Splitting of 3d quaternion dimensions into 2d-sells and a "world screen technology"

TL;DR

This paper explores the algebraic and geometric structures underlying 3D space dimensions, introducing a 'world screen' concept that captures key kinematic properties through 2D surface representations.

Contribution

It establishes a mathematical link between quaternion units and elementary 2D surfaces, proposing a novel 'world screen' framework for understanding 3D space.

Findings

Quaternion units relate to 2D surface structures.

A 'world screen' concept reflects 3D kinematics.

Mathematical link between 3D frames and 2D surfaces.

Abstract

A set of basic vectors locally describing metric properties of an arbitrary 2-dimensional (2D) surface is used for construction of fundamental algebraic objects having nilpotent and idempotent properties. It is shown that all possible linear combinations of the objects when multiplied behave as a set of hypercomples (in particular, quaternion) units; thus interior structure of the 3D space dimensions pointed by the vector units is exposed. Geometric representations of elementary surfaces (2D-sells) structuring the dimensions are studied in detail. Established mathematical link between a vector quaternion triad treated as a frame in 3D space and elementary 2D-sells prompts to raise an idea of "world screen" having 1/2 of a space dimension but adequately reflecting kinematical properties of an ensemble of 3D frames.