1. ## [SOLVED] Koch Snowflake

- URGENT -

Using appropriate mathematical formulae and patterns to determine if teh perimeter and area of teh snowflake are finite or infinite quantities. Decisions much be supported clearly!!

Please don't complicate it too much!!

Edit: THANKS.

2. Originally Posted by summna09
URGENT -

Using appropriate mathematical formulae and patterns to determine if teh perimeter and area of teh snowflake are finite or infinite quantities. Decisions much be supported clearly!!

Please don't complicate it too much!!

THANKS.
Read this: The Koch Curve Unruled Notebook

3. Hello, summna09!

Use appropriate mathematical formulae and patterns to determine
if the perimeter and area of the Koch snowflake are finite
or infinite quantities. .Decisions much be supported clearly.

Given equilateral triangle $\displaystyle ABC$, each side is trisected
and an equilateral triangle is constructed outward.
Code:
A
*
/ \
/   \
/     \
* - - - *       * - - - *
\     /         \     /
\   /           \   /
\ /             \ /
*               *
/                 \
/                   \
/                     \
B * - - - * - - - * - - - * C
\     /
\   /
\ /
*

The process is repeated on the 12 sides of the new "snowflake".
Code:
A
*
* - / \ - *
*   \       /   *
/ \   /     \   / \
* -   - *       * -   - *
\                     /
/                       \
* - \                 / - *
*               *
* - /                 \ - *
\                       /
/                     \
B * -   - *       * -   - * C
\ /   \     /   \ /
*   /       \   *
* - \ / - *
*
And the process is repeated again ... and again.

It can be seen that the perimeter increases by $\displaystyle \tfrac{1}{3}$ at each stage.

If $\displaystyle P_o$ is the initial perimeter, the $\displaystyle n^{th}$ perimeter is: .$\displaystyle P_n \:=\:\left(\tfrac{4}{3}\right)^nP_o$

This is a geometric sequence with $\displaystyle r > 1$.
. . Hence, $\displaystyle P_n$ diverges as $\displaystyle n \to \infty.$

It can be seen that the area increases by $\displaystyle \tfrac{1}{3}$ at each stage.

If $\displaystyle A_o$ is the initial area, the $\displaystyle n^{th}$ area is: .$\displaystyle A_n \;=\;A_o + A_o\left(\tfrac{1}{3}\right) + A_o\left(\tfrac{1}{3}\right)^2 + A_o\left(\tfrac{1}{3}\right)^3 + \hdots$

This is a geometric sequence with $\displaystyle r < 1$
. . Hence, $\displaystyle A_n$ converges as $\displaystyle n \to\infty$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This is yet another joke perpetrated by the Math Gods.

We have a "polygon" with a finite area (it can fit on a postage stamp)
. . yet has an infinite perimeter.