Famous Fractals

Gosper Island

Gosper Island is a base-motif fractal formed by using the following base and motif:

Performing generator iteration, we substitute every side of the hexagon with the motif:

We then again substitute every line segment with the motif and continue the iteration:

In addition to the side of this figure being self-similar, the Gosper Island itself can be broken down into 7 small versions of itself:

Similarly to the Koch Snowflake, which can be filled by Snowflake Sweeps, the Gosper Island can also be filled by a fractal like this:

FRACTAL DIMENSION

The fractal dimension of Gosper Island can be easily found using the similarity method. The motif of the fractal is composed of 3 identical line segments, each of which is 1/sqr(7) long. The fractal dimension is thus log 3 / log (sqr(7)), which is approximately 1.129.

LENGTH

With every iteration, the length of this curve becomes 3/sqr(7) times longer. After an infinite number of iterations, the curve, just like all fractal curves will become infinitely long.

AREA

If you substitute a line segment with the motif of the Gosper Island, you will see that equal areas will be extending in and out:

Thus, using this motif will not change the area, and the area of the fractal will be equal to the area of the base, which is a regular hexagon. If the area of the hexagon is 1, the area of Gosper Island will be 1 as well.