2012/04/06 by Edward Mieczkowski, Mieczkowski, Edward
Computer Science · Mathematics · #11A99 #11Z99 #52C99 #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #FOS: Mathematics #General Mathematics (math.GM) #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1204.2219
openalex publication_date 2012/04/06 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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 ℝ3= \± =± (x3,x2,x,1); x∈ℝ \ effectively characterizes this family of tetrahedra. The set ℝ3 is a subset of the ring ℝ4 = ℝ × ℝ × ℝ × ℝ = \ (x, y, z, w) ; x, y, z, w ∈ ℝ \, with addition and multiplication defined component-wise. The set ℝ3 supports two operations. Multiplication is inherited directly from the ring ℝ4, while addition is a four-argument operation that reflects geometric transformations such as homothety and translation of elements in ℝ3. A novel form of addition in ℝ3 leads to intriguing properties of multiplication in ℝ3, 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.