginary component of each term in the sequence, and the modulus of the complex number.

    What do you mean "get how it is possible"? You just do exactly what you are told to do. Each term in the Mandelbrot sequence is derived from the previous term by z_{n+1}= z_n^2+ c where c and z_0 are given. Set up a spreadsheet (using Corel Quattro Pro or Microsoft Excel?) Setting up one column that gives just the integers, 1, 2, 3,..., the next column giving the real component of z_n^2+ c, the third column giving the imaginary component of z_n^2+ c, and the fourth column giving the absolute value of z_n^2+ c.

    Exactly how you do that depends upon which spreadsheet program you are using.

    Or this question on the Koch snowflake, where stage 0 is the equilateral triangle:

    Calculate the areas for stages 0 to 4 of a snowflake that starts with an equilateral triangle of side 1 unit.
    Show that the total area of the Koch snowflake will never exceed 8/5 of the original area.

    Can someone show me how to do these? Thanks.
    For the second problem, What is the area of an equilateral triangle with sides of length 1? The next "stage" in constructing the Koch snowflake is to put small triangle on the middle third of each side. That means you are adding three triangles each of length 1/3. What is the area of an equilateral triangle with side length 1/3? What is the total area of the new triangles? What is the total area of the figure? "Stage 2" requires adding equilateral triangels at the middle third the two "outer" edges of those small triangles as well the two parts of each side of the orginal triangle that are still there. It should be easy to see that the side of each of these new triangles is (1/3)(1/3)= 1/9. What is the area of a right triangle with side length 1/9? It should also be easy to see that you have added 12 such triangles. What is the area of all 12 triangles? What the total area of the new figure?

    Do that for stage 3 and 4 as well. By that time, you should see that you are adding a specific multiple of the area each time- so getting a geometric series.
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