Circle Perimeter Without Pi: A Harmonic Projective Integral

This work introduces a novel geometric construction for expressing the circumference of a circle. Instead of relying on the analytical definition of pi, the method uses an internal length Y, defined as the perpendicular projection from a point at half-radius to the edge of the circle, and a harmonic projection constant Pi_Y. The resulting formula expresses the circumference as a product of Pi_Y and Y, where Pi_Y = 4 pi / sqrt(3). This provides a direct geometric interpretation of the perimeter, compatible with classical integral approaches but based on a rational internal structure. The proposed harmonic projective integral offers a new perspective on how to reconstruct the circle’s perimeter from internal lengths, and could serve as a foundation for further developments in harmonic geometry and alternative approximations of pi.