2019/10/23 by Grubb, Thomas, Woll, Christian
#11R18 (Primary) 51M05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1910.10325
By viewing the regular N-gon as the set of Nth roots of unity in the complex plane we transform several questions regarding polygon diagonals into when a polynomial vanishes when evaluated at roots of unity. To study these solutions we implement algorithms in Sage as well as examine a trigonometric diophantine equation. In doing so we classify when a metallic ratio can be realized as a ratio of polygon diagonals, answering a question raised in a PBS Infinite Series broadcast. We then generalize this idea by examining the degree of the number field generated by a given ratio of polygon diagonals.