Emergence of an aperiodic Dirichlet space from the tetrahedral units of an icosahedral internal space

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 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 multivector formalism of 3D Euclidean space. This lays a geometric framework for understanding several physics theories related to $SU(5)$, $E_6$, $E_8$ 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 \textit{three world} model.