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.