Square Root of Natural Numbers as an infinite Staircase of Continued Fractions

When one lists the continued fractions of √ n across the natural numbers, a striking regularity appears: the coefficients rise in discrete steps, forming an arithmetical and visual staircase. This article reveals the algebraic and geometric underpinnings of that pattern. Each segment between consecutive perfect squares yields a constant pair of leading coefficients and a linearly increasing remainder, giving rise to a simple integer structure that unites the periodicity of quadratic irrationals with an illuminating global picture. We include a short proof of the fundamental relation and three complementary visualizations, along with a brief note on the golden ratio as a conceptual ”zeroth stair.