A High-Precision Approximate Construction Method for the Quadrature of the Circle via 45-Degree Rotational Matrix and Peripheral Scanning Circles

This paper presents a novel and highly practical approximate solution to the classical problem of the Quadrature of theCircle (Squaring the Circle) using a pure geometric construction methodology. By employing a 45-degree rotational matrixsystem combined with four peripheral degree-3 shape-preserving scanning circles, our framework establishes a clear andreproducible bridge between transcendental circular boundaries and discrete square geometries. Based on precise digitaland physical measurements, the system yields a calculated side length of 88.4884 mm against the transcendental ideal of88.6227 mm for a 100 mm diameter circle. This achieves an elite structural error rate of just 0.15%, which is completelyabsorbed by the physical stroke weight of drawing lines, thereby demonstrating an effective 100% precision in empiricalgeometric implementation.