The Second International Symposium on Square Bamboos and the Geometree: A report

December 4 and 5, 2023

Introduction

This symposium is the second of a series [1].  The Second International Symposium on Square Bamboos and the Geometree (ISSBG 2023) was held online in a videoconference format on December 4 and 5, 2023. It is online only (Zoom), to greatly simplify the organization and allow everyone to participate at no cost. The aim is also to record the presentations and to open a video (youtube) channel, so an even broader audience can be reached. In any case, the proceedings are published as full Open Access, removing any possible barrier to the distribution of scientific results

ISSBG 2023 brought together botanists, biologists, technologists, physicists, applied mathematicians and geometers. Contributions spanned a wide range of subjects, but two common themes for this Symposium series are:

·       Cutting-edge research on form and shape in geometry and in the natural sciences

·       Transcending boundaries.

The name of the symposium symbolizes the close connection between geometry and the natural sciences.

Geometree comes from a wonderful book - The poetry of the Universe – by Robert Osserman [2]: “We may picture the product of three thousand years of geometric inventiveness in the form of a tree – the “Geometree” - whose roots go back even further and whose branches represent the outcome of centuries of discovery and creation. With or without application, the branches and fruits of this tree are worth contemplating as a remarkable product of human imagination. The Geometree is healthy, vigorous and in full foliage, older than any redwood, and fully as majestic”.  

Square bamboos opened the door to a uniform and unified mathematical description of natural shapes.  In 1993 superellipses were first used to describe the culm cross sections of square bamboos (Chimonobambusa species) and other square shapes in botany.  Starfish and many other natural shapes with different symmetries followed soon, through the generalization of Lamé curves to Gielis Transformations (aka Superformula), as a uniform description of natural shapes and phenomena [3].  

ISSBG, a great annual event

The first Symposium was organized in November 2022, and the Proceedings have been published recently. ISSBG 2022 was divided into three sessions: (1) Geometry; (2) Mathematics; (3) Applications in Biology and Technology. Contributions in Geometry deal with position vectors in submanifold theory, the construction of equilibrium surfaces with symmetry for anisotropic energy functions, Generalized Möbius-Listing bodies and geometric algebra using R-functions. Contributions in Mathematics deal with using nested analytic functions to compute Laplace Transforms, the stability of solutions in mixed differential equations and umbral calculus. Applications in technology involve computational optimization of antennas, applications of the superformula in CAD/CAM and technology, modeling of animal bones using superellipses, and the connection to complexity theory.

As the organizer of this Symposium, it is my goal to bring researchers from various fields together to encourage cooperation and collaboration. It is often simply the case that subjects are very similar, but the mathematical languages used are very different.  With various guests and speakers, I already have a long-term collaboration on various topics.  

DAY 1

Time Hybrids: a new generic model of reality

Prof. em. Van Oystaeyen (University of Antwerp) presented his new generic model of reality, which is based only on Time. After going from Space to Spacetime, only Time is left. With his background in algebraic geometry and non-commutative geometry, our current science can be considered as a shadow of non-commutative space.  In the paper Time-Ordered Momentary States of the Universe and a Dynamic Generic Model of Reality: Étale Over a Dynamic Non-Commutative Geometry [4], and his book, Time Hybrids [5,6], more details can be found.

Fred Van Oystaeyen was the first mathematician I contacted after finding the link between superellipses and square bamboos, around 1994. This led to a master thesis in informatics testing some of our hypotheses, to joint work with Bert Beirinckx (see proceedings [7]. It also resulted in a joint paper on the modeling of plants [8], which defined our/my research program in the past (almost) three decades.  From [8]: 

All plant shapes generated by one deterministic L-system are identical, so stochastic L-systems were introduced in order to produce more variation. The drawback is that, when a desired plant shape is given, one cannot find, in an algorithmic or otherwise mathematically precise way, the L-system that will generate it. This is true for fractals, supershapes, R-functions or any other model (and in all of science). A plant, however, is more than meets the eye. It is a complex organism, the result of the precise interplay between geometry, genes and enzymes, and many other substances and metabolites, growing in specific climatic and edaphic conditions, subject to many biotic and abiotic factors, prone to diseases and in need of pollinators, with limited or abundant availability of water, light, and nutrients. Its genetic constitution is the result of  millions of years of evolution at work and has incorporated all mechanisms necessary to develop and grow successfully.

Hence, our model needs to be much more than a virtual plant that looks nice. The model we need is that of an “abstract plant”, not a virtual plant, where certain model-parts are controlled by suitable parameters. Plants are continuously communicating with an ever-changing environment where macro and micro-effects interact and the parameters in the abstract plant enable us to measure the effect of these interactions, hence introducing parameters and choosing values of them with the aim of obtaining a virtual plant resembling a given specimen contradicts our aim. 

Our “abstract” plant in the study of the morphogenesis is a time sequence of adaptations to internal and external stresses acting on the plants. The model constructed should be generic, i.e. free of in-built constraints depending on results or theories contained in those to be tested in the actual research. An abstract tree will be different from an abstract cactus, but the mathematical model underlying both is the same.

In Exploration of Human Thought Patterns: Roots of Thoughts and Zeros in Mathematics

Matthew He (Nova Southeastern University, Florida, US) is the most recent mathematician with whom I started collaborating. In this collaboration, our focus is on the mind, a long-term project, no doubt. Understanding the true nature of human thoughts and their mathematical patterns will be essential and critical. To explore the mathematical structure of human thought, Ulf Grenander’s work “A Calculus of Ideas: A Mathematical Study of Human Thought” [9] serves as a guide, among others. Matthew He introduced the key concept of Thinking Kernels as the geometric structure of human thought. The Central Dogma of Human Thoughts connecting the internal black box of thinking kernels and the external open box of thinking expressions via the middle gray box of thinking flows, generates the connections between Neural Spikes with Zeros of mathematics via an integration of fuzzy logic and binary logic (Fuzzinary Logic). 

Geometric figures which appear after SS cutting of the GML bodies in the radial cross section

Ilia Tavkelidze (Tbilisi University, Georgia) pioneered the generalization of Möbius bands into Generalized Möbius-listing GML surfaces and bodies. Starting with hollow or filled prisms, the ends are then connected with or without a twist [10]. In the former case a classical toroidal body results, whereas with a twist either a toroidal body results or one with Möbius phenomenon. This generalization aims at two goals: first, to understand complex movements as a superposition of elementary movements (in the spirit of Gaspard Monge), and second, to solve boundary value problems PDE, since knowledge of the domain can greatly aid in understanding both qualitative and quantitative aspects of the PDE and boundary value problems.  

In this sense, cutting of GML has become a proper field of study and the general solution was found some years ago in a joint collaboration. However, the determination of the number and the exact shape of the cross sections after cutting remains an open question. In his presentation Geometric figures which appear after SS cutting of the GML bodies in the radial cross section, Ilia Tavkhelidze presented the regularities of cutting from side to side in GML’s.  It is a work in progress, since last year he presented the solution for Vertex to vertex cutting [11].

Helical Hypersurfaces in Four-Dimensional Minkowski Space

Twisted surfaces and bodies have attracted the interest of researchers in many fields. Professor Erhan Güler(Department of Mathematics Bartin University (Turkey) focuses on helicoidal, rotational, minimal, one-sided, algebraic (hyper)surfaces in space forms, and on elimination theory. In his presentation, he characterized helicoidal hypersurfaces in Minkowski four-space 𝔼^4_1 based on their axis of rotation, giving rise to three distinct types of helicoidal hypersurfaces. Expressions for their curvatures, including Gaussian and mean curvatures, are derived and he established a theorem that classifies helicoidal hypersurfaces with timelike axes, meeting the condition Δ𝐇=𝐴𝐇[12].

A general model for wildfire propagation with wind and slope

M.A. Javaloyes Victoria is professor at the Department of Mathematics, University of Murcia, Spain. I have visited Murcia University various times, but last time we met was in Eindhoven, in 2022 at a meeting on Finsler Geometry.  The presentations then were mainly very technical, but Miguel Angel proposed Finsler geometry for wildfire propagation with wind and slope. He presented joint work with his PhD students , on a geometric model for the computation of the forefront of a forest wildfire which takes into account several effects [13,14].  Fires travel faster uphill, and wind may be very time-dependent. The model relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework.  It will be interesting to test these models in real-world wildfires and I hope a collaboration will emerge with other groups, working in Greece to understand wildfires and there spread. Drones and AI will be very important tools.

THE SUPERFORMULA AND MODEL QUANTUM SYSTEMS AS TOOLS FOR LEARNING
Dave Thompson’s background is in chemical physics and theoretical chemistry, focusing on numerical methods for electronic structure calculations.  In 2022 I wrote an email to him after learning he had published a presentation on model quantum systems as tools for learning, including the superformula. A collaboration emerged and developed. Quantum theory is mainly based in isotropic methods, in which directions are all measured the same, with the Euclidean circle as the basic shape. The movement of particles in a supercircular confinement is less straightforward.

Teaching and understanding of quantum phenomena often begin with simple ‘particle-in-a-box’ style problems, already in the very early beginning of Quantum theory (by Nobel laureate Sir Neville Mott). The solutions to these problems introduce the student to foundational quantum concepts such as degeneracy and quantization. Through consideration of the shape element of the syntax, the superformula – a simple extension of the equation describing a circle – is introduced and discussed [15]. This collaboration is also expected to evolve into real cross-boundary collaborations, since there are several overlaps with other studies, e.g. the solution of BVP and sounds.

ELASTIC SURFACES WITH ELASTIC BOUNDARY

One of the first mathematicians to use the Superformula is Bennett Palmer (Idaho State University, USA). With Miyuki Koiso (Nara Women University, Japan), he used these shapes as examples to extend constant mean curvature CAMC surfaces to the anisotropic case [16]. CAMC surfaces occurs in the study of soap bubbles and films, and the anisotropic case extends this to anisotropic functionals, a.k.a. Wulff shapes (in crystallography). Here, in joint work with A. Pampano, the case of Elastic Surfaces with Elastic Boundary is presented, which finds its roots in the study of membrane and the Euler-Helfrich energy functional, which contains the squared L2 norm of the sum of the mean curvature H and the spontaneous curvature, coupled to the elastic energy of the boundary curve. This functional is used to model the morphology of lipid bilayers with pores. When the spontaneous curvature is non-negative, those cases are characterized where the infimum of this energy is finite for topological annuli and we find the minimizer in the cases which it exists [17, 18]. Results for topological discs are also given and the bifurcation of solutions of the Helfrich model were also discussed [19].

 Day 2

The generalized performance equation and the rotated Lorenz curve

My collaboration with Peijian Shi (Nanjing Forestry University) goes back to about 2016, after they published a great paper showing that superellipses are far superior to circles when modeling tree rings in softwoods. The work of Peijian and his team has been of immense value for me, since in total they have tested more than 40000 biological specimens, showing that, except for a few cases, superellipses and the superformula are an excellent tool. Software has been made available to a broad audience of researchers.

Two years ago, I introduced Gini coefficients and Lorenz equations to him. These are well-known in the domain of inequality research in the economy (for example in inequality in household income per country). They can also be used to quantify the inequality of leaf areas on trees and plants [20, 21] and fruits on melon plants [22].  Remarkably however, Peijian, with his background in the study of insects, could link Lorenz curves in economy and biology, with performance equations via a 135° rotation (which I have named the Shi Rotation).  From this a whole range of indicators could be understood as variations, which can also be used to assess the diversity of trees versus size in forests [23]. This opens many doors between separate areas of the natural and the human sciences.

Unusual exponential functions and applications to population dynamics models

Paolo Emilio Ricci, without doubt one of the nicest and friendliest mathematicians on our earth, uses his insights to couple exponential functions to population dynamics. Logistic functions use the classic exponential function but different types of exponential functions, such as the fractional exponential function or the Laguerre-type exponential functions can be used. An exponential function has the property of invariance concerning a given differential operator since it is an eigenfunction of that one. This works for fractional derivatives as well.  The Laguerre exponential function is invariant concerning the Laguerre derivative. With this function oscillations show up in logistic models in a natural way [24, 25].

Low factorial polynomials, difference equations, and discrete logistic processes

Giuseppe “Pino” Dattoli was at ENEA Researcher Research Center Frascati,  Italy, and has done seminal work high energy accelerators, free electron lasers, and applied mathematics. We met many years ago, and it is always amazing (not only to me) how he can unveil so many connections in maths.  

Originally the subject was low factorial polynomials and difference/differential equations [26], but the talk went further than that. With a very broad range of applications of logistic functions, there are different forms of functions and distributions that are recognized as generalized logistics.  Dr. Dattoli reported on criteria might be to infer the associated non-linear differential equations, useful to guess “hidden” evolution mechanisms. He analyzed different forms of logistic functions using simple methods to reconstruct the differential equation they satisfy. These examples also involved differential equations containing non-standard forms of derivative operators, like those of the Laguerre type [27].  He also promised he would present further results on our 2024 symposium.

Exponential analysis, sparse interpolation and superresolution

Annie Cuyt and I met many years ago, when I was a member of some IOF fund at the University of Antwerp. She and Wen Shin Lee founded the Expower group (Expower.eu)

In her presentation, she discussed how sparse interpolation in computer algebra and exponential analysis in digital signal processing can cross-fertilize and lead to new results. The Nyquist constraint has governed digital signal processing since the beginning of the 20th century. In the past two decades, this constraint was first broken with the use of randomly collected signal samples and later for use with uniform samples. The latter method closely relates to the original version of the exponential data fitting algorithm published in 1795 by the French mathematician de Prony, which is often cited in sparse interpolation research. Besides avoiding the Nyquist constraint, the new result in [28] also solves several remaining open problems in exponential analysis.  All of the above can be generalized in several ways, to the use of more functions besides the exponential on the one hand, and to the solution of multidimensional versions of this inverse problem on the other [29].

Pythagorean Submanifolds in Euclidean spaces

In 2004 I was introduced to many differential geometers, following my contact with Prof. L. Verstraelen [3]. Among them were Adela Mihai and Ion Mihai, professors of mathematics in Bucharest. A copy of my book on their desks led to inspiring their son Radu-Ion.  Adela and Ion have organized the conferences Riemannian Geometry and Application in recent years, in which I gladly participated.  But what caught my attention for this conference was an article on Pythagorean Submanifolds in Euclidean Space.  The a, b and c as sides of a rectangular triangle are replaced by the first, second and third fundamental forms (I, II, and III respectively) [30].  In the most recent joint work with M. E. Aydin and C. Ozgur [31], they introduced the notion of a Pythagorean submanifold isometrically immersed into a Euclidean space, based on the shape operator of the submanifold. Adela and her coworkers proved that any Pythagorean submanifold in an Euclidean space is isoparametric, the principal curvatures along any parallel normal vector field being given in terms of the Golden or conjugate Golden Ratios. Pure Mathematical Beauty. 

Audiovisual Applications of Supershape Polygons and Complementary N-gon Waves

I came into contact with Dominik Chapman, a sound engineer and musician about four years ago, and talked him into testing supershaped sounds.  To our surprise, the sounds were polyphonic [32]. More recently he constructed a supershape audio database, for which he used a shape database we developed twenty years ago.  One example was a composition by musician Rexleigh Bunyard, with only supershaped sounds. 

In his talk, Dominik describes supershape polygons as audiovisual oscillators. When complementary n-gon waves are used to (re-)construct supershape polygons a four-waveform oscillator is created. Approaches to such a music theory covering timbre, pitch, duration, and scales of waveforms of supershape oscillators, non-linear interpolation between vertices of supershape polygons, shuffled samples noise and other signal processing methods applied to supershapes and supershape polygons open up a space for further discussions and audiovisual explorations.  

The two last speakers of the conference deal with mathematical methods to deal with dispersive materials (soils, tissues, bones,…)

Electromagnetic Modeling for Ground Penetrating Radar

Diego Caratelli, a dedicated student of Paolo Emilio Ricci, besides his work in antennas, has a long-standing interest in electromagnetic modeling of tissues and soils, and Dispersive Dielectric materials in general. In his talk, he applies this to ground penetration radar (GPR) to detect utilities, land mines, and structural parts, or to prevent avalanches. Successful application of GPR requires accurate knowledge of the electric field distribution within the soil (amount and nature of water, chemical composition, physical properties, geometrical structure, and density).  Using various models (Cole-Cole. Cole-Davidson , Havriilliak-Negami, Raicu) with relaxation times and relative permittivity, and making use of fractional calculus, an FDTD formulation has been developed [32], strengthened by various numerical validations. This is closely related to the development of specific antennas [33, 34].

Fractional time derivative and Herglotz-Nevanlinna functions

In the Expower project, I met Yvonne Ou (professor of mathematics, University of Delaware), with her interesting work on a time-domain simulation for the application of poroelastic wave equations in biomedical and geological research [35]. The dispersive nature of a poroelastic material is encoded in the memory term of the relevant equations. The presence of the memory term poses a significant challenge in developing a time-domain numerical solver for the poroelastic equations. She showed how this challenge can be handled by the method of auxiliary variables as long as a high-precision approximation of the memory kernel function can be obtained. Utilizing the link between the Laplace transform of the memory kernel and the Herglotz-Nevanlinna functions [36, 37], she showed how to obtain a highly accurate pole-residue approximation, which makes it possible to replace the memory terms with sums of auxiliary variables that satisfy ordinary differential equations.

This ended the Symposium. In summary, our Second Symposium was, like the first, very inspiring. There are so many possibilities for cooperation, incredible.  The third Symposium will be organized in November-December 2024 timeframe, and various speakers have already confirmed.

References

[1] Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)https://www.athena-publishing.com/series/atmps/issbg-22

[2] Osserman, R. (2011). Poetry of the Universe. Anchor.

[3] 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.

[4] Van Oystaeyen, F. (2023) Time-Ordered Momentary States of the Universe and a Dynamic Generic Model of Reality: Étale Over a Dynamic Non-Commutative Geometry. Growth and Form. https://doi.org/10.55060/j.gandf.231208.001

[5] Van Oystaeyen, F. (2021) Time Hybrids: a new generic theory of reality. Nova Publishers  https://doi.org/10.52305/TOTN2173

[6] Gielis, J. (2023). Book review: Time hybrids: a new generic theory of reality. Symmetry, Culture and Science, 34(3), 347-351.

[7] Beirinckx, B. (2023) From Superellipses to Superformula and Technology. In: Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022), 123-142 (2023), doi.org/10.55060/s.atmps.231115.011

[8] Van Oystaeyen, F., Gielis, J., & Ceulemans, R. (1996). Mathematical aspects of real plant modelling. Scripta Botanica Belgica, 13, 7-27.

[9] Grenander, U. (2012) A Calculus of Ideas: A Mathematical Study of Human Thought” (in 2012).

[10] Gielis, J., & Tavkhelidze, I. (2023). A Note on generalized Möbius-Listing bodies. In Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022) (pp.31-39).

[11] Tavkhelidze, I. (2023) Geometric figures which appear after VV cutting of the GML bodies in the radial cross section. In: Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022),

[12] Güler E. Helical Hypersurfaces in Minkowski Geometry 𝔼^4_1. Symmetry. 2020; 12(8) : 1206. https://doi.org/10.3390/sym12081206

[13] Markvorsen, S., & Pendás-Recondo, E. (2023). Snell’s law revisited and generalized via Finsler geometry. International Journal of Geometric Methods in Modern Physics, 20(08), 2350138.

[14] Ángel Javaloyes, M., Pendás-Recondo, E., & Sánchez, M. (2023). A General Model for Wildfire Propagation with Wind and Slope. SIAM Journal on Applied Algebra and Geometry, 7(2), 414-439.

[15] Thompson, D., & Gielis, J. (2023) The Superformula and model quantum systems as tools for learning.https://chemrxiv.org/

[16] Koiso M., & Palmer, B. (2008). Rolling construction for anisotropic Delaunay surfaces. Pacific Journal of Mathematics, 234(2), 345-378.nett Pacific
[17] Palmer, B., Pámpano, Á. (2021) Minimizing configurations for elastic surface energies with elastic boundaries. Journal of Nonlinear Science 31.1 (2021): 1-36.

[18] Palmer, B., Pámpano, Á. (2022) The Euler Helfrich functional. Calculus of Variations and Partial Differential Equations 61, no. 3 (2022): 1-28.

[19] Palmer, B., Pámpano, Á. (2024) Symmetry breaking bifurcation of membranes with boundary." Nonlinear Analysis 238

[20] Huang, L.C., Ratkowsky, D.A., Hui, C., Gielis, J., Lian, M., Yao, W.H., Li, Q.Y., Zhang, L.Y., Shi, P.J., 2023. Inequality measure of leaf area distribution for a drought-tolerant landscape plant. Plants, 12, 3143.

[21] Lian, M., Shi P.J., Zhang, L.Y., Yao, W.H., Gielis, J., Niklas, K.J., 2023. A generalized performance equation and its application in measuring the Gini index of leaf size inequality. Trees − Structure and Function, 37:1555–1565.

[22] Wang, L., Hui, C., Ratkowsky, D.A., Yao, W.H., Lian, M., Wang, J.F., Shi, P.J. 2023, Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution in bamboo and muskmelon. Ecological Modelling, in preparation.

[23] Zhang, L., Quinn, B.K., Hui, C., Lian, M., Gielis, J., Gao, J., Shi, P.J., 2023. New indices to balance α-diversity against tree size inequality. Journal of Forestry Research, in press.

[24] Caratelli, D., Natalini, P., & Ricci, P. E. (2023). Examples of expansions in fractional powers, and applications. Symmetry, 15(9), 1702.

[25] Bretti, G, Ricci, P.E. (2007) Laguerre-type special functions and population dynamics, Applied mathematics and computation 187.1 (2007): 89

[26] Dattoli, G., & Garra, R. (2023). A note on differential equations of logistic type. arXiv preprint arXiv:2308.06135.

[27] Cuyt, A., Lee, W-S. (2018) Multivariate exponential analysis from the minimal number of samples. Adv. Comput. Math., 44:987. 

[28] Cuyt, A., Lee, W-S How to get high resolution results from sparse and coarsely sampled data. Appl. Comput. Harmon. Anal., 48:1066-1087.

[29] Aydin, M. E., & Mihai, A. (2020). A note on surfaces in space forms with Pythagorean fundamental forms. Mathematics, 8(3), 444.

[30] Mihai A; Aydin, M. E., Mihai, A., & Ozgur, C. (2023). Pythagorean Submanifolds in Euclidean Spaces. Results in Mathematics, 78(6), 211.

[31] Chapman and Gielis SCS CHAPMAN D., GIELIS J. (2021) Gielis transformations for the audiovisual database. Symmetry, Science and Culture. 

[32] Mescia, L., Bia, P., & Caratelli, D. (2022). FDTD-based electromagnetic modeling of dielectric materials with fractional dispersive response. Electronics, 11(10), 1588.

[33] Paraforou, V., Caratelli, D., & Tran, D. (2014, June). A novel low-profile SWB unidirectional supershaped antenna for advanced ground penetrating radar applications. In Proceedings of the 15th International Conference on Ground Penetrating Radar (pp. 828-832). IEEE.

[34] Loutridis, A., Kazıcı, S., Stukach, O. V., Mirmanov, A. B., & Caratelli, D. (2020). A novel class of super-elliptical Vivaldi antennas for ultra-wideband Applications. Advanced Radio Frequency Antennas for Modern Communication and Medical Systems.

[35] Ou, M. J. Y., Plecháč, P., & Xie, J. (2022). On the time-domain full waveform inversion for time-dissipative and dispersive poroelastic media. Applicable Analysis, 101(11), 3895-3911.

[36] Luger, A., & Ou, M. J. Y. (2022). On applications of Herglotz-Nevanlinna functions in material sciences, I: classical theory and applications of sum rules. In Research in Mathematics of Materials Science (pp. 433-459). Cham: Springer International Publishing.

[37] Ou, M. J. Y., & Luger, A. (2022). On Applications of Herglotz–Nevanlinna Functions in Material Sciences, II: Extended Applications and Generalized Theory. In Research in Mathematics of Materials Science (pp. 461-499). Cham: Springer International Publishing.