A Complex Hyper-Spherical Geometry Founded on the Platonic Solids' Abelian Orthogonal Sign Transformation Group

Abstract This study develops a finite complex hyper-spherical (3-sphere S³R(t)) geometry for representing edge-midpoint configurations and discrete sign transformations associated with the Platonic solids. The construction separates a finite algebraic sign field from its geometric carrier, a single two-sided, time-dependent 3-sphere. Main-diagonal matrices are assigned to the outer orientation, while secondary-diagonal matrices provide the inner mirror realization. Beginning with the tetrahedral sign group of order eight, whose six non-observer states label the six edge midpoints, the construction extends to an observer-extended elementary Abelian group of order sixty-four. Sixty non-observer states are assigned to two 30-element edge-midpoint realizations of the icosahedron-dodecahedron dual structure, while four states define two observer axes. Incidence-placement maps distinguish algebraic labels from positions on complex 3-sphere S³R(t). Subgroup decompositions, orthogonal midpoint relations, chiral observer multiplication, and matrix-labelled line transformations establish the finite construction. A prescribed four-stage sequence of half-turn transformations returns an oriented line state to its initial state after an accumulated 720 degrees. The algebraic structure is invariant under changes of the geometric scale. The construction is discrete and is not identified with the classical rotation groups of the Platonic solids or with the full continuous rotation group of the 3-sphere.