Koch curve

A Koch curveMathworldPlanetmath is a fractalMathworldPlanetmath generated by a replacement rule. This rule is, at each step, to replace the middle 1/3 of each line segmentMathworldPlanetmath with two sides of a right triangleMathworldPlanetmath 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 LK of the Koch curve at the nth step will be


This quantity increases without bound; hence the Koch curve has infiniteMathworldPlanetmathPlanetmath 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 trianglesMathworldPlanetmath we have just created. We can bound the area of each triangle of side length s by s2 (the square containing the triangle.) Hence, at step n, the area AK “under” the Koch curve (assuming l=1) is

AK < (13)2+3(19)2+9(127)2+
= i=1n13i-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