Box Counting Dimension Sierpinski Carpet

This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane.

Box counting dimension sierpinski carpet.

Fractal dimension box counting method. 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. 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. But not all natural fractals are so easy to measure.

Box counting analysis results of multifractal objects. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. 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.

For the sierpinski gasket we obtain d b log 3 log 2 1 58996. To calculate this dimension for a fractal. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. 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. A for the bifractal structure two regions were identified. The hausdorff dimension of the carpet is log 8 log 3 1 8928.

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. Fractal dimension of the menger sponge. Sierpiński demonstrated that his carpet is a universal plane curve.