Box Counting Dimension Sierpinski Carpet

Fractal dimension of the menger sponge.

Box counting dimension sierpinski carpet.

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. The gasket is more than 1 dimensional but less than 2 dimensional. 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. To calculate this dimension for a fractal.

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. Fractal dimension box counting method. This leads to the definition of the box counting dimension. 111log8 1 893 383log3 d f.

4 2 box counting method draw a lattice of squares of different sizes e. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. But not all natural fractals are so easy to measure.

We learned in the last section how to compute the dimension of a coastline. Box counting analysis results of multifractal objects. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle.

This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. A for the bifractal structure two regions were identified. Sierpiński demonstrated that his carpet is a universal plane curve.