  Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional
pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades
rather than vectors. This discovery allows points to be factored into $d$
orthogonal hyperplanes, establishing points as pseudoscalars of a local
geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this
factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric
gauge group at each point. Moreover, a point can alternatively be factored into
a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$,
which are traditionally used to label spinor representations. Therefore, points
reveal previously hidden geometric foundations for some of quantum field
theory's mysteries. This work outlines the impact of PGA on the study of spinor
representations in any number of dimensions, and is the first in a research
programme exploring the consequences of this insight.
