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.

Dimension of sierpinski carpet.

The sierpinski carpet is self similar with 8 non overlapping copies of itself each scaled by the factor r 1. 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. Remember it is a 2d fractal.

1 the theorem is proved in section 2. First take a rough guess at what you might think the dimension will be. Let s see if this is true. 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.

In section 3 we recall the. Between 1 and 2. Notion of metric dimension and discuss the following result. Let s use the formula for scaling to determine the dimension of the sierpinski triangle fractal.

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. 3x3 9x9 27x27 or 81x81. A very challenging extension is to ask students to find the perimeter of each figure in the task. The sierpinski carpet is a plane fractal curve i e.

Sierpiήski carpetrform 2 n 3 andr 0 0 1 1 2 0. 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. It was first described by waclaw sierpinski in 1916. Sierpiński demonstrated that his carpet is a universal plane curve.

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. The hausdorff dimension of the carpet is log 8 log 3 1 8928. A curve that is homeomorphic to a subspace of plane. These options will be used automatically if you select this example.

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. Since the sierpinski triangle fits in plane but doesn t fill it completely its dimension should be less than 2. First you have to decide which scale your sierpinski carpet should be.