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Symmetry: Culture and Science
Volume 32, Number 2, pages 141-144 (2021)
https://doi.org/10.26830/symmetry_2021_2_141

STANDARD PLANES AND SPACES EMERGE WHEN SEQUENCING PAIRS OF NATURAL NUMBERS

Karl Javorszky^*

^* Institut für angewandte Statistik, Loeblichgasse 13/16, A-1090 Wien, Austria
E-mail: karl.javorszky@gmail.com
Web: http://www.tautomat.com
ORCID: 0000-0002-4751-2682

Abstract: Sequencing and resequencing a collection of logical objects of the form (a,b), where a,b Î N, a,b ≤ 16, a ≤ b, creates patterns of movements of the elements, known as ‘cycles’. Of some of the cycles, (namely A: 1. a+b,a ↔ b-2a,a; 2. b-2a,a ↔ a-2b,b-2a; 3. a-2b,b-2a ↔ a+b,a; and B: 1. a+b,b ↔ b-2a,a-2b; 2. b-2a,a-2b ↔ a-2b,a; 3. a-2b,a ↔ a+b,b; these we call ‘standard cycles’, they appear on resorting arg_α,arg_β ↔ arg_γ,arg_δ), two Descartes type spaces can be constructed. The relation of the two spaces to each other invites research into their symmetrical properties, as are the pairwise symmetries that appear when relating the planes to each other which make up the 3D spaces. The spaces are furthermore transcended by another two, almost perfectly symmetrical, planes. There is some importance to the theme of a priori existing logical spaces, determining laws of physics, because the syntax of the cycles agrees to the syntax Nature uses when de-linearizing and re-linearizing the genetic information from and into the DNA. Symmetry is evidently a sharp tool in the shed of Nature, so this simple report on results of sorting and ordering the first few incarnated variants of a+b=c is an alert for attention and professional investigation: symmetry evidently plays an important part also in the transfer of genetic information!

Keywords: arithmetic, cycles, information
MSC 2010: 03E05, 03-02, 03A99, 05E99, 13F99
PACS 2010: 02.10.De, 02.30.Lf, 02.60.Cb, 04.50.Gh, 04.60.Cf, 81.16.Dn, 82.20.Fd, 82.30.Nr

References:
Javorszky, K. (2013) Learn to Count in Twelve Easy Steps. www.tautomat.com, also in Webinar in FIS: http://listas.unizar.es/cgi-bin/mailman/listinfo/fis;

Javorszky, K. (2016) Natural Orders, De Ordinibus Naturalibus. Morawa: Vienna, Austria. ISBN: 978-3-99057-139-2

Javorszky, K. (2017) Transfer of genetic information: An innovative model. Proceedings, 1, 3, 222. https://doi.org/10.3390/IS4SI-2017-04030

OEIS A235647. Available online: www.oeis.org/A235647.