Author Description

The encapsulated work is structured as an axiomatic extension of Euclid's elements relevant to the geometry of proportion infused with notions common to Pythagorean number theory; This provides the logical foundation necessary to analyze the structure of the natural dimensional proportion of Pascal's triangle. A novel modulus nine analysis method is deployed enabling full geometric characterization of the complementary symmetry shown to mediate the consecutive geometric transformations of such numeric proportion. Key elements of conventional quantum theory are placed within the context of the proposed dimensional framework: The nature of spherical harmonics, thought to control the periodic order of atomic energy states, is shown to be symmetrically governed by the complementary cuboctahedral dipole field produced by the rotational differentiation of space and time. Through this lens atomic energy is seen as the absolute measure of electron-proton dipole field-potential accumulated and bound-neutral via spatial/temporal differentiation. The base field symmetry is geometrically proven to culminate with the in-circle perspective of the 3-4-5 triangle, which produces an integral spectrum of electromagnetic prime-monopole proportion posited to be characteristic of photons. Such proportion are proven to be carried upon the diagonals of sequential triple right-triangles ordered by the constant variance between in-circle and ex-circle radii (this variable is shown to modulate the frequency and amplitude of such proportion).