A Note on Affinely Regular Polygons

A characterization of affinely regular polygons

In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $λ\geq 0$ such that $p_{j+2}-p_{j-1} = λ(p_{j+1}-p_j)$ for $j=1,2,\ldots, n$. The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\ldots,p_n \in \mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \in \mathbb{C}$ and $m_1,m_2,k \in \mathbb{Z}$. In particular, we show that in `most' cases this implies that the polygon is …

余维2子流形上的多边形与共面点

Polygons and coplanar points on submanifolds of codimension 2

本文解决格林第73个公开问题，证明n≥6时ℝⁿ中余维2光滑子流形的非空相对开子集含仿射正五边形顶点，n≥7时得6个共面点，核心方法是让图映射低傅里叶模式沿圆消失。

A note on the maximal perimeter and maximal width of a convex small polygon

The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for the maximum perimeter and the maximum width.

Errata to: the limiting curve of Jarnik's polygons

In this note we point out that the limit shape of the Jarnk polygonal curve [1] is the circle, not a curve consisting of arcs of parabolas as it has been stated in the main result of [2] (Section 2). The correct result has been established and proven in [4].

Note on the chromatic number of Minkowski planes: the regular polygon case

The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice-sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.

A note on the pentagram map and tropical geometry

In this note we study the action of the pentagram map on the moduli space of twisted polygons. The action with respect to the canonical coordinate turns out to be applicable to the framework of tropical geometry. As an application, we induce quasi-recursiveness of the pentagram map on some domain, which is induced from dynamical property of the pentagram automaton.