The exact measures of the Sierpiński d-dimensional tetrahedron in connection with a Diophantine nonlinear system

The Sierpiński d-dimensional tetrahedron Δd is the generalization of the most known Sierpiński gasket which appears in many fields of mathematics. Considering the sequences of polytopes {Δnd}n that generate Δd, we find closed formulas for the sum vnd,k of the measures of the k-dimensional elements of Δnd, deducing the behavior of the sequences {vnd,k}n. It becomes quite clear that traditional analysis does not have the adequate language and notations to go further, in an easy and manageable way, in the study of the previous sequences and their limit values; contrariwise, by adopting the new computational system for infinities and infinitesimals developed by Y.D. Sergeyev, we achieve precise evaluations for every k-dimensional measure related to each Δd, obtaining a set W={v◯1d,k}d,k of values expressed in the new system, which leads us to a Diophantine problem in terms of classical number theory. To solve it, we work with traditional tools from algebra and mathematical analysis. In particular, we define two kinds of equivalence relations on W and we get a detailed description of the partition of various of its subsets together with the exact composition of the corresponding classes of equivalence. Finally, we also show as the unique Sierpiński tetrahedron for each dimension d, is replaced, if we adopt Sergeyev's framework, by a whole family of infinitely many Sierpiński d-dimensional tetrahedrons.