  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.
