Box Counting Dimension Sierpinski Carpet

To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle.

Box counting dimension sierpinski carpet.

Box counting analysis results of multifractal objects. But not all natural fractals are so easy to measure. 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.

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. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. We learned in the last section how to compute the dimension of a coastline.

111log8 1 893 383log3 d f. Sierpiński demonstrated that his carpet is a universal plane curve. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve.

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. This leads to the definition of the box counting dimension. 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.

A for the bifractal structure two regions were identified. The hausdorff dimension of the carpet is log 8 log 3 1 8928. To calculate this dimension for a fractal.