Some Insights into the Sierpiński Triangle Paradox

Abstract

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 SAC⊂SG. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve PC⊂0,12 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 dlSAC=d=1, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, dlSG=D. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted.