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

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Abstract

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 \({\overline{H}}\) 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.

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Notes

  1. The basic proposition of elementary calculus often called necessary Cauchy condition for convergence states that “if \(\sum _n a_n\) converges, then \(\{a_n\}_n\) is infinitesimal”. But the word “infinitesimal” does not refer, obviously, to the elements \(a_n\) which are, in this case, ordinary real or complex numbers.

  2. Note that \(\displaystyle \frac{6^{\tiny \textcircled {{\tiny 1}}}}{4^{\tiny \textcircled {{\tiny 1}}}-1} = \left( \frac{3}{2}\right) ^{\tiny \!\!\textcircled {{\tiny 1}}} + \frac{\left( 3/2\right) ^{\tiny \textcircled {{\tiny 1}}}}{4^{\tiny \textcircled {{\tiny 1}}}-1}\).

  3. We give here some explanations. First of all, for any \(\varepsilon >0\) and \(P\in \mathbb {R}^2\) denote by \(D_{\varepsilon }(P)\) the open disc of radius \(\varepsilon \) centered at P. Consider, for instance, the curve \(M_N\), with N any number of \({\widehat{\mathbb {N}}}\), and a point \(P\in M_N\). Then note that \(M_N\cap D_{\varepsilon }(P)\subseteq L_h\cup L_v\), where \(L_h\) and \(L_v\) are two suitable lines of \(\mathbb {R}^2\) (\(L_h\) horizontal and \(L_v\) vertical, and one of them at least contains P).

  4. The interested reader can easily and in different ways give more formal and precise meanings to the term “approximation” used here; for instance, he could see [12, Definition 2.1] or simply start from the observation

    $$\begin{aligned} l\big (H^t_{N}\big )= l\big (P^d_{M}\big ) \quad \text{(up } \text{ to } \text{ infinitesimals) } \quad \ \Leftrightarrow \quad \ \frac{(t-1)N}{(d-1)M}=\log _23. \end{aligned}$$

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Authors and Affiliations

  1. Via Pinnelli, 51, 31100, Treviso, TV, Italy

    Luigi Antoniotti

  2. Department of Mathematics and Computer Science, Università della Calabria, Cubo 31/B, ponte Bucci, 87036, Arcavacata di Rende, CS, Italy

    Fabio Caldarola

  3. Department of Environmental and Chemical Engineering, Università della Calabria, Cubo 42/B, ponte Bucci, 87036, Arcavacata di Rende, CS, Italy

    Mario Maiolo

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  1. Luigi Antoniotti
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  2. Fabio Caldarola
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Correspondence to Fabio Caldarola.

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This work was partially supported by the research project “I-BEST”, CUP B28I17000290008, PON “Innovazione e competitività” 2014/2020 MiSE Horizon 2020.

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Antoniotti, L., Caldarola, F. & Maiolo, M. Infinite Numerical Computing Applied to Hilbert’s, Peano’s, and Moore’s Curves . Mediterr. J. Math. 17, 99 (2020). https://doi.org/10.1007/s00009-020-01531-5

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