Inscribing a regularm-gon in a regular n-gon
نویسندگان
چکیده
We find necessary and sufficient conditions on m and n for inscribing a regular m-gon in a regular n-gon. Mathematics Subject Classification (2010). Primary 51M20; Secondary 52C15.
منابع مشابه
Classification of First-order Flexible Regular Bicycle Polygons
A bicycle (n, k)-gon is an equilateral n-gon whose k-diagonals are equal. S. Tabachnikov proved that a regular n-gon is first-order flexible as a bicycle (n, k)-gon if and only if there is an integer 2 ≤ r ≤ n − 2 such that tan(π/n) tan(krπ/n) = tan(kπ/n) tan(rπ/n). In the present paper, we solve this trigonometric diophantine equation. In particular, we describe the family of first order flexi...
متن کاملInscribing Smooth Knots with Regular Polygons
A knot K : S → R is said to be inscribed by a regular n-gon if there is a set of points x0, ..., xn−1 lying on K in a cyclic order, such that the distances ‖xi−1 − xi‖ between xi−1 and xi are the same for i = 1, ..., n, where xn = x0. Jon Simon asked the question of whether every smooth knot K is inscribed by a regular n-gon for all n. Up to rescaling we may assume that the length of K is 1. It...
متن کاملUsing Benthic Diatoms as a Bioindicator to Assess Rural-urban River Conditions in Tropical Area: A Case Study in the Sai Gon River, Vietnam
The changes in diatom assemblages along an urban-to-rural gradient were characterized to assess the ecological status of the Sai Gon River, Vietnam. Diatoms and physico-chemical variables were measured at 10 stations during dry and rainy season. One-way ANOVA showed that diatom metrics and physicochemical variables were significantly different (p < 0.05) between the upper course sites and both ...
متن کاملThe Minimum Distance Energy for Polygonal Unknots
This paper investigates the energy UMD of polygonal unknots. It provides equations for finding the energy for any planar regular n-gon and for any m-gon, where the vertices lie on the vertices of a regular m 2 -gon and on the midpoints of each edge. In addition, we show that a regular 4-gon minimizes the energy for any quadrilateral. Finally, this paper includes a proof showing that if we have ...
متن کاملNapoleon Polygons
An n-gon is called Napoleon if the centers of the regular ngons erected outwardly on its sides are vertices of a regular n-gon. In this note we obtain a new geometric characterization of Napoleon ngons and give a new proof of the well-known theorem of Barlotti Greber ([1], [3]) that an n-gon is Napoleon if and only if it is affine regular. Moreover, we generalize this theorem by obtaining an an...
متن کامل