# Edit Edit an Image Page: Koch Snowflake

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 Image Title*: Upload a Math Image The image is an example of a Koch Snowflake, a fractal that first appeared in a paper by Swede Niels Fabian Helge von Koch in 1904. It is made by the infinite iteration of the Koch curve. Curve Construction''' [[Image:Koch_Construction.jpg|center|400px]] The curve begins as a line segment and is divided into three equal parts. A equilateral triangle is than created, using the middle section of the line as its base, and the middle section is removed. [[Image:KochAnimation.gif|thumb|left|200px|First 7 iterations]] The Koch Snowflake is an '''iterated process'''. It is created by repeating the process of the Koch Curve on the three sides of an equilateral triangle an infinite amount of times in a process referred to as '''iteration''' (however, as seen with the animation, a complex snowflake can be created with only seven iterations - this is due to the butterfly effect of iterative processes). Thus, each iteration produces additional sides that in turn produce additional sides in subsequent iterations. An interesting observation to note about this fractal is that although the snowflake has an ever-increasing number of sides, its perimeter lengthens infinitely while its area is finite. The Koch Snowflake has perimeter that increases by 4/3 of the previous perimeter for each iteration and an area that is 8/5 of the original triangle. Click here, for more information about [[Iterated Functions]]. ==Fractal Properties== [[Image:Koch_Perimeter.gif|right|frame|Infinite Perimeter of the Koch Snowflake|250px]] '''Self-similarity''' The Koch Snowflake displays a property known as '''self-similarity''', emphasized in the animation. This means that as we continue to magnify the Koch Snowflake, each magnified section continues to look similar to the larger perspective. '''Fractal Dimension''' [[Image:KochFractalDimension.png|left|thumb|200px|2nd iteration of Koch Snowflake]] The [[Fractal Dimension|fractal dimension]] of a Koch Snowflake can be calculated using the formula for fractal dimension: $\frac{logN}{loge}$. Taking the image shown to the left, the top diagram shows that the new new Koch Curve lengths are a third of the previous iteration's length after the second iteration, and so e = 3. The bottom diagram shows that there are now a total of 4 Koch Curves, so that N = 4. Using the formula for fractal dimension: $\frac{logN}{loge} = \frac{log4}{log3} \approx 1.26\,$. ==Demonstration== ==Other Properties== Image:Koch1.jpg Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other Cynthia Lanius, [http://math.rice.edu/~lanius/frac/koch.html Cynthia Lanius' Fractal Unit:Koch Snowflake] Larry Riddle, [http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm Koch Curve] Yes, it is.