Infinite Numerical Computing Applied to Hilbert’s, Peano’s, and Moore’s Curves

The Peano and the Hilbert curves, denoted by P and H respectively, are historically the first and some of the best known space-filling curves. They have a fractal structure, many variants (as the well-known Moore curve M or a probably new “looped” version H ¯ of H ), and a huge number of applications in the most diverse fields of mathematics and experimental sciences. In this paper, we employ a recently proposed computational system, allowing numerical calculations with infinite and infinitesimal numbers, to investigate the behavior of such curves and to highlight the differences with the classical treatment. In particular, we perform several types of computations and give many examples based not only on the curves H and P , but also on their d -dimensional versions H d and P d , respectively. Following our approach, it is easy to apply this new computational methodology to many other geometrical contexts, with interesting advantages such as summarizing in a single (infinite) number, representing the final result of a sequence of computations, much information both on the geometrical meaning of such a sequence and on the base geometrical structure itself.