Showing papers on "n-flake published in 1998"
TL;DR: In this paper, the authors calculate the box-counting dimension of a self-affine version of the Sierpinski triangle by investigating the singular values of the affine transformations.
Abstract: We calculate the box-counting dimension of a self-affine version of the Sierpinski triangle. This is done by investigating the singular values of the affine transformations. We also investigate multifractal features of self-affine measures supported by certain generalized Sierpinski triangles.
20 citations
TL;DR: In this paper, the authors studied the geometry of the cut locus of a separating fractal set A in a Riemannian manifold, and showed that every point of A is a limit point of C{A] of A, and the Hausdorff dimension of C {A} is greater than or equal to that of A.
Abstract: We study the geometry of the cut locus of a separating fractal set A in a Riemannian manifold. In particular, we prove that every point of A is a limit point of the cut locus C{A) of A, and the Hausdorff dimension of C{A) is greater than or equal to that of A. Furthermore, we study the cut locus of the well-known Koch snowflake, and show the Hausdorff dimension of its cut locus is log 6/log 3 which is greater than the Hausdorff dimension, Iog4/log3, of the Koch snowflake itself. We also give another example for which the Hausdorff dimension of the cut locus stays the same. These two new examples are new fractal objects which are of interest on their own right.
1 citations
TL;DR: For the Sierpinski family of fractals, it has been conjectured that, where d is the dimension of the Euclidean space in which the Sierspinski fractal is embedded as mentioned in this paper.
Abstract: Random walks on some fractals can be analysed by renormalization procedures. These techniques make it possible to obtain the Laplace transform of the first-passage time probability density function of a random walker that moves in the fractal. The calculation depends on a function that is particular to each kind of fractal. For the Sierpinski family of fractals, it has been conjectured that , where d is the dimension of the Euclidean space in which the Sierpinski fractal is embedded. We provide a proof of the conjecture that is based on the symmetries of the Sierpinski fractal.