The arithmetic of simplices

TL;DR

This paper explores the algebraic structure of sets of similar tetrahedra and simplices, introducing a novel addition operation and generalizing to higher dimensions, revealing new geometric and algebraic properties.

Contribution

It introduces a new addition operation on sets of similar simplices and generalizes the algebraic framework to higher dimensions, expanding understanding of geometric transformations.

Findings

Characterization of tetrahedra via the set $\\mathbb{R}_3$

Introduction of a novel addition operation reflecting geometric transformations

Generalization to higher-dimensional simplices

Abstract

This paper continues the study initiated in "The aithmetic of Triangles." We begin by examining a set of similar tetrahedra with parallel sides, together with a set of points in three-dimensional space. It turns out that the set $\mathbb{R}_3= \{\pm <x >=\pm (x^3,x^2,x,1); x\in\mathbb{R} \}$ effectively characterizes this family of tetrahedra. The set $\mathbb{R}_3$ is a subset of the ring $\mathbb{R}^4 = \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} = \{ (x, y, z, w) ; x, y, z, w \in \mathbb{R} \}$, with addition and multiplication defined component-wise. The set $\mathbb{R}_3$ supports two operations. Multiplication is inherited directly from the ring $\mathbb{R}^4$, while addition is a four-argument operation that reflects geometric transformations such as homothety and translation of elements in $\mathbb{R}_3$. A novel form of addition in $\mathbb{R}_3$ leads to intriguing properties of multiplication in $\mathbb{R}_3$, which are examined in a dedicated chapter. We then generalize this approach to sets of $k$-dimensional similar simplices with parallel sides, along with corresponding sets of points in $k$-dimensional space.