Abstract

Full paper prices: €32 / €24 / €60

Open Access price: €100

Symmetry: Culture and Science
Volume 36, Number 1, pages 083-095 (2025)
https://doi.org/10.26830/symmetry_2025_1_083

A NOTE ON LAGUERRE-TYPE AND EULER-TYPE GRANDI’S ROSES

Diego Caratelli^1,2, Primo Brandi³, Paolo Emilio Ricci^*4

¹ Department of Electrical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands

² Department of Research and Development, The Antenna Company, High Tech Campus 29, 5656 AE - Eindhoven, The Netherlands
Email: d.caratelli@tue.nl

³ Università di Perugia, Dipartimento di Matematica e Informatica, Via Vanvitelli, 1,06123 - Perugia, Italia
Email: primo.brandi@gmail.com

⁴ International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 - Roma, Italia
Email: paoloemilioricci@gmail.com

^* corresponding author

Abstract: Starting from the connection between the cartesian and the polar equation of a given curve, we introduce two different types of Grandi's roses (also called rhodoneas curves) with integer degree. The first extension is connected with the Laguerre-type exponentials, and the second one deals with the Euler's nearly cosine series. Several graphs of the relevant curves are shown, derived by the first author using the Mathematica computer algebra program.

AMS 2010 Mathematics Subject Classifications: 33C99; 12E10; 42C10; 42C05.

Keywords: Spirals, Grandi's roses, pseudo-Chebyshev functions, Laguerre-type exponentials, Euler's nearly cosine series.

References:
Bini, D., Cherubini, C., Filippi, S., Gizzi, A., Ricci, P.E. (2010) On the universality of spiral waves, Comm. Computat. Physics, 8, 610-622. https://doi.org/10.4208/cicp.240909.170210a

Brandi, P., Ricci, P.E. (2019) Some properties of the pseudo-Chebyshev polynomials of half-integer degree, Tbilisi Math. J., 12(4), 111-121. https://doi.org/10.32513/tbilisi/1578020571

Caratelli, D., Ricci, P.E. (2020) A note on the orthogonality properties of the pseudo-Chebyshev functions, Symmetry, 5. https://doi.org/10.3390/sym12081273

Cesarano, C., Ricci, P.E. (2019) Orthogonality properties of the pseudo-Chebyshev functions (Variations on a Chebyshev's theme), Mathematics, 7, 180. https://doi.org/10.3390/math7020180

Cesarano, C., Pinelas, S., Ricci, P.E. (2019) The third and fourth kind pseudo-Chebyshev polynomials of half-integer degree, Symmetry, 11, 274. https://doi.org/10.3390/sym11020274

Curcio, A., Dattoli, G., Di Palma, E., Natalini, P., Ricci, P.E. (2025) Euler nearly cosine series and generalized trigonometric functions, Georgian Math. Journal, (to appear).

D'Arcy Thompson, W. (1942) On Growth and Form, Reprint of the University Press, Cambridge.

Dattoli, G., Ricci, P.E. (2003) Laguerre-type exponentials, and the relevant L-circular and L-hyperbolic functions, Georgian Math. Journal, 10, 481-494. https://doi.org/10.1515/GMJ.2003.481

Gielis, J. (2011) The Geometry of Universal Natural Shapes. https://doi.org/10.1142/9789814355247_0016

Gielis, J., Caratelli, D., Shi, P., Ricci, P.E. (2020) A note on spirals and curvature, Growth and Form, 1(1), 1-8. https://doi.org/10.2991/gaf.k.200124.001

Heath, T.L. (1897) The Works of Archimedes, Cambridge Univ. Press., Google Books.

Lockwood, E.H. (1961) A book of curves, Cambridge Univ. Press, Cambridge, 1961. https://doi.org/10.1017/CBO9780511569340

Ricci, P.E. (2018) Complex spirals and pseudo-Chebyshev polynomials of fractional degree, Symmetry, 10, 671. https://doi.org/10.3390/sym10120671

Ricci, P.E. (2020a) A Survey on pseudo-Chebyshev Functions, 4Open, 3 2. https://doi.org/10.1051/fopen/2020001

Ricci, P.E (2020b) Laguerre-type Exponentials, Laguerre Derivative and Applications. A Survey, Mathematics, 8, 2054. https://doi.org/10.3390/math8112054