Topological Dimension Of Sierpinski Carpet

Sierpinski used the carpet to catalogue all compact one dimensional objects in the plane from a topological point of view.

Topological dimension of sierpinski carpet.

What this basically means is the sierpinski carpet contains a topologically equivalent copy of any compact one dimensional object in the plane. 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. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. In this letter the analytical expression of topological hausdorff dimension d t h is derived for some kinds of infinitely ramified sierpiński carpets.

Fractal dimension of the menger sponge. Begin with a solid square. The measurement of the surface vanishes as the resolution gets refined. Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve.

To build the sierpinski carpet you take a square cut it into 9 equal sized smaller squares and remove the central smaller square. In the case of the sierpinsky carpet figure 2 and since it is a surface we have. This makes sense because the sierpinski triangle does a better job filling up a 2 dimensional plane. The hausdorff dimension of the carpet is log 8 log 3 1 8928.

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. Sierpiński demonstrated that his carpet is a universal plane curve. Sierpinski carpet as another example of this process we will look at another fractal due to sierpinski. 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.

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. Figure 4 presents another example with a topological dimension and a fractal dimension. That is one reason why area is not a useful dimension for this set. 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.