Box Counting Dimension Sierpinski Carpet

To calculate this dimension for a fractal.

Box counting dimension sierpinski carpet.

For the sierpinski gasket we obtain d b log 3 log 2 1 58996. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. The gasket is more than 1 dimensional but less than 2 dimensional. A for the bifractal structure two regions were identified.

This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. Sierpiński demonstrated that his carpet is a universal plane curve. But not all natural fractals are so easy to measure. This leads to the definition of the box counting dimension.

Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. 4 2 box counting method draw a lattice of squares of different sizes e. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. We learned in the last section how to compute the dimension of a coastline.

Fractal dimension box counting method. Fractal dimension of the menger sponge. 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 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. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. 111log8 1 893 383log3 d f.