(von Koch snowflake) To construct the famous fractal known as the vo Koch snowflake curve, use an
equilateral triangle for S0. Then do the von Koch procedure.
i) draw s2 and s3 (i can do this bit i think)
S0
S1
S2
S3
ii) the von Koch snowflake is the limiting curve S = lim n->infinity Sn. Show that it has infinite arc
length.
iii) Find the area of the region enclosed by S.
iv) Find the similarity dimension of S.
Let
be the number of sides,
be the length of a single side,
(that is Sn) be the
length of the perimeter, and
the snowflake's area after the nth iteration. Further,
denote the area of the initial n = 0 triangle
, and the length of an initial n = 0 side 1.
Then
LimSn = Lim3(4/3)n since 4/3>1, then Lim3(4/3)n =
n->
n->
So it has infinite arc length.
.
Solutions of (iii)(iv) are the following:
The capacity dimension is then
n->
Now compute the area explicitly,
so as
,