1.Introduction 

For a long time, I have believed that fractal geometry and continued fractions are related in some fundamental way.

where n is a finite positive integer and does not tend to infinity.
Fractal geometry is recursion in geometry, and continued fractions are recursion in arithmetic;
the two share the same structural origin and are isomorphic in form.

Under infinite iteration, the area of a fractal tends to zero,
so the area is not directly equal to a continued fraction.
However, the similarity ratio that defines the essence of a fractal
is precisely expressed by a finite-order continued fraction.

This is a vivid manifestation of the unity of mathematics and physics:
Form is number, and number is form.

To this end, Fractal Continued-Fraction Geometry（FCFG）is established.

2 Core Definitions and Foundational Axiom of FCFG

2.1 Definition of Finite-Order Simple Continued Fractions

Given a positive integer sequence \{a_1,a_2,\dots,a_n\}, the n-th order finite continued fraction reads:

r_n=\frac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\dots+\dfrac{1}{a_n}}}}

where r_n\in\mathbb Q is a rational number and n\in\mathbb{N}^+. No infinite limiting process is required at this stage.

2.2 Definition of Finite n-th Order Self-Similar Fractals

An n-th order finite self-similar fractal F_n is generated recursively via n rounds of geometric scaling: an initial primitive shape undergoes successive proportional subdivision and nested duplication over n iterations to form a finite fractal skeleton. We define its global equivalent similarity ratio S_n, a characteristic constant governing the overall scaling transformation of F_n.

2.3 Foundational Axiom of FCFG (First Axiom of the Discipline)

Isomorphism Axiom: For any positive integer n, the value r_n of the n-th order continued fraction is identically equal to the global equivalent similarity ratio S_n of its associated n-th order finite self-similar fractal:

\boldsymbol{r_n=S_n}.

Corollary 1: A continued fraction serves as the algebraic-arithmetic representation of a self-similar fractal, while the self-similar fractal is the geometric realization of a continued fraction. Arithmetic recursion and geometric recursion share identical structural origin and isomorphic configuration.

Corollary 2: Within the finite-order framework, rational continued fractions correspond bijectively to finite self-similar fractals with rational similarity ratios.

2.4 Extended Definition for Infinite Limiting Case

As n\to\infty, if the sequence \{r_n\} converges to an irrational number \alpha, the corresponding sequence of finite fractals \{F_n\} converges to a complete infinite self-similar fractal F_\infty, with \alpha being its global limiting similarity ratio.

Remark: Fractal measures (length, area) degenerate under infinite iteration and hence do not match the numerical value of the continued fraction; nevertheless, the similarity ratio that determines intrinsic fractal topology is precisely quantified by the continued fraction at all times.