From Pythagoras to Fourier and From Geometry to Nature

DOI: https://doi.org/10.55060/b.p2fg2n.ch010.220215.013

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The Gielis formula as generalization of Lamé curves and superellipses uses trigonometric functions which are transcendental. These equations can be rewritten in terms of Chebyshev polynomials [41, 46], for *m* = 1 and *n*_{1,2,3} = *p* with *p* an integer, this describes supercircles and superellipses, and circles or ellipses when *m* = 1 and *n*_{1,2,3} = 2:

There exists also a direct connection between pseudo-Chebyshev functions and flowers. Using Gielis transformations on Grandi or Rhodonea curves, but with

Equation (10.2) with

It is remarkable that flowers and leaves are connected to these polynomials and functions. Their multiple uses in mathematics can serve as a guide in the study of botany. For the study of flowers one can make use of the orthogonality properties of pseudo-Chebyshev functions [6].

This directly leads to Fourier series solutions using pseudo-Chebyshev functions of the fourth kind. Consider for an *L*_{1}[*−π*, *π*], 2*π*-periodic function [6]:

Here *W _{n}*(

The flowers in Figure 27 can be understood as a single-term Grandi curve inscribed in a polygon defined by Equation (9.23). As a generalization of Fourier series (or trigonometric series in general) each individual term of a Fourier series can be transformed. One can define a generalized series, with *ρ*_{0}, *ρ _{k}* defining a stretchable radius, as [45]:

Since the first constant term of the series *ρ*_{0}*a*_{0} can be associated with a particular transformation *ρ*_{0}, any shape described by Equation (9.23) is described precisely in one term of this generalized series *ρ*_{0}*a*_{0} without any need for additional terms. Figure 28 shows a sum of three terms.

Equation (9.23) and the generalized Fourier series of Equation (10.8) can be used to generate waves. One example is the generation of square waves. A square wave may be generated in various ways, e.g. with reference to step functions such as the Heaviside step function (Equation 10.9). Note that the Dirac delta function is the distributional derivative of the Heaviside function [43].

An alternative method is synthesis via Fourier series. One well-known disadvantage of this 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

In order to generate a square wave which is differentiable everywhere, all exponents in Equation (9.24) are equal to 1, *m* = 4 and *A* is very large, so that the cosine term becomes very small, *ɛ*. In Figure 29, the shape of the sine wave is given for various values of *ɛ* [43].

As long as *ɛ* is finite and not zero, the function is differentiable everywhere. In this way, Gibbs phenomena are avoided and differentiability can be ensured everywhere. These curves can also be framed in a window, e.g. the interval [*−*1; 1] or in a Gaussian window *ɛ* (see Figure 30).

Second and higher order trigonometric functions based on Equation (9.23) can be generated. Given a shape *ρ*(*ϑ*) defined by Equation (9.23), the polar plot is generated by:

The functions *c*(*ϑ*) and *s*(*ϑ*) are displayed in Figure 31 with values *A* = 2, *B* = 1, *m*_{1} = 1.5, *m*_{2} = 0.5, *n*_{1} = 1, *n*_{2} = 2 and *n*_{3} = 3. This can be continued to any order.

Figure 32 displays rational Gielis curves with *m* a rational number (*m* = 4/5). Increasing the values of the *n*_{2}, *n*_{3} parameters lowers the magnitude of the main *q*-th harmonic partial (5th in this example) and raises the magnitudes of the other partials, as can be seen in the two harmonic spectra of the *X*-coordinates waveforms. The spectra of the *Y*-coordinates waveforms (not shown) have similar polyphonic spectral characteristics [27].

Equation (9.23) can be extended to any dimension. In 3D, both spherical coordinates (Equation 10.13) [9] and parametric representations (Equation 10.14) can be used:

The parametric equation (10.14) is based on two perpendicular cross sections *ρ*_{1}, *ρ*_{2} which can be of different sizes leading to toroidal structures (see Figures 33 & 34) [41]:

This can be further extended within the framework of Generalized Möbius-Listing surfaces and bodies with Equations (10.15) and (10.16) [49, 96, 97]. Both the basic line and the cross section can be defined by any of the curves discussed earlier. The lower index of the notation

The main advantage of such general coordinate systems, as also observed by Lamé, is that to each physical shape or phenomenon a best fitting coordinate system can be found. The goal of mathematical physics then is to solve the boundary value problems associated with the problem under investigation [41, 55].

Both the cross section and the basic line can be defined by Equation (9.24) and a time component can operate on cross section, basic line and twisting parameters for dynamical studies.