Two Forms of Proximal Physical Geometry. Axioms, Sewing Regions Together, Classes of Regions, Duality, and Parallel Fibre Bundles

This paper introduces two proximal forms of Lenzen physical geometry, namely, an \emph{axiomatized strongly proximal physical geometry} that is built on simplicial complexes with the dualities and sewing operations derived from string geometry and an \emph{axiomatized descriptive proximal physical geometry} in which spatial regions are described based on their features and the descriptive proximities between regions. This is a computational proximity approach to a Lenzen geometry of physical space. In both forms of physical geometry, region is a primitive. Intuitively, a region is a set of connected subregions. The primitive in this geometry is \emph{region}, instead of \emph{point}. Each description of a region with $n$ features is a mapping from the region to a feature vector in $\mathbb{R}^n$. In the feature space, proximal physical geometry has the look and feel of either Euclidean, Riemannian, or non-Euclidean geometry, since we freely work with the relations between points in the feature space. The focus in these new forms of geometry is the relation between individual regions with their own distinctive features such as shape, area, perimeter and diameter and the relation between nonempty sets of regions. The axioms for physical geometry as well as the axioms for proximal physical geometry are given and illustrated. Results for parallel classes of regions, descriptive fibre bundles and BreMiller-Sloyer sheaves are given. In addition, a region-based Borsuk-Ulam Theorem as well as a Wired Friend Theorem are given in the context of both forms of physical geometry.