Famous Fractals   

Koch Island

Koch Island is a name given to any base-motif fractal whose base is a polygon and whose motif is any fractals curve (a fractal with fractal dimension from 1 to 2). Although the "shorelines" of Koch Islands can be very twisted, they never intersect themselves at any point. Since all of them have fractal dimension between 1 and 2, all shorelines are also infinite in length. The area of Koch Islands, though, is always a finite number. Two very famous Koch Islands are the Koch Snowflake and the Fudgeflake. Both of them use an equilateral triangle as the base:

Another famous Koch Island is the Gosper Island, which uses a hexagon for the base:

Koch Islands which use a square for the base are called quadric Koch Islands. Below are three examples of motifs that you can use to generate them with their fractal dimensions (D). The second motif without a square creates a fractal curve called a Minkowski Sausage.

	D = log 3 / log (sqr(5)) = 1.37
	D = log 8 / log 4 = 1.5
	D = log 18 / log 6 = 1.61

These motifs generate the following fractals:

As you can see with the first example, the islands themselves, not just their shorelines are self-similar and can be divided into identical versions of themselves. Similarly to the Koch Snowflake, which can be filled by Snowflake Sweeps, the first Koch Island can also be filled by a fractal which is called the quartet:

FRACTAL DIMENSION

The fractal dimensions above are calculated using the similarity method. The variable N is the number of identical segments in the motif and e is the magnification required to make every one of them as big as the entire picture. Notice that as we make the motif more complicated, the fractal dimension increases. We could continue to make the motifs even more twisted until the fractal dimension becomes 2 and a Peano curve is created.

LENGTH

Just like all Koch Island shorelines and fractal curves in general, the length of the perimeter of every fractal above is infinite in length.

AREA

If you draw a horizontal line through every one of the motifs above, you will see that identical areas are separated above and below the line. This means that with every iteration, the same exact areas stick inside and outside the fractal, which doesn’t change the overall areas. In fact, the areas remain equal to the area of the original square, which is 1.