Exam 12: Fractal Geometry: the Kinky Nature of Nature

The following is called the Chaos Game. Start with a square ABCD in which A is located at (0,0), B is located at (27,0), C is located at (27,27) , and D is located at (0,27) . Then, roll a fair die. We will say that A is the winner if we roll a 1, B is the winner if we roll a 2, C is the winner if we roll a 3, and D is the winner if we roll a 4 . If we roll a 5 or 6 , we disregard the roll and roll again. Each roll of the die generates a point inside or on the boundary of the square according to the following rules. - Start: Roll the die. Mark the winning vertex and call it P1. - Step 1: Roll the die again. From P1 move two-thirds of the way straight towards the next winning vertex. Mark this point and call it P2 . - Steps 2,3, etc.: Continue rolling the die, each time moving to a point two-thirds of the way from the last position to the winning vertex. The grid below show the square ABCD . What would be the resulting coordinates of P3 if the sequence of (4,1,4) was rolled?

To answer the following question, refer to the Mandelbrot replacement process described by: Start: Choose an arbitrary complex number s , called the seed of the Mandelbrot sequence. Set the seed s as the initial term of the sequence: Recursive Procedure: To find the next term in the sequence, square the preceding term and add the seed: If the seed is , then find the term

-Refer to the information from table shown above ; what is the perimeter of the Koch snowflake created if the Step process were to continue indefinitely?

You plan on constructing a Sierpinski gasket by starting with a seed triangle whose area is 6720 in2. The table below gives the following information where R is the number of triangles removed at a Particular step, S is the area of each removed triangle, and T is the total area of the newly created figure. What is the total area of the figure after step 2 ?

-Refer to the table shown above ; what is the total area of the figure after Step 3?

To answer the following question, refer to the Mandelbrot replacement process described by: - Start: Choose an arbitrary complex number s , called the seed of the Mandelbrot sequence. Set the seed as the initial term of the sequence: - Recursive Procedure: To find the next term in the sequence, square the preceding term and add the seed: If the seed is , then find the term .

-Refer to the information from table shown above ; what is the total length of the darkened boundary of the Sierpinski gasket created if the Step process were to continue indefinitely?

-Refer to the table shown above ; what is the total area of the figure after Step 3?

To answer the following question, refer to the Mandelbrot replacement process described by: - Start: Choose an arbitrary complex number s, called the seed of the Mandelbrot sequence. Set the seed s to be the initial term of the sequence: - Recursive Procedure: To find the next term in the sequence, square the preceding term and add the seed: Which of the statements below is correct?

You plan on constructing a Koch snowflake by starting with a seed triangle whose perimeter is 432 in. The table below gives the following information where M is the number of sides at a particular step, L Is the length of each side, and P is the total perimeter of the figure. What is the length of each side of the figure in Step 1?

The following is called the Chaos Game. Start with a square ABCD in which A is located at (0,0), B is located at (27,0), C is located at (27,27) , and D is located at (0,27) . Then, roll a fair die. We will say that A is the winner if we roll a 1, B is the winner if we roll a 2, C is the winner if we roll a 3 , and D is the winner if we roll a 4 . If we roll a 5 or 6 , we disregard the roll and roll again. Each roll of the die generates a point inside or on the boundary of the square according to the following rules. - Start: Roll the die. Mark the winning vertex and call it P1 . - Step 1: Roll the die again. From P1 move two-thirds of the way straight towards the next winning vertex. Mark this point and call it P2 . - Steps 2,3, etc.: Continue rolling the die, each time moving to a point two-thirds of the way from the last position to the winning vertex. The grid below show the square ABCD . What would be the sequence (1,2,3) was rolled?

You plan on constructing a Sierpinski gasket by starting with a seed triangle whose perimeter is X. The table below gives the following information where U is the number of darkened triangles at a particular step, V is the perimeter of each darkened triangle, and W is the length of the darkened boundary of the figure (gasket) obtained in a particular step.

Which of the following is an example of fractal behavior in nature?

-Refer to the information in the table shown above ; how many darkened triangles are there in Step 3?

-Refer to the information in the table above ; what is the area of the Koch snowflake created if the Step process were to continue indefinitely?

-Refer to the table shown above ; what is the total perimeter length in Step 1? (a) 324 in (b) 576 in (c) 648 in (d) 1296 in (e) None of the above.

-Refer to the table shown above ; what is the area of each of the triangles removed during Step 3 ?

To answer the following question, refer to the Mandelbrot replacement process described by: a. Start: Choose an arbitrary complex number s , called the seed of the Mandelbrot sequence. Set the seed s as the initial term of the sequence: b. Recursive Procedure: To find the next term in the sequence, square the preceding term and add the seed: Which of the statements below is correct?

You plan on constructing a Koch snowflake by starting with a seed triangle whose perimeter is X. The table below gives the following information where M is the number of sides at a particular step, L is the length of each side, and P is the total perimeter of the figure.