Coastline

It is difficult to accept that a coastline would be infinite. We can't see infinity on the map. This must be a mathematical trick.


D＞1
D is fractal dimensions.

As you can see that a coastline is very complicated,　it is difficult to measure the line. It depends on dividers. If you had high quality dividers, the line would be longer because you divide the line smaller and smaller.

ε is a length of a divider which you measure.
n(ε) is the sum of ε.

L(ε) is an approximate length of the coastline.
∴ L(ε)=εn(ε)



F is the length of the coastline.

If F=100(meter),D=1.2, and ε=3(meter),
n(ε)=26.75805.
Therefore, you need about 26.76 small lines to measure 100(meter) of the coastline.

L(ε)=80.27416
This is far from 100.

If ε=2(meter),
n(ε)=43.52753
L(ε)=87.05506.
This is close to 100.

However, If ε=0.1,
n(ε)=1584.893
L(ε)=158.4893.
This is over 100.

If ε=0.0001,
n(ε)=6309573.445
L(ε)=630.9573445.
This is far from 100.

Therefore, you see that you can't measure the coastline because of innovation of dividers.
If D=1, F is equal to L(ε) so there is no approximation.
However, in this case, D is more than 1.
This is chaos.