Box Counting Dimension Sierpinski Carpet

Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve.

Box counting dimension sierpinski carpet.

The hausdorff dimension of the carpet is log 8 log 3 1 8928. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. To calculate this dimension for a fractal. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle.

Box counting analysis results of multifractal objects. A for the bifractal structure two regions were identified. Fractal dimension box counting method. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle.

111log8 1 893 383log3 d f. 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. We learned in the last section how to compute the dimension of a coastline.

But not all natural fractals are so easy to measure. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. 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.

Fractal dimension of the menger sponge. The gasket is more than 1 dimensional but less than 2 dimensional. 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.