Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic Characterizations of Angle Multisections over Rings

Takashi Hirotsu

Abstract

Let $m,$ $n \geq 2$ be integers, and let $R$ be a subring of $\mathbb R$ with field of fractions $F.$ In this article, we generalize the rational angle bisection problem previously proposed by the author as follows: characterize the pairs of linearly independent vectors $\vec{a},$ $\vec{b} \in R^n$ that form angles with sequences of $m$-sector vectors lying in $R^n.$ When $\vec{a}$ and $\vec{b}$ are nonorthogonal, we prove that this condition is equivalent to the existence of a root in $F$ of a certain $m$-th degree polynomial over $R.$ In particular, when $R = \mathbb Z,$ the condition holds if and only if the polynomial has a root among the divisors of its constant term. When $m = 2^e$ with integer $e \geq 1,$ we also prove that the condition is equivalent to $\cos (θ/2^{e-1}) \in F,$ where $θ$ is the angle between $\vec{a}$ and $\vec{b}.$

Algebraic Characterizations of Angle Multisections over Rings

Abstract

Let be integers, and let be a subring of with field of fractions In this article, we generalize the rational angle bisection problem previously proposed by the author as follows: characterize the pairs of linearly independent vectors that form angles with sequences of -sector vectors lying in When and are nonorthogonal, we prove that this condition is equivalent to the existence of a root in of a certain -th degree polynomial over In particular, when the condition holds if and only if the polynomial has a root among the divisors of its constant term. When with integer we also prove that the condition is equivalent to where is the angle between and
Paper Structure (4 sections, 11 theorems, 42 equations, 1 figure)

This paper contains 4 sections, 11 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Extending Sequences of Angle Equisectors
  3. Angle Multisection
  4. Angle $2^e$-section

Key Result

Proposition 1

Let $\bm{a},$$\bm{c} \in R^n$ be linearly independent vectors. Then there exists a nonzero vector $\bm{b} \in R\bm{a}+R\bm{c}$ such that $\bm{c}$ is a bisector vector of $\angle (\bm{a},\bm{b}).$ Such a vector $\bm{b}$ is parallel to $2\langle\bm{a},\bm{c}\rangle\bm{c}-|\bm{c}|^2\bm{a}.$

Figures (1)

  • Figure 1: A sequence of nonasector lines of the angle between $y = (1/7)x$ and $y = -(29/278)x.$

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Example 1
  • Theorem 1: Hir25
  • Remark 2
  • Theorem 2
  • Corollary 1
  • Example 2
  • Theorem 3
  • ...and 18 more