A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH ...

A TUBE FORMULA FOR THE KOCH SNOWFLAKE CURVE, WITH APPLICATIONS TO COMPLEX DIMENSIONS.

MICHEL L. LAPIDUS AND ERIN P.J. PEARSE

Current (i.e., unfinished) draught of the full version is available at .

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K0 K1 K2 K3 K4

K Figure 1. The Koch curve K (left) and the Koch snowflake (right).

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...

Goal: derive a formula for the -neighbourhood of the Koch curve (and snowflake).

?{

?{ Figure 2. The -neighbourhood of the Koch curve, for two different values of .

We want to find a formula for V () = area of shaded region = vol2{x : d(x, ) < }

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Q: What use is V ()? A: A precise formula for V () will help towards extending the theory of fractal strings into higher dimensions.

A fractal string is any bounded open subset of R

L := {lj} j=1, with lj < .

j=1

l1 l2 l3 . . . , or distinctly (with multiplicity):

l1 > l2 > l3 > . . . . Idea/origin: comes from studying fractal subsets

L R.

Figure 3. The Cantor Set

Figure 4. The Cantor String

l1=1/3 l2=1/9

/ l3=1 27

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The Cantor String example has lengths 3-(n+1)

with multiplicities w3-(n+1) = 2n.

l2

l4

l5

l1 l6

l3 l7

CS =

1 3

,

1 9

,

19,

1 27

,

217,

1 27

,

217,

.

.

.

The geometric zeta function of a string

L (s) = ljs = wlls

j=1

l

encodes all this information.

Example:

CS(s) =

2n3-(n+1)s

=

1

3-s -2?

3-s

.

n=0

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