The arithmetic of simplices
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2012
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| _version_ | 1866911158375546880 |
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| author | Mieczkowski, Edward |
| author_facet | Mieczkowski, Edward |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1204_2219 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | The arithmetic of simplices Mieczkowski, Edward General Mathematics 52C99, 11A99, 11Z99 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. |
| title | The arithmetic of simplices |
| topic | General Mathematics 52C99, 11A99, 11Z99 |
| url | https://arxiv.org/abs/1204.2219 |