Box Counting Dimension Sierpinski Carpet

Fractal dimension box counting method.

Box counting dimension sierpinski carpet.

In fractal geometry the minkowski bouligand dimension also known as minkowski dimension or box counting dimension is a way of determining the fractal dimension of a set s in a euclidean space r n or more generally in a metric space x d it is named after the german mathematician hermann minkowski and the french mathematician georges bouligand. Fractal dimension of the menger sponge. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. The values of these slopes are 1 8927892607 and 1 2618595071 which are respectively the fractal dimension of the sierpinski carpet and the two dimensional cantor set.

We learned in the last section how to compute the dimension of a coastline. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. 4 2 box counting method draw a lattice of squares of different sizes e. This leads to the definition of the box counting dimension.

Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. Sierpiński demonstrated that his carpet is a universal plane curve. A for the bifractal structure two regions were identified. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle.

To calculate this dimension for a fractal. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. The gasket is more than 1 dimensional but less than 2 dimensional. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle.

Box counting analysis results of multifractal objects. 111log8 1 893 383log3 d f. The hausdorff dimension of the carpet is log 8 log 3 1 8928.