From Pythagoras to Fourier and From Geometry to Nature



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AndrewsL.C. Special Functions of Mathematics for Engineers. Oxford University Press, Oxford – New York, 1998.
AntonelliP.L.ZastawniakT.J. Preface: Lagrange Differential Geometry, Finsler Spaces and Noise Applied in Biology and Physics. Mathematical and Computer Modelling, Vol. 20(4-5), pp. xixiii, 1994. DOI: DOI:
ArrowK.J.CheneryH.B.MinhasB.S.SolowR.M. Capital-Labor Substitution and Economic Efficiency. The Review of Economics and Statistics, Vol. 43, pp. 225250, 1961.
BiaP.CaratelliD.MesciaL.GielisJ. Analysis and Synthesis of Supershaped Dielectric Lens Antennas. IET Microwaves, Antennas & Propagation, Vol. 9(14), pp. 14971504, 2015. DOI: DOI:
BracewellR.N. The Life of Joseph Fourier. In: PriceJ.F. (eds.), Fourier Techniques and Applications. Springer, Boston (MA), 1985.
BrandiP.RicciP.E. Some Properties of the Pseudo-Chebyshev Polynomials of Half-Integer Degree. Tbilisi Mathematical Journal, Vol. 12(4), pp. 111121, 2019.
BrandiP.SalvadoriA. A Magic Formula of Nature and Art. APLIMAT 2018, 17th Conference on Applied Mathematics, Bratislava, Slovakia, 6–8 February 2018.
Van BrummelenG. Jamshīd al-Kāshī: Calculating Genius. Mathematics in School, Vol. 27(4), pp. 4044, 1998. URL: URL:
CaratelliD.GermanoB.GielisJ.HeM.X.NataliniP.RicciP.E. Fourier Solution of the Dirichlet Problem for the Laplace and Helmholtz Equations in Starlike Domains. Lecture Notes of TICMI, Vol. 10, pp. 164, 2009.
CaratelliD.GermanoB.HeM.X.RicciP.E. Solution of the Dirichlet Problem for the Laplace Equation in a General Cylinder. Lecture Notes of TICMI, Vol. 10, pp. 2034, 2009.
CaratelliD.GielisJ.NataliniP.RicciP.E.TavkhelidzeI. The Robin Problem for the Helmholtz Equation in a Starlike Planar Domain. Georgian Mathematical Journal, Vol. 18(3), pp. 465479, 2011. DOI: DOI:
CaratelliD.GielisJ.RicciP.E. Fourier-Like Solution of the Dirichlet Problem for the Laplace Equation in k-Type Gielis Domains. Journal of Pure and Applied Mathematics: Advances and Applications, Vol. 5(2), pp. 99111, 2011.
CaratelliD.GielisJ.RicciP.E. The Robin Problem for the Helmholtz Equation in a Three-Dimensional Starlike Domain. Applied Mathematics, Informatics and Mechanics, Vol. 21(1), pp. 517, 2016.
CaratelliD.GielisJ.TavkhelidzeI.RicciP.E. Fourier-Hankel Solution of the Robin Problem for the Helmholtz Equation in Supershaped Annular Domains. Boundary Value Problems, Volume 2013, 253, 2013. DOI: DOI:
CaratelliD.GielisJ.TavkhelidzeI.RicciP.E. Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell. Applied Mathematics, Vol. 4(1A), pp. 263270, 2013. DOI: DOI:
CaratelliD.NataliniP.RicciP.E. Fourier Solution of the Wave Equation for a Star-Like-Shaped Vibrating Membrane. Computers & Mathematics With Applications, Vol. 59(1), pp. 176184, 2010. DOI: DOI:
CaratelliD.NataliniP.RicciP.E.YarovoyA. Fourier Solution of the 2D Neumann Problem for the Helmholtz Equation. Lecture Notes of Seminario Interdisciplinare di Matematica, Vol. 9, pp. 163172, 2010.
CaratelliD.NataliniP.RicciP.E.YarovoyA. The Neumann Problem for the Helmholtz Equation in a Starlike Planar Domain. Applied Mathematics and Computation, Vol. 216(2), pp. 556564, 2010. DOI: DOI:
CaratelliD.RicciP.E. The Dirichlet Problem for the Helmholtz Equation in a Starlike Domain. Lecture Notes of TICMI, Vol. 10, pp. 5059, 2009.
CaratelliD.RicciP.E. The Dirichlet Problem for the Laplace Equation in a Starlike Domain. Lecture Notes of TICMI, Vol. 10, pp. 3549, 2009.
CaratelliD.RicciP.E.GielisJ. The Robin Problem for the Laplace Equation in a Three-Dimensional Starlike Domain. Applied Mathematics and Computation, Vol. 218(3), pp. 713719, 2011. DOI: DOI:
CarlesonL. On Convergence and Growth of Partial Sums of Fourier Series. Acta Mathematica, Vol. 116, pp. 135157, 1966. DOI: DOI:
CesaranoC.PinelasS.RicciP.E. The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree. Symmetry, Vol. 11(2), 274, 2019. DOI: DOI:
CesaranoC.RicciP.E. Orthogonality Properties of the Pseudo-Chebyshev Functions (Variations on a Chebyshev’s Theme). Mathematics, Vol. 7(2), 180, 2019. DOI: DOI:
ChaconR. Using Jacobian Elliptic Functions to Model Natural Shapes. International Journal of Bifurcation and Chaos, Vol. 22(1), 1230005, 2012. DOI: DOI:
ChakrabartyD. Non-Parametric Deprojection of Surface Brightness Profiles of Galaxies in Generalised Geometries. Astronomy and Astrophysics, Vol. 510, A45, 2010. DOI: DOI:
ChapmanD.GielisJ. Gielis Transformations for the Audiovisual Database. Symmetry Festival 2021, Sofia, Bulgaria. Extended abstract in: Symmetry, Culture and Science, 2021.
Chen.B.Y. Total Mean Curvature and Submanifolds of Finite Type. Series in Pure Mathematics, Vol. 1, World Scientific, 1984. DOI: DOI:
ChenB.Y.DecuS.VerstraelenL. Notes on Isotropic Geometry of Production Models. Kragujevac Journal of Mathematics, Vol. 38(1), pp. 2333, 2014.
ChernS.-S. Finsler Geometry Is Just Riemannian Geometry Without the Quadratic Restriction. Notices of the American Mathematical Society, Vol. 43(9), pp. 959963, 1996.
ChiharaT. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.
DattoliG.MiglioratiM.RicciP.E. The Parabolic Trigonometric Functions and Chebyshev Radicals. ENEA Report RT/2007/21/FIM, 2007.
DattoliG.Di PalmaE.GielisJ.LicciardiS. Parabolic Trigonometry. International Journal of Applied and Computational Mathematics, Vol. 6(2), 37, 2020. DOI: DOI:
FeynmanR. The Feynman Lectures on Physics, Vol. 2, 1963.
FicheraG.De VitoL. Funzioni Analitiche di Una Variabile Complessa. Veschi, Rome, 1964.
FougerolleY.D.GribokA.FoufouS.TruchetetF.AbidiM.A. Boolean Operations With Implicit and Parametric Representation of Primitives Using R-Functions. IEEE Transactions on Visualization and Computer Graphics, Vol. 11(5), pp. 529539, 2005. DOI: DOI:
GardnerM. Piet Hein’s Superellipse. In: Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American, pp. 240254, 1977.
GautschiW. On Mean Convergence of Extended Lagrange Interpolation. Journal of Computational and Applied Mathematics, Vol. 43(1-2), pp. 1935, 1992. DOI: DOI:
GielisF.GielisJ. Gielis Transformations and Their Impact on Science and Technology. Technical Report on ResearchGate, 2021. DOI: DOI:
GielisJ. A Generic Geometric Transformation That Unifies a Wide Range of Natural and Abstract Shapes. American Journal of Botany, Vol. 90(3), pp. 333338, 2003. DOI: DOI:
GielisJ. The Geometrical Beauty of Plants. Atlantis Press, Paris, 2017.
GielisJ. De Uitvinding van de Cirkel. Geniaal Press, Antwerp, 2001 (ISBN 90-6215-792-0). Translated into English: J. Gielis. Inventing the Circle. Geniaal Press, Antwerp, 2003.
GielisJ.CaratelliD.van CoevordenC.M.D.J.RicciP.E. The Common Descent of Biological Shape Description and Special Functions. In: International Conference on Differential & Difference Equations and Applications, pp. 119131, Springer, Cham, 2017.
GielisJ.CaratelliD.FougerolleY.RicciP.E.GeratsT. A Biogeometrical Model for Corolla Fusion in Asclepiad Flowers. In: Atlantis Transactions in Geometry, Vol. 2, Modeling in Mathematics: Proceedings of the Second Tbilisi-Salerno Workshop on Modeling in Mathematics, pp. 83106. Atlantis Press, Paris, 2017.
GielisJ.CaratelliD.FougerolleY.RicciP.E.TavkelidzeI.GeratsT. Universal Natural Shapes: From Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems. PLOS One, Vol. 7(9), e29324, 2012. DOI: DOI:
GielisJ.CaratelliD.ShiP.RicciP.E. A Note on Spirals and Curvature. Growth and Form, Vol. 1(1), pp. 18, 2020. DOI: DOI:
GielisJ.HaesenS.VerstraelenL. Universal Natural Shapes: From the Super Eggs of Piet Hein to the Cosmic Egg of Georges Lemaître. Kragujevac Journal of Mathematics, Vol. 28, pp. 5768, 2005.
GielisJ.NataliniP.RicciP.E. A Note About Generalized Forms of the Gielis Formula. In: Atlantis Transactions in Geometry, Vol. 2, Modeling in Mathematics: Proceedings of the Second Tbilisi-Salerno Workshop on Modeling in Mathematics, pp. 107116. Atlantis Press, Paris, 2017.
GielisJ.RicciP.E.TavkhelidzeI. The Möbius Phenomenon in Generalized Möbius-Listing Surfaces and Bodies, and Arnold’s Cat Phenomenon. Tbilisi Mathematical Journal, article in press, 2021.
GielisJ.VerhulstR.CaratelliD.RicciP.E.TavkhelidzeI. On Means, Polynomials and Special Functions. The Teaching of Mathematics, Vol. 17(1), pp. 120, 2014.
GouzévitchI.GouzévitchD. Gabriel Lamé à Saint Pétersbourg (1820–1831). Bulletin de la Sabix - Société des Amis de la Bibliothèque et de l’Histoire de l’École Polytechnique, Vol. 44, pp. 2043, 2009. DOI: DOI:
GrahamA.W.SpitlerL.R.ForbesD.A.LiskerT.MooreB.JanzJ. LEDA 074886: A Remarkable Rectangular-Looking Galaxy. The Astrophysical Journal, Vol. 750(2), 121, 2012.
GrammelR. Eine Verallgemeinerung der Kreis- und Hyper-belfunktionen. Ingenieur-Archiv, Vol. 16(3-4), pp. 188200, 1948.
GuidettiM.CaratelliD.RoystonT.J. Converging Super-Elliptic Torsional Shear Waves in a Bounded Transverse Isotropic Viscoelastic Material with Nonhomogeneous Outer Boundary. The Journal of the Acoustical Society of America, Vol. 146(5), EL451, 2019. DOI: DOI:
GuitartR. Les Coordonnées Curvilignes de Gabriel Lamé, Réprésentations des Situation Physiques et Nouveaux Objets Mathématiques. Bulletin de la Sabix - Société des Amis de la Bibliothèque et de l’Histoire de l’École Polytechnique, Vol. 44, pp. 119129, 2009. DOI: DOI:
HaesenS.NistorA.I.VerstraelenL. On Growth and Form and Geometry I. Kragujevac Journal of Mathematics, Vol. 36(1), pp. 525, 2012.
HalmosP.R. Finite-Dimensional Vector Spaces. Springer-Verlag, Berlin - New York, 1974.
HowellK.B. Principles of Fourier Analysis. CRC Press, Boca Raton (FL), 2001.
HuangW.LiY.NiklasK.J.GielisJ.DingY.CaoL.ShiP. A Superellipse With Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo. Symmetry, Vol. 12(12), 2073, 2020. DOI: DOI:
JeffreyA. Mathematics for Engineers and Scientists, 6th Edition. CRC Press, Boca Raton (FL), 2004.
KaganV.F. Riemann’s Geometric Ideas. The American Mathematical Monthly, Vol. 112(1), pp. 7986, 2005.
KnoppK. Infinite Sequences and Series. Dover Publications, New York (NY), 1956.
KoisoM.PalmerB. Equilibria for Anisotropic Surface Energies and the Gielis Formula. Forma, Vol. 23(1), pp. 18, 2008.
KoisoM.PalmerB. Rolling Construction for Anisotropic Delaunay Surfaces. Pacific Journal of Mathematics, Vol. 234(2), pp. 345378, 2008. DOI: DOI:
KrallG. Meccanica Tecnica Delle Vibrazioni, Vol. II. Veschi, Rome, 1970.
LaméG. Examen des Différentes Méthodes Employées pour Résoudre les Problèmes de Géométrie. Mme. Ve. Courcier, Imprimeur-Libraire, 1818.
LenjouK. Krommen en Oppervlakken van Lamé en Gielis: Van de Formule van Pythagoras tot de Superformule. Master’s Thesis, University of Louvain, Department of Mathematics, 2005.
LinS.ZhangL.ReddyG.V.P.HuiC.GielisJ.DingY.ShiP. A Geometrical Model for Testing Bilateral Symmetry of Bamboo Leaf With a Simplified Gielis Equation. Ecology and Evolution, Vol. 6(19), pp. 67986806, 2016. DOI: DOI:
LucasÉ. Recherche sur Plusieurs Ouvrages de Léonarde de Pise. Bullettino di Bibliografia e di Storia Delle Scienze Matematiche e Fisiche, Vol. 10, March, April and May, 1877. Imprimerie des Sciences Mathématiques et Physiques, Rome, 1877.
MarksRobert J.II Handbook of Fourier Analysis & Its Applications. Oxford University Press, 2009. DOI: DOI:
MasonJ.C.HandscombD.C. Chebyshev Polynomials. Chapman and Hall, New York (NY) / CRC Press, Boca Raton (FL), 2003.
MatsuuraM. Gielis’ Superformula and Regular Polygons. Journal of Geometry, Vol. 106(2), pp. 383403, 2015. DOI: DOI:
NataliniP.PatriziR.RicciP.E. Heat Problems for a Starlike Shaped Plate. Applied Mathematics and Computation, Vol. 215(2), pp. 495502, 2009. DOI: DOI:
NataliniP.RicciP.E. The Laplacian in Stretched Polar Coordinates and Applications. Lecture Notes of TICMI, Vol. 10, pp. 719, 2009.
NikiforovA.F.UvarovV.B. Special Functions of Mathematical Physics. Birkhäuser Verlag, Basel, 1988.
RicciP.E. Alcune Osservazioni Sulle Potenze Delle Matrici del Secondo Ordine e Sui Polinomi di Tchebycheff di Seconda Specie. Atti della Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali, Vol. 109, pp. 405410, 1975.
RicciP.E. Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree. Symmetry, Vol. 10(12), 671, 2018. DOI: DOI:
RicciP.E. A Note on the D-Trigonometry and the Relevant D-Fourier Expansions. Growth and Form, Vol. 2(1), pp. 1116, 2021. DOI: DOI:
RicciP.E. A Note on Golden Ratio and Higher Order Fibonacci Sequences. Turkish Journal of Analysis and Number Theory, Vol. 8(1), pp. 15, 2020. DOI: DOI:
RicciP.E. I Polinomi di Tchebycheff in Più Variabili. Rendiconti di Matematica, Vol. 11, pp. 295327, 1978.
RicciP.E. Sulle Potenze di Una Matrice. Rendiconti di Matematica, Vol. 9, pp. 179194, 1976.
RicciP.E. A Survey on Pseudo-Chebyshev Functions. 4Open, Vol. 3, 2, 2020. DOI: DOI:
RieszF. Untersuchungen über Systeme Integrierbarer Funktionen. Mathematische Annalen, Vol. 69(4), pp. 449497, 1910.
RivlinT.J. The Chebyshev Polynomials. John Wiley & Sons, New York, 1974.
Rodríguez-OliverosR.Sánchez-GilJ.A. Gold Nanostars as Thermoplasmonic Nanoparticles for Optical Heating. Optics Express, Vol. 20(1), pp. 621626, 2012. DOI: DOI:
RudinW. Functional Analysis, 2nd Edition. McGraw-Hill, New York (NY), 1991.
ShiP.HuangJ.G.HuiC.Grissino-MayerH.D.TardifJ.C.ZhaiL.H.WangF.S.LiB.L. Capturing Spiral Radial Growth of Conifers Using the Superellipse to Model Tree-Ring Geometric Shape. Frontiers in Plant Science, Vol. 6, 856, 2015. DOI: DOI:
ShiP.RatkowskyD.A.GielisJ. The Generalized Gielis Geometric Equation and Its Application. Symmetry, Vol. 12(4), 645, 2020. DOI: DOI:
ShiP.RatkowskyD.A.LiY.ZhangL.LinS.GielisJ. A General Leaf Area Geometric Formula Exists for Plants – Evidence from the Simplified Gielis Equation. Forests, Vol. 9(11), 714, 2018. DOI: (Best Annual Paper Award in Forests in 2019) DOI:
SpíchalL. Gielisova Transformace Logaritmické Spirály. Pokroky Matematiky, Fyziky a Astronomie. Vol. 65(2), pp. 7689, 2020.
SpíchalL. Jednotková Parabola, Zlatý Řez a Parabolickeé π. Preprint on Researchgate, 2021.
SpíchalL. Superelipsa a Superformule. Matematika-Fyzika-Informatika, Vol. 29(1), pp. 5469, 2020.
SrivastavaH.M.ManochaH.L. A Treatise on Generating Functions (Mathematics and Its Applications). Halsted Press / Ellis Horwood, Chichester, 1984.
StruckC. Natural Orbit Approximations in Single Power-Law Potentials. Monthly Notices of the Royal Astronomical Society, Vol. 446(3), pp. 31393149, 2015. DOI: DOI:
SwintonJ.OchuE. MSI Turing’s Sunflower Consortium. Novel Fibonacci and Non-Fibonacci Structure in the Sunflower: Results of a Citizen Science Experiment. Royal Society Open Science, Vol. 3(5), 160091, 2016. DOI: DOI:
TavkhelidzeI.CaratelliD.GielisJ.RicciP.E.RogavaM.TransiricoM. On a Geometric Model of Bodies With “Complex” Configuration and Some Movements. In: Atlantis Transactions in Geometry, Vol. 2, Modeling in Mathematics: Proceedings of the Second Tbilisi-Salerno Workshop on Modeling in Mathematics, pp. 129158. Atlantis Press, Paris, 2017.
TavkhelidzeI.CassisaC.GielisJ.RicciP.E. About “Bulky” Links, Generated by Generalized Möbius Listing’s Bodies GML 3n. Rendiconti Lincei – Matematica e Applicazioni, Vol. 24(1), pp. 1138, 2013.
ThomR. Paraboles et Catastrophes: Entretiens sur les Mathématiques, la Science et la Philosophie. Flammarion, 1983.
ThompsonA.C. Minkowski Geometry. Cambridge University Press, 1996. DOI: DOI:
TolstovG.P. Fourier Series (reprint). Dover Publications, New York (NY), 2012.
TrefethenL.N. Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (PA), 2013.
VerstraelenL. A Concise Mini History of Geometry. Kragujevac Journal of Mathematics, Vol. 38(1), pp. 521, 2014.
VerstraelenL. Curves and Surfaces of Finite Chen Type. In: Geometry and Topology of Submanifolds, III, pp. 304311, World Scientific, 1991.
WeiQ.JiaoC.GuoL.DingY.CaoJ.FengJ.DongX.MaoL.SunH.YuF.YangG.ShiP.RenG.FeiZ. Exploring Key Cellular Processes and Candidate Genes Regulating the Primary Thickening Growth of Moso Underground Shoots. New Phytologist, Vol. 214(1), pp. 8196, 2016. DOI: DOI:
WeilA.ChernS.S. Geometer and Friend. As cited in: S.-S. Chern, Selected Papers. Springer Verlag, 1978.
WilcoxH.J.MyersD.L. An Introduction to Lebesgue Integration and Fourier Series (reprint). Dover Publications, New York (NY), 1994.