Box Counting Dimension Sierpinski Carpet

We learned in the last section how to compute the dimension of a coastline.

Box counting dimension sierpinski carpet.

Fractal dimension of the menger sponge. Box counting analysis results of multifractal objects. 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. 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.

111log8 1 893 383log3 d f. Sierpiński demonstrated that his carpet is a universal plane curve. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic.

To calculate this dimension for a fractal. 4 2 box counting method draw a lattice of squares of different sizes e. The hausdorff dimension of the carpet is log 8 log 3 1 8928. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane.

The gasket is more than 1 dimensional but less than 2 dimensional. But not all natural fractals are so easy to measure. This leads to the definition of the box counting dimension. A for the bifractal structure two regions were identified.

Fractal dimension box counting method. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve.