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Binódic periodic system: a mathematical approach

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Maybe we can even be at the point of a revolution in representation of the periodic system, which is being marked by current developments.

—Eric R. Scerri (2004)

Abstract

This article discusses the mathematizing of the chemical periodic system as a grid, which leads to a quadratic function or “binódica function” formed by pairs of periods or binodos (dyads). We describe the periodic law as an increasing function of the principal quantum number (n). It works subject to the dialectical laws that generate; first: gradual quantitative changes: (2n2), with “duplication” of periods: (2n2, 2n2). Second: radical quantitative changes: (4n2), with the emergence of new quantum transitions, growth and a change in the format of the binodos (qualitative change). We use analytical and graphic methods to mathematize Mendeleev’s law, making the size of the binodos (Y) and the continuous series (Z) depend on the number of the binode (B), which is the same quantum number (n). Likewise, we show a graphic representation, in 2D, of a continuous and self-similar spiral function of the elements, in a geometric growth pattern.

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Notes

  1. Rule of Madelung-Klechkowski, also of Pao Fang Yi, R. Hakala, rule (n + l), rule of the handsaw, etc. It is an empirical rule for electron filling in atomic orbitals. It is not exact, there are 19 exceptions. E. Scerri, and especially Eugene Schwarz, have clarified the critical points at which this rule, which is not derived from quantum mechanics, is inconsistent with empirical results. Schwarz believes that the Madelung Rule is incorrect and that “it has never been and cannot be proven by quantum theory”; personal correspondence, June 13, 2012. (Ostrovsky denies this way of thinking).

  2. This concept of spirals rolled in a cone was developed from an idea of Baca Mendoza (1953), where he comments on the development of concatenated stages that form enveloping spirals like spindles, on a cone.

  3. That is, in “a final equation whose contemplation produces the same effect as the culmination of a work of art.” In this symmetrical way, something is hidden in the parabolic equation or in the spirals: "The beauty that inspires a melody, a painting or a poetry” (Álvarez Vita, Beauty as a Guide for Science, undated monograph).

  4. Mandelbrot (1982) formulates a geometry of fractals which has characteristics of self-similarity. An object is self-similar if parts have the same shape as the whole structure, but may be in a different scale and slightly deformed. The fractal dimension is generated by mathematical algorithms or iterated function systems.

  5. In 1862, Chancourtois (1830–1889) devised a telluric helix rolled in a cylinder in which he distinguished the periodicity of the second binode with (8, 8) elements. His proposal was not fulfilled for the following periods or dyads—(18, 18), (32, 32)—because the sequence of elements was incomplete.

  6. Libedinsky (1938) postulates a singular spiral classification or "dialectical grouping of the elements" that we find published in: https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=909.

  7. The spectral distribution exhibits numerous exceptions noted by many authors; however, in the proton sequence there can be no such exceptions because the identity of the element or the atomic number completely identifies it.

  8. “Clearly, there is a general desire for a sleeker periodic system that fits well with modern physical theories of matter. Maybe we can even be at the point of a revolution in representation of the periodic system, which is being marked by current developments. In addition, the new field of philosophy of chemistry has led to renewed interest in the fundamental aspects of the central model of chemistry, the periodic table. Perhaps the time is ripe for a new careful assessment of the best possible representation”; Scerri (2004). “The best representation for the periodic system: the role of the rule n + l and the concept of an element as basic substance”, in: D.H. Rouvray and R.B. King (eds.), The Periodic Table in the XXI century. Research Studies Press, Chapter 5, 2004.

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Acknowledgements

In memory D.I. Mendeleev in the 150th Anniversary of the publication of the Periodic Table. In memory of Dr. Oswaldo Baca Mendoza, whose work inspired the author of this study. With gratitude to Dr. Eric Scerri, for his example, support and encouragement that he gave me for this work. My thanks to Dr. Conal Boyce for providing constructive criticism during an early phase of the project, in 2018, and to Mark Leach, Milagros Paredes, Lucas Pierce Paredes, Ana María Enciso, Dimitri Weise, G. Restrepo, Alfio Zambón, Rolando Alfaro, M. Labarca, and the group “Periodic Table mailing list” (moderated by E. Scerri). In memory of Ray Hefferlin and Rubén Darío Osorio.

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  1. National University of San Antonio Abad of Cusco, Cusco, Peru

    Julio Antonio Gutiérrez Samanez

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Appendices

Appendices

Appendix 1: New periodic system of the elements of Dr. O. Baca Mendoza (1953)

figure a

Appendix 2: ‘Left step’ periodic table of Charles Janet (1929[1928])

figure b

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Gutiérrez Samanez, J.A. Binódic periodic system: a mathematical approach. Found Chem 22, 235–266 (2020). https://doi.org/10.1007/s10698-020-09359-3

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