Area Of Sierpinski Carpet

As with the gasket the area tends to zero and the total perimeter of the holes tend to infinity.

Area of sierpinski carpet.

Created by math nerds from team browserling. Just press a button and you ll automatically get a sierpinski carpet fractal. Whats people lookup in this blog. This means that we have removed all of the area of the original square in constructing the sierpinski carpet.

Notice that these areas go to 0 as n goes to infinity. The hausdorff dimension of the carpet is log 8 log 3 1 8928. Closely related to the gasket is the sierpinski carpet. A curve that is homeomorphic to a subspace of plane.

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 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. But of course there are many points still left in the carpet. Free online sierpinski carpet generator.

A very challenging extension is to ask students to find the perimeter of each figure in the task. 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. Creating one is an iterative procedure. A sierpinksi carpet is one of the more famous fractal objects in mathematics.

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. It was first described by waclaw sierpinski in 1916. This means that the area of c n is 8 9 n for all n. Sierpiński demonstrated that his carpet is a universal plane curve.

We have proved that the white portion s area is 1 and being the area of the whole square 1 the are of the black portion must be 1 1 0 so when infinitely many iterations are made the black part doesn t exist and the surface of the sierpinski s carpet is 0. The sierpinski carpet is a plane fractal curve i e. Instead of removing the central third of a triangle the central square piece is removed from a square sliced into thirds horizontally and vertically. That is one reason why area is not a useful dimension for this.