| Previous abstract | Back to issue content | Next abstract |
Volume 36, Number 4, pages 417-448 (2025)
https://doi.org/10.26830/symmetry_2025_4_417
FROM EUCLID TO EUCLID-PLUS
Johan Gielis
Geniaal BV, Antwerp, Belgium
Department of Biosciences Engineering, University of Antwerp, Belgium
Email: johan.gielis@gmail.com
ORCID: 0000-0002-4436-3839
Abstract: Building on our work with generalized conic sections, with the Superformula, we show how this can lead to a generalization of Euclidean geometry when flexibility (Ultraflex) is considered instead of rigidity as encoded in the geometry of Euclid and Descartes. By considering points as ultra-extensible primitives, the most important topics in Euclid, namely the Pythagorean theorem and commensurability (Books V and X), are generalized in a purely geometric way. In Section 1, the Superformula and its stretchable radii are used as a concept to illustrate Ultraflex geometry. Section 2 deals with shapes defined by the Superformula, and what happens when shapes are shrunk to zero (the classical points). It is shown how the information about the shape is preserved even when the size is reduced to zero. The strategy of sections 1 and 2 allows us to look at classical results in mathematics, logic and physics from a different perspective. Section 3 discusses how this different perspective also leads us to rethink notions such as complexity and classical paradoxes. In Section 4, arguments are made for studying nature with tools other than circles, thus providing an answer to one of the deepest questions in the sciences. In the final section, we discuss a possible path forward, with a minimal set of rules.
Keywords: Superformula, flexibility, commensurability, natural shapes.
References:
Arnold, V.I., Oleinik, O. (1979). Topology of real algebraic manifolds. Moscow Bulletin,34, 9–17.
Caratelli, D., et al. (2009) The Dirichlet problem for the Laplace equation in a starlike domain. Lecture Notes TICMI, Tbilisi, Georgia, 10, 34-49.
Chapman, D., Bunyard, R., & Gielis, J. (2024) Pitch and Timbre of Supershape Oscillators. Symmetry: Culture & Science, 34(3). https://doi.org/10.26830/symmetry_2024_3_299
Chen BY (1984) Total mean curvature and submanifolds and finite type curves. Series in Pure Mathematics. https://doi.org/10.1142/0065
Chern Foreword to Handbook of Differential Geometry, Volume I (eds. Dillen and Verstraelen), North-Holland
Feynman, R. P. (1963) Feynman lectures. CRC Press.
Gallison P. (2003) Einstein’s Clocks, Poincaré’s maps: Empires of Time. W.W. Norton & Company, New York.
Gielis, J. (2003) A generic geometric transformation that unifies a wide range of natural and abstract shapes. American journal of botany, 90(3), 333-338. https://doi.org/10.3732/ajb.90.3.333
Gielis, J. De uitvinding van de Cirkel (Dutch, 2001) Inventing the Circle (English, 2003). Geniaal Publishing, Antwerp, 2001.
Gielis, J., Haesen, S., & Verstraelen, L. (2004) Universal natural shapes: From the super eggs of Piet Hein to the cosmic egg of Georges Lemaître. Kragujevac Journal of Mathematics (28), 47-68.
Gielis, J., Caratelli, D., Fougerolle, Y., Ricci, P. E., Tavkelidze, I., & Gerats, T. (2012). Universal natural shapes: From unifying shape description to simple methods for shape analysis and boundary value problems. https://doi.org/10.1371/journal.pone.0029324
Gielis, J., Natalini, P., & Ricci, P. E. (2017) A note about generalized forms of the Gielis formula. In Modeling in Mathematics: Proceedings of the Second Tbilisi-Salerno Workshop on Modeling in Mathematics (pp. 107-116), Atlantis Press. https://doi.org/10.2991/978-94-6239-261-8_8
Gielis, J. (2017) The geometrical beauty of plants. Atlantis/ Springer, 2017. https://doi.org/10.2991/978-94-6239-151-2
Gielis, J., Caratelli, D., & Tavkhelidze, I. (2024). Rational Science, Unique and Simple. Proceedings of the ISSBG 2023. Geniaal Press, Antwerpen, Belgium
Gielis, J. (2025) A Point-Theory of Morphogenesis. Mathematics 13(19), 3076. https://doi.org/10.3390/math13193076
Hilger, S. (1990). Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics, 18(1), 18-46. https://doi.org/10.1007/BF03323153
Huang, W., et al. (2020) A superellipse with deformation and its application in describing the cross-sectional shapes of a square bamboo. Symmetry, 12(12), 2073. https://doi.org/10.3390/sym12122073
Huang, W., et al. (2024a) Ellipse or superellipse for tree-ring geometries? Evidence from six conifer species. Trees, 1-11; 2024. https://doi.org/10.1007/s00468-024-02561-2
Huang, W., et al. (2024b) Superellipse Equation Describing the Geometries of Abies alba Tree Rings. Plants, 13(24), 3487, 2024. https://doi.org/10.3390/plants13243487
Javaloyes, M. Á., Pendás-Recondo, E., & Sánchez, M. (2024). Gielis superformula and wildfire models. Proceedings of the ISSBG 2023. Geniaal Press, Antwerpen, Belgium (arXiv preprint arXiv:2406.06831).
Lamé, G. (1818). Examen des différentes méthodes employées pour résoudre les problèmes de géométrie. Mme. Ve. Courcier, imprimeur-libraire.
Manin Y.I. (2019) Time and Periodicity from Ptolomy to Schrödinger: Paradigm shifts versus continuity in the history of Mathematics. Geometry in history, 129-138. https://doi.org/10.1007/978-3-030-13609-3_3
Natalini, P., Patrizi, R., & Ricci, P. E. (2008) The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface. Numerical Algorithms, 49, 299-313, 2008. https://doi.org/10.1007/s11075-008-9201-z
Özdemir, Z., & Parlak, E. (2025). Superquadric Motion and Superquadric Hyperbolic Split Quaternion Algebra Via Gielis Formula. Mathematical Methods in the Applied Sciences, 48(16), 15035-15048. https://doi.org/10.1002/mma.11229
Parlak, E., & Özdemir, Z. (2025). Superelliptic motion and superelliptic quaternions via Gielis formula. Physica Scripta, 100(9), 095212. https://doi.org/10.1088/1402-4896/adffbc
Penrose, R. (2010) Cycles of Time. An extraordinary new view of the universe. The Bodley Head, London
Plato, VIIth Letter. http://classics.mit.edu/Plato/seventh_letter.html
Ricci, P. E., & Gielis, J. (2022). From Pythagoras to Fourier and from geometry to nature. Athena Publishing, ISBN 978-90-832323-1-7
Russell, B. (1897) An essay on the foundations of geometry. Routledge 1996, originally published by Cambridge: At the University Press
Shi, P., Ratkowsky, D. A., & Gielis, J. (2020). The generalized Gielis geometric equation and its application. Symmetry, 12(4), 644. https://doi.org/10.3390/sym12040645
Shi, P. J., et al. (2014) Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape. Frontiers in plant science, 6, 846. https://doi.org/10.3389/fpls.2015.00856
Sloterdijk P. Sferen. Boom (Dutch edition of Sphären I -Blasen-Microsphärologie and Sphären II -Globen-Macrosphärologie.
Spíchal, L. (2020). Superelipsa a superformule. Matematika–fyzika–informatika, 29(1), 44-69.
Stevin, S. (1584) De Thiende.
Tavkhelidze, I. (2000). On the Some Properties of One Class of Geometrical Figures. TICMI, 4, 41–44.
Tavkhelidze, I., Ricci, P.E. (2006). Classification of a wide set of Geometric figures, surfaces and lines (Trajectories), Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica e Applicazioni, 124o, vol. XXX, fasc. 1,. 191–212.
Thom, R. (2018) Structural Stability and Morphogenesis. CRC Press, 2018 edition
Verstraelen, L. (2014) A concise mini history of geometry. Kragujevac Journal of Mathematics, 38(1), 4-21. https://doi.org/10.5937/KgJMath1401005V
Verstraelen L (1991) Curves and surfaces of finite Chen type. Geometry and Topology of Submanifolds III, World Scientific, Singapore, 304–311. https://doi.org/10.1142/9789814540124
Wei, Q. et al. (2017). Exploring key cellular processes and candidate genes regulating the primary thickening growth of Moso underground shoots. New Phytologist 214: 81–96. https://doi.org/10.1111/nph.14284
Yajima T. and Nagahama H. (2009) Finsler geometry of seismic ray path in anisotropic media. Proc. R. Soc. A 469, 1763–1777. https://doi.org/10.1098/rspa.2008.0453
| Previous abstract | Back to issue content | Next abstract |