A factral, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.
The Koch snowflake (also known as the Koch star and Koch island) is a mathematical curve and one of the earliest fractal curves to have been described.
Construction
The Koch snow flake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:
1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3.remove the line segment that is the base of the triangle from step 2.
After one iteration of this process, the resulting shape is the outline of a hexagram.
The Koch snowflake is the limit approached as the above steps are followed over and over again. The Koch curve originally described by Koch is constructed with only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake.
Start with an equilateral triangle T. Scale T by a factor of 1/3 and place 3 copies along each of the three sides of T as illustrated in the diagram below to form a new image S(1). Next scale T by a factor of 1/9 = (1/3)^2 and place 12=4*3 copies along the sides of T(1) as illustrated to form the image S(2). For the next iteration, take 48=4*12 copies of Tscaled by a factor of 1/27=(1/3)^3 and place them around the sides of S(2) to form the image S(3). Continue this construction. The Koch Snowflake is the limiting image of the construction.
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