Dimension Of Sierpinski Carpet

A very challenging extension is to ask students to find the perimeter of each figure in the task.

Dimension of sierpinski carpet.

The figures students are generating at each step are the figures whose limit is called sierpinski s carpet this is a fractal whose area is 0 and perimeter is infinite. First take a rough guess at what you might think the dimension will be. 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 sierpinski carpet 1 is a well known hierarchical decomposition of the square plane tiling associated with that is pairs of integers consider the sierpinski graph 2 which is the adjacency graph of the complement of in where is one of the hierarchical subsets of gray squares are used to depict the intersection of with a subset of.

Let s see if this is true. A curve that is homeomorphic to a subspace of plane. Sierpiήski carpetrform 2 n 3 andr 0 0 1 1 2 0. First you have to decide which scale your sierpinski carpet should be.

3x3 9x9 27x27 or 81x81. Since the sierpinski triangle fits in plane but doesn t fill it completely its dimension should be less than 2. In section 3 we recall the. The hausdorff dimension of the carpet is log 8 log 3 1 8928.

Remember it is a 2d fractal. The sierpinski carpet is self similar with 8 non overlapping copies of itself each scaled by the factor r 1. Therefore the similarity dimension d of the unique attractor of the ifs is the solution to 8 k 1rd 1 d log 1 8 log r log 1 8 log 1 3 log 8 log 3 1 89279. It was first described by waclaw sierpinski in 1916.

Sierpiński demonstrated that his carpet is a universal plane curve. The sierpinski carpet is a plane fractal curve i e. Whats people lookup in this blog. Possible sizes are powers of 3 squared.

1 the theorem is proved in section 2. Here bright colors are used on a canvas size of 558x558px. This tool draws the sierpinski carpet fractal with three different sizes of squares as the number of iterations is equal to 3. Let s use the formula for scaling to determine the dimension of the sierpinski triangle fractal.

In these type of fractals a shape is divided into a smaller copy of itself removing some of the new copies and leaving the remaining copies in specific order to form new shapes of fractals. These options will be used automatically if you select this example. The metric dimension of r is given by.