Box Counting Dimension Sierpinski Carpet

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

Box counting dimension sierpinski carpet.

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. Fractal dimension of the menger sponge. Sierpiński demonstrated that his carpet is a universal plane curve.

The gasket is more than 1 dimensional but less than 2 dimensional. We learned in the last section how to compute the dimension of a coastline. This leads to the definition of the box counting dimension. The hausdorff dimension of the carpet is log 8 log 3 1 8928.

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. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. For the sierpinski gasket we obtain d b log 3 log 2 1 58996.

But not all natural fractals are so easy to measure. 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. 111log8 1 893 383log3 d f.

To calculate this dimension for a fractal. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. Fractal dimension box counting method.