Topological Dimension Of Sierpinski Carpet

Begin with a solid square.

Topological dimension of sierpinski carpet.

Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. Figure 4 presents another example with a topological dimension and a fractal dimension. The sierpinski carpet is the set of points in the unit square whose coordinates written in base. What this basically means is the sierpinski carpet contains a topologically equivalent copy of any compact one dimensional object in the plane.

Furthermore we deduce that the hausdorff dimension of the union of all self avoiding paths admitted on the infinitely ramified sierpiński carpet has the hausdorff dimension d h s a d t h we also put forward a phenomenological relation for. Make 8 copies of the square each scaled by a factor of 1 3 both vertically and horizontally and arrange them to form a new square the same size as the original with a hole in the middle. That is one reason why area is not a useful dimension for this set. Sierpinski carpet as another example of this process we will look at another fractal due to sierpinski.

This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. Cavalier projection of five iterations in the construction of a curve in three dimensions. In this letter the analytical expression of topological hausdorff dimension d t h is derived for some kinds of infinitely ramified sierpiński carpets. Sierpinski used the carpet to catalogue all compact one dimensional objects in the plane from a topological point of view.

Sierpiński demonstrated that his carpet is a universal plane curve. Dimensions of intersections of the sierpinski carpet with lines of rational slopes volume 50 issue 2 qing hui liu li feng xi yan fen zhao. In the case of the sierpinsky carpet figure 2 and since it is a surface we have. For instance the menger sponge the three dimensional analogue of the sierpiński carpet see plate 145 is universal for all compact metrizable spaces of topological dimension one and thus for all jordan curves in space.

The hausdorff dimension of the carpet is log 8 log 3 1 8928. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. Then you apply the same procedure to the remaining 8 subsquares and repeat this ad infinitum this image by noon silk shows the first six stages of the procedure. 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.