Exam 12: Fractal Geometry: the Kinky Nature of Nature

-Refer to the information from the table shown above ; what is the total length of the darkened boundary of the resulting figure (gasket) in Step 3?

-Refer to the table shown above ; what is the total area T in Step 3?

-Refer to the table shown above ; how many triangles will be added in Step 3?

You plan on constructing a Koch snowflake by starting with a seed triangle whose area is 7290 in². The table below gives the following information where R is the number of triangles added at a particular step, S is the area of each added triangle, and T is the total area of the newly created figure. What is the area of each added triangle in Step 2 ?

-Refer to the table shown above; how many darkened triangles are there in Step 2?

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

-Refer to the table shown above ; what is the area of each added triangle in Step 2?

-Refer to the table shown above ; how many triangles are removed during Step 3?

You plan on constructing a Koch snowflake by starting with a seed triangle whose area is 2187 in2. The table below gives the following information where R is the number of triangles added at a particular step, S is the area of each added triangle, and T is the total area of the newly created figure.

-Refer to the information from table shown above ; what is the total length of the darkened boundary of the resulting figure (Sierpinski gasket) in Step 3?

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.

-Refer to the table shown above ; how many triangles will be added in Step 3?

-Refer to the table shown above ; how many sides does the figure have in Step 2?

You plan on constructing a Sierpinski gasket by starting with a seed triangle whose perimeter is 3360 in. 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 total length of the darkened boundary of the figure (gasket) obtained in a particular step.

The following is called the Chaos Game. Start with rectangle ABCD in which A is located at (0, 0), B is located at (0, 8), C is located at (3, 8), and D is located at (3, 0). 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 one-half of the way straight toward 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 one-half of the way from the last position to the winning vertex. The grid below shows rectangle ABCD. . What sequence of rolls would produce the following sequence of marked points? P1: (0,8), P2 : (1.5,8), P3 : (0.75, 4) (a) (2,3,1) (b) (1,4,2) (c) (2,1,4) (d) (2,4,3) (e) None of the above.

-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 2304 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.

Refer to the information shown above ; what is the total area of the Sierpinski gasket created if the Step process were to continue indefinitely?

Two Koch snowflakes are shown below. One was created by starting with a seed triangle whose perimeter was 10 units while the other was created by starting with a seed triangle whose perimeter was 30 units. Assuming that the iterative process continues indefinitely, which of the snowflakes below has the larger perimeter?

-Refer to the table shown above ; what is the original perimeter X of the seed triangle