This paper defines the volume of a solid as the cumulative sum of infinitesimal volume elements expanding or contracting diagonally from the origin, treating volume not as a static entity but as a dynamic manifold. Using spherical coordinates, the integration of the volume with respect to the diagonal length is performed, transforming the differential volume element via the Jacobian determinant. By calculating the integrand, the final volume integral value of 𝜋𝑅4 is derived, demonstrating how the dynamic flow of three-dimensional space can be explained by orderly mathematical rules. This is consistent with the essential insight of the Hodge Conjecture, which suggests that the shapes of complex high-dimensional spaces can eventually be defined as the sum of simple algebraic pieces.
SungKun Kim (Fri,) studied this question.