SOLUTION: How could i make an equation for Von Koch curve? I attempted it but no luck :/ I do not know where to start, so a help would be very nice :D I need to make an equation fo

Question 1012663: How could i make an equation for Von Koch curve?
I attempted it but no luck :/
I do not know where to start, so a help would be very nice :D
I need to make an equation for perimeter and the area, please write each steps if possible.
Im so dumb T.T
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The Van Koch curve is a famous fractal construction
:
First, a straight line(unit length) is divided into three segments. The middle segment is removed and replaced by two segments of equal length that form an equilateral triangle.
:
The above process is repeated for the four segments in the first step.
:
If we start with one line segment for iteration one and look at the the next two iterations we notice the following:
iteration------segment length-----segment number------curve length
:---0-----------------1------------------1------------------1
:---1----------------1/3-----------------4------------------1.33
:---2----------------1/9----------------16------------------1.77
:---3----------------1/27---------------64------------------2.37
:
The perimeter increases by 4/3 for each iteration, therefore
perimeter = (4/3)^n
This shows that as the number of iterations increases, the curve length(perimeter) tends to infinity while it encloses a finite area
:
Now take a look at the area
We take the first triangle formed on iteration 2 of unit area, then on the next iteration we have four small triangles added. Each triangle is 1/9 the area of the unit triangle.
:
The next iteration produces 16 triangles but each trianlge is 81 times smaller than the unit triangle.
:
iteration-------triangle number-------triangle area-------curve area
:---0------------------0---------------------0----------------0.00
:---1------------------1---------------------1----------------1.00
:---2------------------4---------------------0.44444----------1.44444
:---3------------------16--------------------0.19753----------1.64197
:
Area = 1 + (4/9)^n where n is number of iteration after the iteration with 1 triangle.
:
Note that this equation converges to 0.8 as n goes to infinity, that is
(4/9) / (1 - (4/9)) = 0.8

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