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.