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 , 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 at each stage.
If is the initial perimeter, the perimeter is: .
This is a geometric sequence with .
. . Hence, diverges as
It can be seen that the area increases by at each stage.
If is the initial area, the area is: .
This is a geometric sequence with
. . Hence, converges as
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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.