Abstract

Full paper prices: €32 / €24 / €60

Open Access price: €100

Symmetry: Culture and Science
Volume 35, Number 3, pages 237-240 (2024)
https://doi.org/10.26830/symmetry_2024_3_237

UNIVERSAL NATURAL SHAPES AND THEIR SYMMETRIES

Johan Gielis

University of Antwerp, Belgium
Email: johan.gielis@gmail.com

Abstract: As a generalization of circles and superellipses (or Lamé curves), the Superformula is a geometric transformation that unifies a wide range of natural and abstract shapes (Gielis, 2003). Various mathematicians rebranded and renamed the Superformula as Gielis Formula (or Gielis Transformations GT) giving rise to Gielis curves, surfaces and submanifolds (curves are embedded in 2 or 3 dimensions, surfaces and hypersurfaces are embedded in higher dimensions or in higher dimensional manifolds, hence the name submanifolds) (Verstraelen, 2004; Gielis et al., 2005; Mishra, 2006; Morales and Bobadilla, 2008; Arslan et al., 2009; Matsuura, 2015). It allows for unification and for coupling shape description to understanding why shapes form the way they do. The action performed by Gielis transformations on the equations of Lamé’s superellipses or superellipsoids, to arrive at Gielis curves and surfaces, are basically the same as the deformations of the Minkowskian theorem of Pythagoras which lead to the metrics of relativistic space-times. A direct link between flowers, starfish and many other living organisms, with spacetime itself is established. In analogy with the universe as a whole being an optimum of some simple equation many creatures living in this universe assume their shapes following similar principles (Gielis et al., 2005). Shapes that we observe in nature (from molecules to spacetimes) comply with natural curvature conditions, the simplest of which cause a projectile to follow a parabolic trajectory or a planet an elliptic one. The ultimate challenge is to extend these novel insights (a unifying geometrical approach) to living and non-living systems.

References:
Arslan, K., Bulca, B., Bayram, B., Ozturk, G., & Ugail, H. (2009, September). On spherical product surfaces in E3. In 2009 International Conference on CyberWorlds (pp. 132-137). DOI: 10.1109/CW.2009.64

Darvas, G. (2008) Symmetry, Birkhäuser, Basel.

De Tommasi, E., Gielis, J., & Rogato, A. (2017). Diatom frustule morphogenesis and function: a multidisciplinary survey. Marine genomics, 35, 1-18. DOI: 10.1016/j.margen.2017.07.001

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. DOI: 10.3732/ajb.90.3.333

Gielis, J. (2017). The Geometrical Beauty of Plants. Atlantis/Springer. DOI:10.2991/978-94-6239-151-2

Gielis, J., Haesen, S., & Verstraelen, L. (2005) Universal natural shapes: From the super eggs of Piet Hein to the cosmic egg of Georges Lemaître, Kragujevac Journal of Mathematics, (28), 57-68.

Gielis, J., Shi, P., & Caratelli, D. (2022) Universal equations–A fresh perspective, Growth and Form, 3(1-2), 27-44. DOI: 10.55060/j.gandf.220817.001

Kindratenko, V. V., Van Espen, P. J., Treiger, B. A., & Van Grieken, R. E. (1996) Characterisation of the shape of microparticles via fractal and Fourier analyses of scanning electron microscope images, Microchimica Acta, 1996, suppl. 13, pp. 355–361. DOI: 10.1007/978-3-7091-6555-3_28

Koiso, M., & Palmer, B. (2008) Equilibria for anisotropic surface energies and the Gielis formula, Forma (Japanese Society on Form): 23:1–8.

Libbrecht, K. G. (2005) The physics of snow crystals, Reports on progress in physics, 68(4), 855.

Matsuura, M. (2015) Gielis’ superformula and regular polygons. Journal of Geometry, 106(2), 1–21. DOI: 10.1007/s00022-015-0269-z

Mishra, S. K. (2006) On estimation of the parameters of Gielis Superformula from empirical data. Available at SSRN 905051. DOI: 10.2139/ssrn.905051

Morales, A. K., & Bobadilla, E. (2008, February) Clustering with an n-dimensional extension of Gielis superformula. In Proceedings of 7th WSEAS Int. Conf. on Artificial Intelligence, Knowledge Engineering and Data Bases (AIKED’08).

Shi, P., Ratkowsky, D. A., & Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry, 12(4), 645. DOI: 10.3390/sym12040645

Shi, P., Gielis, J., Quinn, B. K., Niklas, K. J., Ratkowsky, D. A., Schrader, J., ... & Niinemets, Ü. (2022) ‘biogeom’: An R package for simulating and fitting natural shapes (Vol. 1516, 1, pp. 123-134).

Verstraelen, L. (2004) Universal natural shapes, Journal for Mathematics and Informatics of the Serbian Mathematical Society, 40, 13-20.