Dimension Of Sierpinski Carpet

Between 1 and 2.

Dimension of sierpinski carpet.

Possible sizes are powers of 3 squared. A curve that is homeomorphic to a subspace of plane. Sierpiński demonstrated that his carpet is a universal plane curve. First take a rough guess at what you might think the dimension will be.

The sierpinski carpet is a plane fractal curve i e. The metric dimension of r is given by. Sierpiήski carpetrform 2 n 3 andr 0 0 1 1 2 0. This tool draws the sierpinski carpet fractal with three different sizes of squares as the number of iterations is equal to 3.

Remember it is a 2d fractal. Since the sierpinski triangle fits in plane but doesn t fill it completely its dimension should be less than 2. 1 the theorem is proved in section 2. Notion of metric dimension and discuss the following result.

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. These options will be used automatically if you select this example. A very challenging extension is to ask students to find the perimeter of each figure in the task. Let s use the formula for scaling to determine the dimension of the sierpinski triangle fractal.

Here bright colors are used on a canvas size of 558x558px. In section 3 we recall the. Let s see if this is true. Whats people lookup in this blog.

The sierpinski carpet is self similar with 8 non overlapping copies of itself each scaled by the factor r 1. It was first described by waclaw sierpinski in 1916. Solved now we can apply this formula for dimension to fra the sierpinski triangle area and perimeter of a you fractal explorer solved finding carpet see exer its decompositions scientific sierpiński sieve from wolfram mathworld oftenpaper net htm as constructed by removing center. 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.

3x3 9x9 27x27 or 81x81. 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. 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.