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FCFG and Euler's Formula

Zhang Suhang

Luoyang School of Mathematics

Abstract

Euler's formula, e^{i\pi}+1=0, is hailed as an exquisitely beautiful equation in classical mathematics, establishing a connection between exponential operations and planar rotation. Within the framework of complex analysis, it presents a static relationship after the operation is completed. Fractal-Continued-Fraction Geometry (FCFG), on the other hand, offers an alternative perspective. Starting from the recursive structure of continued fractions, it examines the fractal and topological forms corresponding to e and \pi themselves, in order to understand the respective dynamical characteristics of these two transcendental numbers. By combining the two, one can study the operational relationships of complex analysis and the recursive geometric landscape of numbers side by side.

I. Euler's Formula: The Static Connection between Exponentiation and Rotation

Euler's formula brings together five fundamental constants—e, \pi, i, 0, 1—revealing that exponential operations can equivalently describe rotation in the complex plane.

In traditional complex analysis:

· e is used to characterize the base of continuous growth evolution;
· \pi represents the phase scale of circular rotation;
· i establishes the correspondence between algebraic operations and rotational geometry.

However, this system focuses on describing how numbers operate with one another, and does not explore a question: do the transcendental numbers e and \pi themselves possess distinguishable internal recursive structures, and corresponding geometric forms? In conventional understanding, they are merely numerical values substituted into formulas, carrying no inherent geometric shape of their own.

II. FCFG: The Fractal Landscape Corresponding to Continued Fractions of Transcendental Numbers

FCFG relies on the recursive structure of continued fractions to construct corresponding geometric images for transcendental numbers.

The Geometric Landscape of \pi (FCFG Perspective)

The continued fraction expansion of \pi is infinite and non-periodic, with no simple recurrence pattern yet discovered. Under the FCFG framework, it corresponds to a pure rotational fractal manifold on an infinite-origin space. Its characteristics: it maintains scale invariance, with sustained rotation as its primary behavior, and carries no accompanying scale dilation effect. This echoes the properties of circular geometry and phase rotation.

The Geometric Landscape of e (FCFG Perspective)

The continued fraction expansion of e is likewise infinite and non-periodic, but possesses a clear recursive pattern:

e=[2;1,2,1,1,4,1,1,6,1,1,8\cdots]

FCFG maps it to a rotational fractal manifold on an infinite-origin space that carries a scale dilation effect. Its characteristics: in addition to rotational behavior, it superimposes an exponential-type scale evolution, corresponding to the dynamical behavior of continuous growth or decay.

III. Contrast: The Complementary Perspectives of Euler's Formula and FCFG

What Euler's formula demonstrates is the algebraic result obtained after the exponential evolution of e and the rotational phase of \pi are coupled through the imaginary unit. FCFG focuses on the state before this coupling: the inherent recursive structures of e and \pi, and the two distinct types of rotational fractals generated thereby.

The complementary relationship between the two can be understood as follows: Euler's formula characterizes the algebraic result after the interaction of the two; FCFG characterizes the inherent geometric structure each transcendental number carries. e possesses fractal characteristics with scale evolution; \pi possesses fractal characteristics of pure rotation. Through the imaginary unit, orthogonal coupling is achieved, ultimately yielding the closed-loop relationship described by Euler's formula.

This is not an attempt to use FCFG to prove Euler's formula. Euler's formula already has rigorous mathematical proof. FCFG merely provides a new avenue for geometric interpretation, visualizing the inherent recursive structures of the two transcendental numbers.

IV. Research Significance

1. Distinguishing the Dynamical Characteristics of Two Classes of Transcendental Numbers
In Euler's formula, e and \pi are merely two constants, and no difference in their intrinsic nature can be seen. With FCFG, a clear distinction can be made: one class leans toward pure rotation, while the other is rotation superimposed with scale evolution.
2. Building a Comparative Channel between Number Theory, Fractal Geometry, and Complex Analysis
Continued fractions belong to number theory, self-similar structures belong to fractal geometry, and complex plane rotation belongs to complex analysis. FCFG places them within the same framework for comparative study, providing a description combining number and form for the study of ordered and quasi-periodic dynamical systems.
3. Providing Geometric Images for Continuous Dynamical Systems
Phenomena such as waves, rotation, and continuous evolution can be viewed as formed by different combinations of e-type and \pi-type fractal structures.

Conclusion

Euler's formula provides a concise and beautiful algebraic relationship between exponentiation and rotation; FCFG, starting from continued fraction recursion, depicts the rotational fractal landscapes corresponding to e and \pi respectively. Placed together, the two perspectives complement each other, enriching our understanding of these two fundamental transcendental numbers.