# 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

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 $SU\left(5\right)$ , ${E}_{6}$ , ${E}_{8}$ Lie algebras and their composition with the algebra associated with the even unimodular lattice in ${\mathbb{R}}^{3,1}$ . The construction presented here is inspired by Penrose’s three world model.