The common descent of biological shape description

Springer Proceedings in Mathematics & Statistics; Conference: Differential and Difference Equations with Applications: ICDDEA, Amadora, Portugal, June 2017, in the press

The common descent of biological shape description and special functions Gielis J*, **., Caratelli D**,°., Moreno de Jong van Coevorden C. **, Ricci P.E°°.

*

University of Antwerp, Groenenborgerlaan 171, B2020 - Antwerp, Belgium

**

The Antenna Company, High Tech Campus, 5656 AE - Eindhoven, the Netherlands ° Tomsk Polytechnic University, 84/3 Sovetskaya Street, 634050 - Tomsk, Russia °° International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 - Roma, Italia

ABSTRACT Gielis transformations, with their origin in botany, are used to define square waves and trigonometric functions of higher order. They are rewritten in terms of Chebyshev polynomials. The origin of both, a uniform descriptor and the origin of orthogonal polynomials, can be traced back to a letter of Guido Grandi to Leibniz in 1713 on the mathematical description of the shape of flowers. In this way geometrical description and analytical tools are seamlessly combined.

KEYWORDS 54C56, 57N25, 92C80, 33C45



1

Gielis transformations Gielis transformations [1] are geometric transformations acting on planar functions 𝑓 𝜗 unifying a wide range of natural and abstract shapes (Equation 1). Since its discovery two decades ago and the initial publications in 2001-2005 [1-3] they have been used in mathematics, biology and various fields of technology. Gielis transformations can morph a classic Euclidean circle or sphere, into an infinite number of shapes, including regular polygons, providing a designated unit circle or unit sphere. For example, it became possible to derive analytic solutions to a wide class of boundary value problems, using Fourier’s classical methods. Heat distribution solved by Fourier on a circular plate, has now been extended Laplace, Helmholtz, wave and heat equations for 2D and 3D domains, including annuli and shells [4-8]. They can be extended in 3 or more dimensions [2; 9]. also in relation to Generalized Möbius-Listing surfaces and bodies [10]. 𝜅 𝜗; 𝑎, 𝑏, 𝑚, 𝑛! , 𝑛! , 𝑛! = 𝑓 𝜗 .

! !

cos

!! !

𝜗

!!

±

! !

sin

!! !

𝜗

! ! ! ! !!

(1)

Following a generalization of constant mean curvature surfaces for anisotropic energy functionals [11;12], snowflakes and flowers can now be studied as minimal surfaces in the same way as soap bubbles and soap films are minimal surfaces for a given energy functional [13]. In general natural shapes can be described in a uniform way and studied via natural curvature conditions [2;13;14]. In biology, it has been used to drastically improve modelling of annual rings in trees [15], leaf shapes [16] and diatoms [17], to model human skin as dielectric materials [18; 19], or to model the backbone of RNA [20]. It was used to study the mechanical efficiency and stability of petioles [21; 22], or for biomechanical studies of knees and multi-dynamics mechanical systems in the body [23]. Vision algorithms developed with Gielis transformation allows for scanning objects or signals and for efficient compression algorithms, medical imaging and datamining [24;25]. This has led to research in biomedical imaging of blood cells, heart, skulls and various organs [26;27]. These algorithms are the first that can recognize selfintersecting curves or the symmetries of polygons and polygrams without prior learning algorithms encoded in the computer [24]. Intersecting curves are found in biomolecules, molecules, ropes and knots and a variety of other natural shapes. In the field of nanotechnology alone Gielis transformations are used in at least 25 papers. For example, to compute the optimal shape of nanoparticles for applications in solar panels, cancer treatments with thermal methods, nano-antennas, in-body telemetry or to shrink sizes of chips in electronics [28-30]. In engineering they have been used to optimize the shapes of wind turbines [31], heat shields in manned space vehicles [32;33], non-circular gears and gear tooth profiles [34;35], or the shape of non-planar wings in aircraft [36]. In the field of telecommunications they has been used to design waveguides and antennas [37-41], whereby design can also be based on botanical shapes [42;43], and to optimize lasers [44;45]. All these developments originate in the study of plants [46].



2

In its original form it has six parameters, but to quantify bamboo leaves or tree rings, the optimization of nanoparticles or in the development of antennas, two or three parameters suffice for size and shape.

Figure 1: a-d: cross sections of plant stems; e-f: starfish; g-i: transformations of logarithmic (gh) and Archimedean spirals (i); j-l: transformations of cosines, as flowers or in wave view [1]

The origin of Gielis Transformations: botany The origin of Gielis transformations is the study of square stems in plants using Lamé curves [1]. To extend this to other symmetries, inspiration was found in D’Arcy Thompson’s On Growth and Form [47] showing the analogy between certain flowers, and Rhodonea curves, the oldest and most useful mathematical representation of flowers. Rhodonea curves were discovered by Guido Grandi and communicated to Leibniz in a letter [48; 49]. The observation that in Rhodonea or Grandi curves 𝜚 𝜗 = cos 𝑚𝜗 or 𝜚 𝜗 = sin 𝑚𝜗 the argument of the angle specified the frequency, was applied to Lamé superellipses

! !

!

+

! !

!

= 1 with n a positive

integer [50], in particular to the polar representation of the Lamé curves or superellipses defined by 𝜚 𝜗 =

! !

! ! ! ! !"# ! ! !!"# ! !

. The pivotal step to Equation 1

was rewriting Lamé curves in polar coordinates and generalizing the symmetry from 4 to any real number. Figure 1 displays the result of transformation on some simple function. In transforming the circle 𝑓 𝜗 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, also regular polygons can result (Equation 2), or self-intersecting shapes, for m a rational number (Figure 2). Such shapes can be found in plant phyllotaxis [1; 46], the symmetry of DNA in planar view [51], or as separation zones in phase spaces with non-linear resonances [52].



3

m=4

m=7

m=5

m=6

m=8

m=9

Figure 2: Regular polygons for 𝑚 =4, 5 and 6; !

!

!

!

Self-intersecting polygons for 𝑚 = ;

m=10

m=11

m=12

𝜚 𝜗 = lim!! →!

!"!

! ! !

!

; ; !

!

!(!!!! !"#! !"#

! ) !

! !"#

! ! !

!(!!!! !"#! !"#

! ) !

!

!!

(2)

In Equation 1 the function 𝑓 𝜗 can be regarded as the developing function DF, the function that want to grow or develop. The second part of Eq. 1, also known as Gielis formula, denotes the constraining function CF, constraining the development of DF. When 𝑓 𝜗 = 𝛿 𝜗; 𝑚, 𝑛! = cos

! !

!

𝜗

!!

, many natural flowers shapes result

(Figure 3; [53]).

𝒏𝟏 2 1 1 3 3 𝒏 1 1 1 3 1 𝟐,𝟑 2 𝒏 4 4 20 5 5 2 𝟒 4Figure 3 a-e: Choripetalous five-petalled flowers with the corresponding constraining , superpolygons and parameters 3

Coordinate functions of first and higher order, and square waves The flower shapes and the wave-like shapes in Figure 1, lower row, and the flowers in Figure 3, defined by Eq.1 are essentially the trigonometric functions associated with the shapes. Generalized trigonometric functions been defined beyond circular functions [54-58]. For Lamé curves with exponent p for example, the half perimeter is defined as 𝜋! . For the Euclidean circle 𝜋!!! = 𝜋. The corresponding trigonometric functions of Lamé curves are Theorem

!cos 𝜗

!

+

!sin 𝜗

!

!cos 𝜗

and

!sin 𝜗 ,

with a Generalized Pythagorean

= 1 [55, 56]. Likewise, the coordinate functions of

shapes defined by Eq.1, are cosine and sine moderated by Eq.1.



4

With Eq.1 we can modulate these or trigonometric functions. One example is the generation of square waves. A square wave may be generated in various ways, e.g. with reference to step functions, e.g. the Heaviside step function (Eq.3). Note that the Dirac delta function is the derivative of the Heaviside function. 𝐻 𝑥 =

0, 𝑥 < 0 1, 𝑥 ≥ 0

(3)

An alternative method is synthesis via Fourier series. One well-known disadvantage is the Fourier-Gibbs phenomenon, whereby oscillations occur in points of measure zero. These phenomena are an inherent feature of the method, but may be mediated in practice by using 𝑠𝑖𝑛𝑐 𝑥 = !

𝑓 𝑥 = 𝑎! + !

!"# !"

(Eq.4):

!" !!! !!!

𝑠𝑖𝑛𝑐 !! 𝑎! cos

!"# !

+ 𝑏! sin

!"# !

(4)

Using Eq.1 the ratio of the sine function and the absolute value of the sine can be taken. This is a special case of Eq.1. In order to generate a square wave which is differentiable everywhere, all exponents in Eq.1 are equal to 1, 𝑚 = 4, and A very large, so that the cosine term becomes very small, 𝜀 (Eq.5). In Figure 4 the shape of the sine wave is given for various values of 𝜀. As long as 𝜀 is finite and not zero, the function is differentiable everywhere. In this way Gibbs phenomena are avoided and differentiability can be ensured everywhere. sin 𝜗 𝜀 + sin 𝜗

(5)

1.0

1.0

0.5

2

0.5

4

6

8

-0.5



-1.0

2.5



3.0



Figure 4: Eq.5 Sines for varying 𝜀 = 10!! with 𝛼 = 0 green; 𝛼 = 1 blue, 𝛼 = 3 red solid; 𝛼 = 5 orange solid and 𝜀=0 in black dashed -0.5



5 -1.0

3.5

0.10

0.10 0.05

0.05

5

-15

-10

5

-5

10

10

15

20

25

30

15

-0.05

-0.05

-0.10

-0.10





Figure 5: Cosines for 𝜀 = 10!! in Gaussian window with n=2 in (6) (left). Decaying square wave with n=1 in (6) (right)

These curves can also be framed in a window, e.g. the interval [-1;1] or in a Gaussian window

! ! ! ! !

!!! !!

!

(6) for various values of 𝜀 in (5) (Figure 5). The Haar wavelet

(Daubechies 1 wavelet) and various other step functions classically based on distributions, can be defined by Eq.5, using the appropriate window and shifts. !

1 𝜓 𝑡 =

−1 0

! !

0≤𝑡≤! ≤𝑡