About the fractal dimension

Fibonacci numbers, the golden ratio, golden rectangle and its fractal, the dimension of the fractal and Moran Equations and the applications used in the fractal dimension measurements are included in this study. First, some information is given about the Fibonacci numbers and the golden ratio. Fibonacci numbers were found by the great mathematician Leonardo Fibonacci who introduced the Indo-Arabic number system to Europe. These numbers are important in terms of expressing beauty in nature by numbers. Successive numbers of Fibonacci numbers are seen in the flower seeds and organization of the leaves of pinecone, daisies and sunflower. The ratio of these numbers gives Golden Ratio. Golden Ratio is an irrational number, it is written in the decimal system as; 1.618033988749894… and represented by Fi, that ∅ or φ symbol. Golden Ratio is important in the classification of art and aesthetics as a criterion of harmony and beauty. Then, golden rectangle was defined and its dimension was calculated. A rectangle with a 1 unit short side and long side ∅ is called as a golden rectangle. Golden rectangles are suitable for the fractal examination since they give different rectangles which resemble them proportionally. To obtain a fractal from the golden rectangle, rectangles with 1×(∅-1) are removed from the two sides of a golden rectangle with 1 and ∅ sides. The same process is applied to the two obtained rectangles. When this operation is repeated many times the obtained figure produces the rectangle fractal. The dimension of this fractal can be calculated. Finally, Moran equations and applications used in fractal dimension measurement were mentioned. Moran Equation is a very important equation used when all the pieces of a fractal are not in the same scale. The dimensions of the different fractals may be calculated by this equation. Key Words Fibonacci Numbers; Golden Ratio; Golden Rectangle Fractal; Fractal Dimension; Moran Equation