Box Counting Dimension Sierpinski Carpet

We learned in the last section how to compute the dimension of a coastline.

Box counting dimension sierpinski carpet.

4 2 box counting method draw a lattice of squares of different sizes e. This leads to the definition of the box counting dimension. 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.

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. It is relatively easy to determine the fractal dimension of geometric fractals such as the sierpinski triangle. To show the box counting dimension agrees with the standard dimension in familiar cases consider the filled in triangle. But not all natural fractals are so easy to measure.

111log8 1 893 383log3 d f. The hausdorff dimension of the carpet is log 8 log 3 1 8928. A for the bifractal structure two regions were identified. To calculate this dimension for a fractal.

Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. 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.

For the sierpinski gasket we obtain d b log 3 log 2 1 58996. The gasket is more than 1 dimensional but less than 2 dimensional. Sierpiński demonstrated that his carpet is a universal plane curve.