Advertisement

The Sequence of Carboncettus Octagons

  • Fabio CaldarolaEmail author
  • Gianfranco d’Atri
  • Mario Maiolo
  • Giuseppe Pirillo
Conference paper
  • 46 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)

Abstract

Considering the classic Fibonacci sequence, we present in this paper a geometric sequence attached to it, where the word “geometric” must be understood in a literal sense: for every Fibonacci number \(F_n\) we will in fact construct an octagon \(C_n\) that we will call the n-th Carboncettus octagon, and in this way we obtain a new sequence \(\big \{C_n \big \}_{n}\) consisting not of numbers but of geometric objects. The idea of this sequence draws inspiration from far away, and in particular from a portal visible today in the Cathedral of Prato, supposed work of Carboncettus marmorarius, and even dating back to the century before that of the writing of the Liber Abaci by Leonardo Pisano called Fibonacci (AD 1202). It is also very important to note that, if other future evidences will be found in support to the historical effectiveness of a Carboncettus-like construction, this would mean that Fibonacci numbers were known and used well before 1202. After the presentation of the sequence \(\big \{C_n\big \}_{n}\), we will give some numerical examples about the metric characteristics of the first few Carboncettus octagons, and we will also begin to discuss some general and peculiar properties of the new sequence.

Keywords

Fibonacci numbers Golden ratio Irrational numbers Isogonal polygons Plane geometric constructions 

Notes

Aknowledgments

This work is partially supported by the research projects “IoT&B, Internet of Things and Blockchain”, CUP J48C17000230006, POR Calabria FESR-FSE 2014–2020.

References

  1. 1.
    Antoniotti, L, Caldarola, F., Maiolo, M.: Infinite numerical computing applied to Hilbert’s, Peano’s, and Moore’s curves. Mediterr. J. Math. (in press)Google Scholar
  2. 2.
    Ball, D.: Equiangular polygons. Math. Gaz. 86(507), 396–407 (2002)CrossRefGoogle Scholar
  3. 3.
    Caldarola, F.: The Sierpiński curve viewed by numerical computations with infinities and infinitesimals. Appl. Math. Comput. 318, 321–328 (2018).  https://doi.org/10.1016/j.amc.2017.06.024CrossRefzbMATHGoogle Scholar
  4. 4.
    Caldarola, F.: The exact measures of the Sierpiński \(d\)-dimensional tetrahedron in connection with a Diophantine nonlinear system. Commun. Nonlin. Sci. Numer. Simul. 63, 228–238 (2018).  https://doi.org/10.1016/j.cnsns.2018.02.026MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caldarola, F., Maiolo, M., Solferino, V.: A new approach to the Z-transform through infinite computation. Commun. Nonlin. Sci. Numer. Simul. 82, 105019 (2020).  https://doi.org/10.1016/j.cnsns.2019.105019CrossRefGoogle Scholar
  6. 6.
    Caldarola, F., Cortese, D., d’Atri, G., Maiolo, M.: Paradoxes of the infinite and ontological dilemmas between ancient philosophy and modern mathematical solutions. In: Sergeyev, Y., Kvasov, D. (eds.) NUMTA 2019. LNCS, vol. 11973, pp. 358–372. Springer, New York (2020)Google Scholar
  7. 7.
    De Villiers, M.: Equiangular cyclic and equilateral circumscribed polygons. Math. Gaz. 95, 102–107 (2011)CrossRefGoogle Scholar
  8. 8.
    Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)CrossRefGoogle Scholar
  9. 9.
    Margenstern, M.: Fibonacci words, hyperbolic tilings and grossone. Commun. Nonlin. Sci. Numer. Simul. 21(1–3), 3–11 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pirillo, G.: A characterization of Fibonacci numbers. Chebyshevskii Sbornik 19(2), 259–271 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pirillo, G.: Figure geometriche su un portale del Duomo di Prato. Prato Storia e Arte 121, 7–16 (2017). (in Italian)Google Scholar
  12. 12.
    Pirillo, G.: La scuola pitagorica ed i numeri di Fibonacci. Archimede 2, 66–71 (2017). (in Italian)Google Scholar
  13. 13.
    Pirillo, G.: L’origine pitagorica dei numeri di Fibonacci. Periodico di Matematiche 9(2), 99–103 (2017). (in Italian)Google Scholar
  14. 14.
    Pirillo, G.: Some recent results of Fibonacci numbers, Fibonacci words and Sturmian words. Southeast Asian Bull. Math. 43(2), 273–286 (2019)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pirillo, G.: Fibonacci numbers and words. Discret. Math. 173(1–3), 197–207 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pirillo, G.: Inequalities characterizing standard Sturmian and episturmian words. Theoret. Comput. Sci. 341(1–3), 276–292 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pirillo, G.: Numeri irrazionali e segmenti incommensurabili. Nuova Secondaria 7, 87–91 (2005). (in Italian)Google Scholar
  18. 18.
    Sergeyev, Y.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, Cosenza (2003)zbMATHGoogle Scholar
  19. 19.
    Sergeyev, Y.D.: Lagrange lecture: methodology of numerical computations with infinities and infinitesimals. Rend. Semin. Matematico Univ. Polit. Torino 68(2), 95–113 (2010)zbMATHGoogle Scholar
  20. 20.
    Sergeyev, Y.D.: Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4(2), 219–320 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversità della CalabriaArcavacata di RendeItaly
  2. 2.Department of Environmental and Chemical Engineering (DIATIC)Università della CalabriaArcavacata di RendeItaly
  3. 3.Department of Mathematics and Computer Science U. DiniUniversity of FlorenceFirenzeItaly

Personalised recommendations