Box Counting Dimension Sierpinski Carpet

4 2 box counting method draw a lattice of squares of different sizes e.

Box counting dimension sierpinski carpet.

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. 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. Fractal dimension box counting method.

To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. 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. Sierpiński demonstrated that his carpet is a universal plane curve. For the sierpinski gasket we obtain d b log 3 log 2 1 58996.

111log8 1 893 383log3 d f. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. To calculate this dimension for a fractal. 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. Box counting analysis results of multifractal objects. But not all natural fractals are so easy to measure. Fractal dimension of the menger sponge.

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. We learned in the last section how to compute the dimension of a coastline. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane.