Box Counting Dimension Sierpinski Carpet

Fractal dimension box counting method.

Box counting dimension sierpinski 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. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle.

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. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. But not all natural fractals are so easy to measure.

This leads to the definition of the box counting dimension. To calculate this dimension for a fractal. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve.

We learned in the last section how to compute the dimension of a coastline. A for the bifractal structure two regions were identified. The hausdorff dimension of the carpet is log 8 log 3 1 8928. 4 2 box counting method draw a lattice of squares of different sizes e.

Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. The gasket is more than 1 dimensional but less than 2 dimensional. Fractal dimension of the menger sponge.