Box Counting Dimension Sierpinski Carpet

The sierpinski carpet is a compact subset of the plane with lebesgue covering dimension 1 and every subset of the plane with these properties is homeomorphic to some subset of the sierpiński carpet.

Box counting dimension sierpinski carpet.

Box counting analysis results of multifractal objects. The gasket is more than 1 dimensional but less than 2 dimensional. 111log8 1 893 383log3 d f. Sierpiński demonstrated that his carpet is a universal plane curve.

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. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. 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.

Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. 4 2 box counting method draw a lattice of squares of different sizes e. Fractal dimension box counting method.

This leads to the definition of the box counting dimension. 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. But not all natural fractals are so easy to measure.

To calculate this dimension for a fractal. Fractal dimension of the menger sponge. The hausdorff dimension of the carpet is log 8 log 3 1 8928.