Box Counting Dimension Sierpinski Carpet

The gasket is more than 1 dimensional but less than 2 dimensional.

Box counting dimension sierpinski carpet.

Box counting analysis results of multifractal objects. To calculate this dimension for a fractal. 4 2 box counting method draw a lattice of squares of different sizes e. 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.

For the sierpinski gasket we obtain d b log 3 log 2 1 58996. The hausdorff dimension of the carpet is log 8 log 3 1 8928. 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. Sierpiński demonstrated that his carpet is a universal plane curve.

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

Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. Fractal dimension of the menger sponge. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane.

Fractal dimension box counting method. 111log8 1 893 383log3 d f. This leads to the definition of the box counting dimension.