Abstract
Advancing from eight imaginary worlds of octonion algebra, for the first time we introduce dodecanion algebra, a mathematical universe made of twelve imaginary worlds one inside another. The difference between eight and twelve imaginary worlds is that the Fano plane that sets the products of imaginary vectors is replaced by a triplet of manifolds that could coexist in three forms. In the proposed algebra product tensors-like quaternion, octonion, dodecanion, and icosanion are deconstructed as a composition of prime dimensional tensors. We propose a generic conformal cylinder of imaginary worlds, similar to modulo or clock arithmetic, using that one could build the group multiplication tables of multinions, which would enable developing the associated algebra. Space-time (st) metric is known, we added two concepts, 15 geometric shapes as topology (T) and 15 primes as symmetry (S) to build a new metric, space-time-topology-prime(stTS) for a self-operating mathematical universe with n nested imaginary worlds. The stTS metric delivers a decision as shape-changing geometry with time, following fractal information theory (FIT) proposed earlier for hypercomputing in the brain. FIT includes two key aspects, the geometric musical language (GML) and the phase prime metric (PPM) that operates using clock architectures spread over 12 dimensions.
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References
Corrochano B (2010) Geometric computing for wavelet transforms, robot vision, learning, control and action. Springer, Heidelberg, Chapter 6, pp 149–183
Rozenfelʹd AB (1988) A history of non-euclidean geometry: evolution of the concept of a geometric space. Springer, Heidelberg, p 373 (1988)
Conway JH, Smith DA (2003) On quaternions and octonions: their geometry, arithmetic, and symmetry, p 9. ISBN 1-56881-134-9
Shoemake K (1985) Animating rotation with quaternion curves. Comput Graphics 19(3):245–254
Shu JJ, Ouw LS (2004) Pairwise alignment of the DNA sequence using hypercomplex number representation. Bull Math Biol 66(5):1423–1438
Leron B, Duminda D, Duff MJ, Hajar E, Williams R (2009) Black holes, qubits and octonions. Phys Rep 471(3–4):113–219
Ghosh et al (2019) JP-2017-150171, World patent WO 2019/026983
Preparata F, Hong SJ. Convex hulls of finite sets of points in two and three dimensions. In: Manacher G, Graham SL (eds) CACM, vol 20, issue 2, p 88
Gardner M (1992) Fractal music, hypercards, and more: mathematical recreations from scientific american. W. H. Freeman, New York, pp 40, 53, and 58–60
Cormen TH, Charles EL, Ronald LR, Stein C (2001) Introduction to algorithms, 2nd edn. MIT Press and McGraw-Hill, pp. 862–868. ISBN 0-262-03293-7. Section 31.3: Modular arithmetic
Reddy S et al (2018) A brain-like computer made of time crystal: could a metric of prime alone replace a user and alleviate programming forever? Stud Comput Intell 761:1–44
Bewersdorff J (2005) Asymmetric dice: are they worth anything? In: Luck, logic, white lies: the mathematics of games. A K Peters, Wellesley, MA, pp 33–36
Cundy HM, Rollett A (1989) 3.11. Deltahedra. Mathematical models, 3rd edn. Tarquin Pub., Stradbroke, England, pp 142–144
Trigg CW (1978) An infinite class of deltahedra. Math Mag 51(1):55–57 (JSTOR 2689647)
Lasenby A (2005) Recent applications of conformal geometric algebra. Computer algebra and geometric algebra with applications. In: Li H, Olver PJ, Sommer G (eds). IWMM 2004, GIAE 2004. Lecture Notes in Computer Science, vol 3519. Springer, Heidelberg
Pugh A (1976) Polyhedra: a visual approach. University of California Press, Berkeley, California. ISBN 0-520-03056-7, pp 35–36
Bandyopadhyay A (2020) Nanobrain. The making of an artificial brain from a time crystal, 1st edn. CRC Press, March 16, 2020 (Forthcoming), ISBN 9781439875490 - CAT# K13502
Kac VG, Moody RV, Wakimoto M (1988) On E10. Differential geometrical methods in theoretical physics (Como, 1987). NATO Adv Sci Inst Ser C Math Phys Sci 250. Kluwer Acad Publ, Dordrecht, pp 109–128. MR 0981374
West P (2001) E11 and M theory. Classical Quant Gravity 18(21):4443–4460
Watrous J, Aaronson S (2009) Closed time like curves make quantum and classical computing equivalent. Proc R Soc A Math Phys Eng Sci 465(2102):631
Hamber HW (2009) Quantum gravitation—the Feynman path integral approach. Springer Nature. ISBN 978-3-540-85292-6
Penrose R (1971) Angular momentum: an approach to combinatorial spacetime. In Bastin T (ed) Quantum theory and beyond. Cambridge University Press, Cambridge
Penrose R (1969) Applications of negative dimensional tensors: combinatorial mathematics and its applications. In: Welsh DJA (ed) (Proc. Conf., Oxford, 1969), Academic Press, pp. 221–244, esp. p. 241. On the origins of twistor theory in: gravitation and geometry, a Volume in Honour of I. Robinson, Biblipolis, Naples 1987
Oeckl R (2003) Generalized lattice gauge theory, spin foams and state sum invariants. J Geometry Phys 46(3–4):308–354
Reimann MW et al (2017) Cliques of neurons bound into cavities provide a missing link between structure and function. Front Comput Neurosci, 12 June 2017. https://doi.org/10.3389/fncom.2017.00048
Singh P, Ray K, Fujita D, Bandyopadhyay A (2019) Complete dielectric resonator model of human brain from mri data: a journey from connectome neural branching to single protein. In: Ray K, Sharan S, Rawat S, Jain S, Srivastava S, Bandyopadhyay A (eds) Engineering vibration, communication and information processing. Lecture Notes in Electrical Engineering, vol 478. Springer, Singapore
Ghosh S, Sahu S, Fujita D, Bandyopadhyay A (2014) Design and operation of a brain like computer: a new class of frequency-fractal computing using wireless communication in a supramolecular organic, inorganic systems. Information 5:28–99
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Authors acknowledge the Asian office of Aerospace R&D (AOARD) a part of United States Air Force (USAF) for the Grant no. FA2386-16-1-0003 (2016–2019) on the electromagnetic resonance based communication and intelligence of biomaterials.
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Appendix
While others would continue to build new algebras using addition, subtraction, multiplication and division of these complex numbers, we advocate operating a new function Ж that is used frequently in the fractal information theory (FIT), which is a combination of geometric musical language (GML) and phase prime metric, (PPM). The new function Ж looks into the topological symmetry of the participating elements of tensors, wherein all elements are geometric shapes. How interacting geometric shapes would bond together building a new geometric shape, FIT is a systematic study for that purpose. Therefore, when we write Q Ж iO Ж jD Ж kI, it consolidates that the self-similar geometries initiate bonding of wide ranges of geometric shapes in a complex 3D architecture. If all geometric shapes of the architecture are connected to clocks or modulo (modulo = number of corners of a geometric shape), the tensor gets an application in physics. The architecture explores all possible dynamics among the participant complex numbers.
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Singh, P. et al. (2021). A Space-Time-Topology-Prime, stTS Metric for a Self-operating Mathematical Universe Uses Dodecanion Geometric Algebra of 2-20 D Complex Vectors. In: Ray, K., Roy, K.C., Toshniwal, S.K., Sharma, H., Bandyopadhyay, A. (eds) Proceedings of International Conference on Data Science and Applications. Lecture Notes in Networks and Systems, vol 148. Springer, Singapore. https://doi.org/10.1007/978-981-15-7561-7_1
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