Box Counting Dimension Sierpinski Carpet

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions.

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

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. This leads to the definition of the box counting dimension. A for the bifractal structure two regions were identified.

To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. To calculate this dimension for a fractal. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. 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.

The gasket is more than 1 dimensional but less than 2 dimensional. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. Sierpiński demonstrated that his carpet is a universal plane curve. Fractal dimension box counting method.

It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. 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.