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We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that S A C ⊂ S G . Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve P C ⊂ 0,1 2 differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension D = ln ⁡ 3 / ln ⁡ 2 , their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension d l S A C = d = 1 , whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, d l S G = D . Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We study the scaling properties of forced folding of thin materials of different geometry. The scaling relations implying the topological crossovers from the folding of three-dimensional plates to the folding of two-dimensional sheets, and further to the packing of one-dimensional strings, are derived for elastic and plastic manifolds. These topological crossovers in the folding of plastic manifolds were observed in experiments with predominantly plastic aluminum strips of different geometry. Elasto-plastic materials, such as paper sheets during the (fast) folding under increasing confinement force, are expected to obey the scaling force–diameter relation derived for elastic manifolds. However, in experiments with paper strips of different geometry, we observed the crossover from packing of one-dimensional strings to folding two dimensional sheets only, because the fractal dimension of the set of folded elasto-plastic sheets is the thickness dependent due to the strain relaxation after a confinement force is withdrawn. [Copyright &y& Elsevier]