Famous Fractals
Fudgeflake
Fudgeflake is a base-motif fractal with the following base and motif:
In order to generate one of its sides, we substitute a line segment with the motif. In the second iteration, however, we substitute the first line segment with the flipped version of the motif, and the second line segment with the original version of the motif:
In every following iteration, we alternate between the flipped and the original versions of the motif. The fractal is, thus, a sweep:
We can now take three of such curves and put them on sides of an equilateral triangle. The fractal formed is the Fudgeflake. It is called the Fudgeflake because alternating between different versions of the motif results in making the fractal non-symmetrical, or "fudged":
FRACTAL DIMENSION
We can easily find the fractal dimension of this fractal using similarity method. In our motif, there are 2 identical line segments, each of which is 1/sqr(3) long. The dimension is log(2) / log(sqr(3)), which is approximately 1.26.
LENGTH
At every iteration, the length of the fractal becomes 2/sqr(3) times longer. Just like all fractal curves, after an infinite number of iterations its length becomes infinite.
AREA
After the first iteration, the figure becomes a regular hexagon whose area is 2, if the area of the original triangle is 1. In all following iterations, half of the motifs are facing inside and half outside. Those facing outside add some area to the figure, but the ones facing inside subtract the same exact area. Thus, after the first iteration the area remains the same and continues being 2 during further iterations.
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