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.

Topological dimension of sierpinski carpet.

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. Sierpinski carpet as another example of this process we will look at another fractal due to sierpinski. Fractal dimension of the menger sponge. 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.

That is one reason why area is not a useful dimension for this set. Begin with a solid square. Sierpinski used the carpet to catalogue all compact one dimensional objects in the plane from a topological point of view. The measurement of the surface vanishes as the resolution gets refined.

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. The sierpinski carpet is the set of points in the unit square whose coordinates written in base. The hausdorff dimension of the carpet is log 8 log 3 1 8928. 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.

Figure 4 presents another example with a topological dimension and a fractal dimension. In this letter the analytical expression of topological hausdorff dimension d t h is derived for some kinds of infinitely ramified sierpiński carpets. Next we ll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions. Sierpiński demonstrated that his carpet is a universal plane curve.

Note that dimension is indeed in between 1 and 2 and it is higher than the value for the koch curve. 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. In the case of the sierpinsky carpet figure 2 and since it is a surface we have. Cavalier projection of five iterations in the construction of a curve in three dimensions.