Quaternion, Octonion to Dodecanion Manifold: Stereographic Projections from Infinity Lead to a Self-operating Mathematic
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 ima
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P. Singh et al.
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. Keywords Geometric algebra · Complex number · Tensor · Quaternion · Octonion · Dodecanion · Stereographic projection · Geometric language · Prime
1 Introduction The journey to find complex numbers like quaternions [1] started with the question, how could one divide 1 into many parts so that the product of those numbers is 1. Later, each of these dimensions was assigned a geometric axis and the product meant a rotation by 90o . Each new axis or dimension is referred to as a world, holding new dynamics and all axes or worlds used to be called together the universe. Worlds are small pockets with a distinct dimension constituting the universe. A 3D Euclidean space is often represented by a quaternion, a complex number with three vectors (q = q0 + q1 i + q2 j + q3 k) where one could consider three orthogonal axes as (i, j, k), widely used in graphics ([2], analyzing higher level codes in DNA [3]. For an octonion ([4], a complex number with eight vectors, additional four dimensions are invisible but they represent new dynamics in addition to motion represented by a quaternion. O = O0 e0 +O1 e1 +O2 e2 +O3 e3 +O4 e4 +O5 e5 +O6 e6 +O7 e7 , Octonions are widely used in astrophysics, e.g., analyzing the dynamics of information in a black hole [5]. Here, we introduce a higher dimension than octonion, namely dodecanion, a complex number with 12 dimensions, d = d0 h 0 + d1 h 1 + d2 h 2 + d3 h 3 + d4 h 4 + d5 h 5 + d6 h 6 + d7 h 7 + d8 h 8 + d9 h 9 + d10 h 10 + d11 h 11 [6]. Introducing a new complex number requires finding new elementary defining parameters, here, we have studied how shifting from octonion to a dodecanion would change normed division algebra over the real space. Just like variables x, y, and z for numbers, the geometric shapes have allowed and restricted transformations based on symmetries they obtain on the spherical Euclidean surface. Since symmetries are finite, so are the elementary geometric shapes and their transitions, finiteness of symmetries is explored by Reddy et al. [7] to build a universal geometric language, namely geometric musical language, GML [8]. Plenty of geometric languages are created [9] but no proposal existed to link geometric shape with the number system, GML is the singular proposal in that respect. The objective was alway
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