Box Counting Dimension Sierpinski Carpet

Box counting analysis results of multifractal objects.

Box counting dimension sierpinski carpet.

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. Random sierpinski carpet deterministic sierpinski carpet the fractal dimension of therandom sierpinski carpet is the same as the deterministic. 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.

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. To calculate this dimension for a fractal. 111log8 1 893 383log3 d f.

Fractal dimension of the menger sponge. For the sierpinski gasket we obtain d b log 3 log 2 1 58996. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. The gasket is more than 1 dimensional but less than 2 dimensional.

4 2 box counting method draw a lattice of squares of different sizes e. But not all natural fractals are so easy to measure. Fractal dimension box counting method. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle.

A for the bifractal structure two regions were identified. This leads to the definition of the box counting dimension. Sierpiński demonstrated that his carpet is a universal plane curve.