The Velocity-Dependent Geometric Ratio: The L1/L2 Conversion Factor Under Lorentz Contraction

A circle of diameter d sits inside a square of side d. The circle's area is πd²/4. The square's area is d². The ratio, circle to bounding square, is π/4. This ratio is called β. β is not a new constant. It is π/4, known since antiquity. What is recent is the recognition that β is a metric conversion factor. Two metrics exist on every circle: the Euclidean metric (L2), which measures distance along curves, and the taxicab metric (L1), which measures distance as the sum of horizontal and vertical displacements. The L2 circumference of a circle of diameter d is πd. The L1 circumference is 4d, the perimeter of the bounding square. Their ratio is πd/4d = π/4 = β. This ratio appears in at least nine domains of applied mathematics and physics: geometry, fluid mechanics, aerodynamics, electromagnetism, RF engineering, optics, thermal physics, probability, and signal processing. In every case the same operation occurs, a circular quantity is evaluated in rectilinear coordinates, and the conversion factor β mediates between them. The unified equation Q = F · β · d² · Z captures all nine domains, where F is a driving term, β · d² is the geometric invariant, and Z is domain-specific impedance. @HOWL-MATH-1-2026 Separately, the L1/L2 framework extends along two additional axes. The first is the Lp axis: β (p) = 2π/Cₚ generalizes the conversion from L1 (p = 1, giving β = π/4) through L2 (p = 2, giving β = 1) to L∞ (giving β = π√2/4). The second is the manifold axis: the circle is the k = 0 member of an elliptic family parametrized by modulus k. At k > 0 the manifold is a torus, the period is the complete elliptic integral K (k) rather than π/2, and the conversion factor generalizes accordingly. Both extensions are established in prior work. All of this work shares a hidden assumption. Every circle in every equation is at rest relative to the observer. The pipe cross-section, the antenna dish, the capacitor plate, the laser beam waist, the diffraction aperture, all stationary. The staircase paradox, the foundation integral, the nine-domain catalog, all rest-frame geometry. For the nine domains cataloged, this assumption is harmless. Pipe flow, drag, capacitance, and thermal radiation involve systems moving at velocities negligible compared to the speed of light. The rest-frame β = π/4 is correct to extraordinary precision. But the assumption is an assumption, not a theorem. A circle in motion is not a circle. It is an ellipse. Its L2 circumference changes. Its L1 bounding perimeter changes. Their ratio, the conversion factor between the two metrics, changes. β is not a constant of geometry. It is a function of the relative velocity between the circle and the observer. This paper removes the rest-frame assumption. It derives the velocity-dependent conversion factor β (v), proves its endpoints (π/4 at rest, 1 at the speed of light), establishes its monotonicity, connects it to the elliptic integral family from the manifold extension, and identifies the physical regimes where the correction matters.