### 科赫曲線的周長

#{{{
3L \times {({4 \over 3} \times {4 \over 3} \times {4 \over 3} \times \cdots)} \to \infty
}}}#

### 科赫曲線的面積

#{{{
\begin{eqnarray}
A \text{ } &+& ({3 \times 4^0})({1 \over 9^1}A) \nonumber \\
&+& ({3 \times 4^1})({1 \over 9^2}A) \nonumber \\
&+& ({3 \times 4^2})({1 \over 9^3}A) \nonumber \\
&+& \cdots
\end{eqnarray}
}}}#

#{{{
\begin{eqnarray}
A + \sum_{n \to \infty} ({3 \times 4^{n-1}} {1 \over {9^n}}) = A + 3A \sum_{n \to \infty} ( {{4^{n-1}} \over {9^n}} )
\end{eqnarray}
}}}#

#{{{
\begin{eqnarray}
\sum_{n \to \infty} ( {{4^{n-1}} \over {9^n}} ) &=& \sum_{n \to \infty} ( {1 \over 9} \times {{4^{n-1}} \over {9^{n-1}}} ) \nonumber \\
&=& {1 \over 9} \sum_{n \to \infty} ({ 4 \over 9 })^{n-1} \nonumber \\
&=& {1 \over 9} \times {{1 - \lim\limits_{n \to \infty} { ({4 \over 9})^n} } \over {1 - {4 \over 9} } } \nonumber \\
&=& {1 \over 9} \times {1 \over { 5 \over 9 } } \nonumber \\
&=& 1 \over 5
\end{eqnarray}
}}}#

#{{{
A + {3 \over 5}A = {8 \over 5}A = {8 \over 5} \times {{\sqrt 3} \over 4}L^2 = {2{\sqrt 3} \over 5}L^2
}}}#