A Regular Quaternion Polygon

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m, n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that dn/2e − 2 ≤ g(4, n) ≤ n 2 + o(n) and dn/4e ≤ g(n, 4) ≤ n 2 + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the min...

Ranking efficient units by regular polygon area in DEA

Set complexity of construction of a regular polygon

Given a subset of $\mathbb C$ containing $x,y$, one can add $x + y,\,x - y,\,xy$ or (when $y\ne0$) $x/y$ or any $z$ such that $z^2=x$. Let $p$ be a prime Fermat number. We prove that it is possible to obtain from $\{1\}$ a set containing all the $p$-th roots of 1 by $12 p^2$ above operations. This result is different from the standard estimation of complexity of an algorithm computing the $p$-t...

The Vortex Filament Equation for a Regular Polygon

Abstract I shall present some recent results obtained in collaboration with V. Banica and F. de la Hoz about the evolution of vortex filaments according to the so-called binormal law. After reviewing the results for filaments with one corner, we will look at the case of general polygons. The dynamics turn out to be quite complex. Among other things numerical evidence on the appearance of multif...

Counting symmetry classes of dissections of a convex regular polygon

This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some kno...