Question

The Koch snowflake is one of the earliest fractal curves discussed by Helge von Koch in 1904. It is a geometric object obtained as the limit of a process of modifying triangles. To construct the Koch snowflake, start with an equilateral triangle of side length 1 (stage 0). To go to stage one, cut out the middle third of each line segment. Fit a triangle of length 1/3 on the remaining gap. To go to stage 2, repeat the process. The Koch snowflake is the object obtained after an infinite number of stages. Below is the snowflake through the 4th stage. a. Let Pn be the perimeter of the snowflake at the nth stage. So P0 = 3. Show that Pn = 4/3 Pn-1. Use this to compute the perimeter of the snowflake. That is, compute P?. b. Let An be the area of the snowflake at the nth stage. First show that the area at the beginning, A0 = ?3/4. Then show that the area of the entire snowflake is 2?3/5. That is, show that A? = 2?3/5. (Hint: first come up with the area of an equilateral triangle of length s, and use this to come up with the additional area added in at each stage. You will need to write out at least 4 stages before a pattern will emerge.)

          The Koch snowflake is one of the earliest fractal curves discussed by Helge von
Koch in 1904. It is a geometric object obtained as the limit of a process of
modifying triangles.
To construct the Koch snowflake, start with an equilateral triangle of side length 1
(stage 0). To go to stage one, cut out the middle third of each line segment. Fit a
triangle of length 1/3 on the remaining gap. To go to stage 2, repeat the process.
The Koch snowflake is the object obtained after an infinite number of stages.
Below is the snowflake through the 4th stage.
a. Let Pn be the perimeter of the snowflake at the nth stage. So P0 = 3. Show
that Pn = 4/3 Pn-1. Use this to compute the perimeter of the snowflake. That
is, compute P?.
b. Let An be the area of the snowflake at the nth stage. First show that the area
at the beginning, A0 = ?3/4. Then show that the area of the entire snowflake is
2?3/5. That is, show that A? = 2?3/5.
(Hint: first come up with the area of an equilateral triangle of length s, and
use this to come up with the additional area added in at each stage. You will
need to write out at least 4 stages before a pattern will emerge.)

$The Koch snowflake is one of the earliest fractal curves discussed by Helge von Koch in 1904. It is a geometric object obtained as the limit of a process of modifying triangles. To construct the Koch snowflake, start with an equilateral triangle of side length 1 (stage 0). To go to stage one, cut out the middle third of each line segment. Fit a triangle of length 1/3 on the remaining gap. To go to stage 2, repeat the process. The Koch snowflake is the object obtained after an infinite number of stages. Below is the snowflake through the 4th stage. a. Let Pn be the perimeter of the snowflake at the nth stage. So P0 = 3. Show that Pn = 4/3 Pn-1. Use this to compute the perimeter of the snowflake. That is, compute P?. b. Let An be the area of the snowflake at the nth stage. First show that the area at the beginning, A0 = ?3/4. Then show that the area of the entire snowflake is 2?3/5. That is, show that A? = 2?3/5. (Hint: first come up with the area of an equilateral triangle of length s, and use this to come up with the additional area added in at each stage. You will need to write out at least 4 stages before a pattern will emerge.)$

Added by Andr-S M.

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