MENGER SPONGE SECTIONS

Abstract

Serpinsky's carpet and Menger's sponge are flat and three-dimensional analogues of the Cantor set, respectively. When these fractals intersect in straight lines and planes, new interesting fractal objects are formed. The article presents an algorithm for arbitrarily accurate construction of sections of the Serpinsky carpet and the Menger sponge by vertical lines and planes, respectively. In particular, it is shown that the vertical sections of the Menger sponge are different combinations of the Serpinsky carpet and the Cantor dust at different stages of construction. Similarly, vertical or horizontal sections of the Serpinsky carpet are combinations of constructions of the Cantor set and segments of appropriate length at different stages. In addition, some other cross-sections of these objects are shown in other straight lines and planes. The construction of such cross-sections is important because the structure of the Menger sponge is very similar to the structure of synthetic activated carbon or artificially created porous alloys NiTi, which are able to remember the shape.