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Mathematics 2017, 5(2), 29; doi:10.3390/math5020029

Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space

Quantum Gravity Research, Los Angeles, CA 90290, USA
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Academic Editor: Lokenath Debnath
Received: 23 February 2017 / Revised: 12 May 2017 / Accepted: 18 May 2017 / Published: 26 May 2017
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Abstract

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to S U ( 5 ) , E 6 , E 8 Lie algebras and their composition with the algebra associated with the even unimodular lattice in R 3 , 1 . The construction presented here is inspired by Penrose’s three world model. View Full-Text
Keywords: aperiodic Dirichlet lattice; icosahedral symmetry; Clifford spinors and Lie algebras aperiodic Dirichlet lattice; icosahedral symmetry; Clifford spinors and Lie algebras
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Sen, A.; Aschheim, R.; Irwin, K. Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space. Mathematics 2017, 5, 29.

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