The arithmetic of 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.