A classiflcation of points on the Sierpinski gasket

Chaos, Solitons & Fractals

In this paper, we aim to construct fractal interpolation function(FIF) on the product of two Sierpiński gaskets. Further, we collect some results regarding smoothness of the constructed FIF. We prove, in particular, that the FIF are Hölder functions under specific conditions. In the final section, we obtain some bounds on the fractal dimension of FIF.

2013

Chapter 6. Definition of Apollonian gasket 6.1. Basic facts Info G. Fibonacci numbers 6.2. Examples of non-bounded Apollonian tiling 6.3. Two interpretations of the set D 6.4. Generalized Descartes theorem 6.5. Integral solutions to Descartes equation Info H. Structure of some groups generated by reflections Chapter 7. Arithmetic properties of Apollonian gaskets 7.1. The structure of Q 7.2. Rational parametrization of circles 7.3. Nice parametrizations of discs tangent to a given disc 7.4. Integral Apollonian gaskets Info I. Möbius inversion formula Chapter 8. Geometric and group-theoretic approach Info J. Hyperbolic (Lobachevsky) plane L 8.1. Action of the group G and Apollonian gaskets 8.2. Action of the group Γ 4 on a Apollonian gasket Chapter 9. Many-dimensional Apollonian gaskets 9.1. General approach 9.2. 3-dimensional Apollonian gasket Bibliography A. Popular books, lectures and surveys B. Books C. Research papers D. Web sites

We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of 2 if f is continuous and 2f is defined as a continuous function. We show that if f is a nonconstant function in the domain of 2, then f 2 is not in the domain of 2. We give two proofs of this fact. The first is based on the analog of the pointwise identity 2f 2 &2f 2f = |{f | 2 , where we show that |{f | 2 does not exist as a continuous function. In fact the correct interpretation of 2f 2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of 2, at a junction point x 0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for | f (x)& f (x 0)| in terms of d(x, x 0) ; for a certain value ;, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of 2.

Mathematische Nachrichten, 2007

We suggest a flexible way to study the self-similar interface of two different fractals. In contrast to previous methods the participating energies are modified in the neighborhood of the intersection of the fractals. In the example of the Vicsek snowflake and the 3-gasket, a variant of the Sierpinski gasket, we calculate the admissible transition constants via the "Short-cut Test". The resulting range of values is reinterpreted in terms of traces of Lipschitz spaces on the intersection. This allows us to describe the interface effects of different transition constants and indicates the techniques necessary to generalize the present interface results.

Mathematical Problems in Engineering, 2012

A fractal lattice is defined by iterative maps on a simplex. In particular, Sierpinski gasket and von Koch flake are explicitly obtained by simplex transformations.

SIAM Undergraduate Research Online, 2013

This paper examines the fractal nature of the Narayana fractal, an object defined by N = {(i, j) ∈ N × N : N (i + j + 1, j + 1) = 1 (mod 2)}. where N (n, k) = 1 n n k n k − 1 are the Narayana numbers. This object closely resembles a fractal derived from Pascal's triangle. This similarity is used to prove that the Hausdorff dimension of the Narayana fractal is log 3/ log 2, and the limit of the Narayana fractal converges to the union of Sierpinski's gasket with one additional point.

Contemporary Mathematics, 2013

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Journal of Mechanics of Materials and Structures, 2009

Self-similar fractals are geometrically stable in the sense that, when generated by a recursive copying process that starts from a basic building block, their final image depends only on the recursive generation process rather than on the shape of the original building block. In this article we show that an analogous stability property can also be applied to fractals as elastic structural elements and used in practice to obtain the stiffnesses of these fractals by means of a rapidly converging numerical procedure. The relative stiffness coefficients in the limit depend on the generation process rather than on their counterparts in the starting unit. The stiffness matrices of the Koch curve, the Sierpiński triangle, and a two-dimensional generalization of the Cantor set are derived and shown to abide by the aforementioned principle.

Fractal and Fractional

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.