FCFG and Trigonometric Functions: Geometric Origins of Wave Motion and Harmonic Analysis

Author: Zhang Suhang, Luoyang School of Mathematics

Abstract

In FCFG and Euler's Formula, we argue that e^{i\pi}+1=0 forms an algebraic closed loop arising from the coupling of the intrinsic recursive structures of transcendental numbers. Nevertheless, this formula is only a static special case where the phase angle takes the specific value \pi. This paper further extends the FCFG framework to examine the general form of Euler's formula, e^{i\theta}=\cos\theta+i\sin\theta. From the FCFG perspective, projections of a rotational fractal manifold onto orthogonal coordinate axes naturally generate trigonometric functions. The isomorphism between continued-fraction recursion and analytic expansions of trigonometric functions (such as Lambert’s continued fraction) reveals that \sin and \cos serve as “coordinate anatomical diagrams” of the rotational fractal manifold within orthogonal coordinate systems. Accordingly, FCFG expands from static constant geometry into the domain of dynamic wave motion and harmonic analysis, offering a new geometric interpretive perspective for dynamical systems and signal evolution.

Keywords: FCFG; trigonometric functions; Euler's formula; continued fractions; harmonic analysis; rotational manifold

I. Introduction: From Static Closed Loop to Dynamic Motion

Within the FCFG framework, e corresponds to a spiral fractal with scale stretching, and \pi corresponds to a manifold fractal of pure rotation. In FCFG and Euler's Formula, we observe that the two are orthogonally coupled via the imaginary unit i, eventually converging to the static closed loop e^{i\pi}+1=0 on the unit circle.

When the phase \theta is no longer fixed at \pi but treated as a continuously varying variable, e^{i\theta} transforms from static rotation into dynamic periodic motion. This naturally leads to the general form of Euler's formula:

e^{i\theta} = \cos\theta + i\sin\theta

In traditional complex analysis, trigonometric functions are defined as coordinates on the unit circle. The question then arises: what are trigonometric functions from the geometric picture of FCFG?

The task of this paper is to identify the geometric entity of trigonometric functions within the FCFG framework, and thereby open the pathway connecting FCFG to harmonic analysis and dynamics.

II. The FCFG Perspective: The Projection Nature of Trigonometric Functions

In FCFG topology, the \pi-type pure rotational manifold presents a continuous rotational field (such as circles and tori) in an infinite-origin space. Projecting this continuous rotational manifold onto the complex plane spanned by orthogonal real and imaginary axes yields two orthogonal components, namely \cos\theta and \sin\theta.

We thus arrive at a geometric definition:

Trigonometric functions are the two projection components of a rotational fractal manifold onto orthogonal coordinate axes.

- \cos\theta: projection component of the rotational manifold on the real axis;

- \sin\theta: projection component of the rotational manifold on the imaginary axis.

The core merit of this interpretation is that it reduces abstract algebraic functions to intuitive geometric structures within the FCFG system. The overall rotation corresponds to e^{i\theta}, while \cos\theta and \sin\theta are the structured decomposition results of the rotational fractal manifold performed by orthogonal coordinate systems.

III. Deep Confirmation via Continued-Fraction Recursion: Lambert’s Continued Fraction

The foundational core of FCFG is continued-fraction recursive structure. If trigonometric functions are essentially orthogonal projections of rotational fractals, their analytic expansions must carry recursive properties. Lambert’s continued fraction perfectly supports this reasoning.

The standard continued-fraction expansion for tangent:

\tan x = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - \cdots}}}}

This expansion acts as a key supporting evidence for FCFG theory:

1. It possesses strict infinite arithmetic recursion, consistent with the core mechanism of FCFG;

2. It describes the ratio of components for a rotation angle, fully matching the geometric property of the \pi-type rotational manifold;

3. It establishes a direct isomorphism between analytic forms of trigonometric functions and continued-fraction recursion.

From the FCFG perspective, Lambert’s continued fraction is not merely an analytic formula. It is an arithmetic recursive trace retained during the orthogonal projection of the rotational fractal manifold, suggesting that trigonometric functions may be interpreted within this continued-fraction fractal geometric system.

IV. FCFG Picture of Dynamical Systems and Wave Motion

The fundamental simple harmonic vibration equation in dynamics:

x(t) = A \cos(\omega t + \phi)

Within the FCFG framework, this equation represents a composite system of two types of transcendental fractal structures:

- Amplitude A: corresponds to the scale-stretching property of e-type fractals, characterizing scaling evolution of geometric structures;

- Frequency \omega and phase \phi: correspond to the \pi-type pure rotational manifold, characterizing periodic evolution over time.

Trigonometric functions undertake the critical coupling role, enabling superposition, interference and resonance between e-type scale evolution and \pi-type periodic rotation, forming the underlying geometry of macroscopic wave phenomena.

Extending to the core system of harmonic analysis: Fourier series decompose arbitrary periodic signals into superpositions of sine and cosine functions. In the FCFG view, Fourier decomposition essentially breaks down complex dynamical systems into linear superpositions of multi-frequency, multi-order projected components of \pi-type rotational fractals, establishing underlying geometric logic for harmonic analysis.

V. Implications and Outlook

1. Connecting FCFG with harmonic analysis

By clarifying trigonometric functions as orthogonal projections of rotational fractals and Fourier decomposition as superposition of fractal projections, FCFG gains access to core research fields including signal analysis and wave theory.

2. Linking dynamical systems and chaos research

Evolution of dynamical systems can be defined as parametric coupling between e-type stretching and \pi-type rotation. Continued-fraction recursive features may be used to characterize fractal dimensions, evolutionary patterns and chaos thresholds of systems.

3. Completing the theoretical landscape of FCFG

A complete logical chain is formed, covering geometric entities of transcendental numbers, static coupling in Euler's formula, and dynamic projection of trigonometric functions. FCFG thus achieves a systematic upgrade from constant geometry to dynamic analysis.

Conclusion

Euler's formula presents an elegant static algebraic relation between exponentiation and rotation; trigonometric functions constitute the full expansion of this relation under dynamic variation. Starting from continued-fraction recursion, FCFG depicts the rotational fractal picture of e and \pi, and clarifies that trigonometric functions are projections of the rotational fractal manifold onto orthogonal coordinate axes. This perspective not only enriches our understanding of fundamental transcendental numbers, but also marks FCFG’s entry into the broad domain of wave motion, dynamics and harmonic analysis.