Quaternion, Octonion to Dodecanion Manifold: Stereographic Projections from Infinity Lead to a Self-operating Mathematical Universe

Abstract

For two hundred years, quaternions and octonions were explored, not a single effort was made on constructing the mathematical universe with more than eight imaginary worlds. We cross that 200 years old jinx and report dodecanion, a universe made of 12 imaginary worlds and show that once a fractal-like system is dynamic with 12 dimensions, it acquires a geometric feature unprecedented at lower dimensions. While the topology of octonion algebra remains an identity, the topology of a dodecanion algebra demands the coexistence of three distinct manifolds at a time and three distinct stereographic projections at a time. We define it as the condition for a self-operational mathematical universe. Earlier, dimensions were only a new dynamic associated with a new orthogonal axis, now, we assign modular or clock arithmetic systems in the singularity points of a system, thus, it assembles a mathematical structure where the systems are assembled one inside the other. The dimensions 12, 18, 20, 24, 30, 36 create a distinct catalog of manifolds. Since the maximum allowed higher dimension in recent physics is 10 (String theory) or 11 (M-theory), the dodecanion algebra with 12D is the simplest multinion that maps the topological variability and the interactions of physical worlds representing different dimensions, i.e., dynamics. We mapped here distinct projections from infinity during stereographic projections while transiting from 2 to 12 imaginary worlds. The dodecanion algebra has the ability to incorporate the manifolds created by multinions of higher dimensions, it is essential and sufficient for a generic self-operating universe.