A Quantized Geometric Model of the Discrete Unit Circle and Its Rational Geometric Constants

Abstract

This work develops a discrete geometric model for the construction and measurement of a unit circle in a quantized metric space that rejects the continuity axiom of the real line, without the infinitesimal tools used to define $\pi$. Numbers are assumed to possess finite spatial extension: each value is an areal unit, a multiple of a reference sample square, whose shape adapts to the discrete domain while preserving its magnitude; these units form a two-dimensional numerical staircase in place of the real line. The circle develops through two coupled discrete progressions, linear growth of the radius and quadratic growth of the area, transferred onto the unit circle as a fan of sectors opening from the vertical axis. Rectifying the staircase with a $45^\circ$ cutting line through the risers' midpoints isolates right trapezoids equivalent to these sectors; when the bounding segments are proportional to the radius subdivisions through an invariant scale factor $\gamma$, their accumulation forms an isosceles right triangle with the circle's area, while for a unit radius the boundary length matches, numerically, the area of a square of side $\gamma$. Together with a confinement condition borrowed from classical Euclidean geometry, external to the quantization axioms, the construction yields the rational area constant $k=\frac{25}{8}$, the area of the unit circle in the quantized metric, and the scale factor $\gamma=\frac{5}{2}$. Since every area of this metric is rational by construction, $k$ cannot be $\pi$: the discrete circle is a distinct object, not a measurement of the Euclidean one. The framework echoes Kronecker's finitist view of numerical structures.