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

## TL;DR

This paper demonstrates how a six-dimensional root system emerges from icosahedral tetrahedral units using Clifford algebra, linking 3D symmetry to higher-dimensional lattice structures relevant in advanced physics theories.

## Contribution

It introduces a novel geometric framework connecting 3D icosahedral symmetry with 6D Dirichlet lattices via Clifford algebra, providing insights into higher-dimensional physics models.

## Key findings

- Emergence of 6D root system from 3D icosahedral tetrahedra
- Connection between 3D icosahedral seed and higher-dimensional lattices
- Framework relevant to $SU(5)$, $E_6$, $E_8$ Lie algebras

## 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.