# Koch curve

A Koch curve^{} is a fractal^{} generated by a replacement rule. This rule is, at each step, to replace the middle $1/3$ of each line segment^{} with two sides of a right triangle^{} having sides of length equal to the replaced segment. Two applications of this rule on a single line segment gives us:

To generate the Koch curve, the rule is applied indefinitely, with a starting line segment. Note that, if the length of the initial line segment is $l$, the length ${L}_{K}$ of the Koch curve at the $n$th step will be

$${L}_{K}={\left(\frac{4}{3}\right)}^{n}l$$ |

This quantity increases without bound; hence the Koch curve has infinite^{} length. However, the curve still bounds a finite area. We can prove this by noting that in each step, we add an amount of area equal to the area of all the equilateral triangles^{} we have just created. We can bound the area of each triangle of side length $s$ by ${s}^{2}$ (the square containing the triangle.) Hence, at step $n$, the area ${A}_{K}$ “under” the Koch curve (assuming $l=1$) is

${A}_{K}$ | $$ | ${\left({\displaystyle \frac{1}{3}}\right)}^{2}+3{\left({\displaystyle \frac{1}{9}}\right)}^{2}+9{\left({\displaystyle \frac{1}{27}}\right)}^{2}+\mathrm{\cdots}$ | ||

$=$ | $\sum _{i=1}^{n}}{\displaystyle \frac{1}{{3}^{i-1}}$ |

but this is a geometric series of ratio less than one, so it converges. Hence a Koch curve has infinite length and bounds a finite area.

A Koch snowflake is the figure generated by applying the Koch replacement rule to an equilateral triangle indefinitely.

Title | Koch curve |
---|---|

Canonical name | KochCurve |

Date of creation | 2013-03-22 12:05:34 |

Last modified on | 2013-03-22 12:05:34 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 8 |

Author | akrowne (2) |

Entry type | Definition |

Classification | msc 28A33 |

Classification | msc 28A80 |

Synonym | Koch snowflake |