Box Counting Dimension Sierpinski Carpet

This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane.

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. Box counting analysis results of multifractal objects. We learned in the last section how to compute the dimension of a coastline. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions.

The hausdorff dimension of the carpet is log 8 log 3 1 8928. Fractal dimension box counting method. 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.

For the sierpinski gasket we obtain d b log 3 log 2 1 58996. Sierpiński demonstrated that his carpet is a universal plane curve. 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. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic.

But not all natural fractals are so easy to measure. 111log8 1 893 383log3 d f. 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.

A for the bifractal structure two regions were identified. This leads to the definition of the box counting dimension. To calculate this dimension for a fractal.