2. A. Ineffective Ways to Measure
Area of the Koch curve
Here is a summary of the data so far, and an extrapolation.
n
area of a triangle
number of triangles
A
n
0
(√3)/12
1
A
0
=
(√3)/12
1
((√3)/12)⋅(1/9)
4
A
1
=
((√3)/12)⋅(4/9)
2
((√3)/12)⋅(1/81)
=
((√3)/12)⋅(1/9)
2
16 =
4
2
A
2
=
((√3)/12)⋅(16/81)
=
((√3)/12)⋅(4/9)
2
...
...
...
...
n
((√3)/12)⋅(1/9)
n
4
n
A
n
=
((√3)/12)⋅(4/9)
n
Here is a graph of A
n
vs n
The area of the Koch curve is less than each A
n
, so we see the Koch curve has zero area.
Return to
Ineffective Ways to Measure
.
