Comments on generalized Heron polynomials and Robbins’ conjectures

AbstractHeron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, (4A)2 is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223–236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523–530. MR 96k:51024] showed that for any cyclic polygon with n edges, (4A)2 satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n=5 and n=6. He conjectured the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371–404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3–24. MR 2004d:51014]. A short independent proof will be shown here

The area of cyclic polygons: Recent progress on Robbins' Conjectures

In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David Robbins

Curves of Finite Total Curvature

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.Comment: 25 pages, 4 figures; final version, to appear in "Discrete Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 200

カガク ギジュツシャ ノ タマゴ オ イクセイ スル カガク ギジュツシャ ノ ハックツ ヨウセイ コウザ ノ テンカイ : スウガク リョウイキ ニオケル プレマスター コース ノ ガクシュウ ナイヨウ

The contents of the lectures and the learner’s reaction in the pre master’s course in a mathematical domain are reported here. Two contents in the lectures are designed, the area of a circle and an isoperimetric problem. All learners were middle school students and have got high motivation for mathematics. Although the learners know the formula of the area of a circle well, they have little experiences which make them realize that the formula does really hold. In the first lecture, an argument − to compare the areas of inscribed regular polygons with the one of a unit circle, whose idea comes to the usual measure theory in the future, was explained. Through the process of calculating the areas of various kind of inscribed regular polygons to a unit circle, the learners appreciated that the area of a unit circle is really equal to π and convinced themselves of the fact. In the latter lecture, they have considered an isoperimetric problem and showed that the equilateral triangle has the largest area among the triangles with constant perimeter. Although a few mathematical concepts and treatments have been required to follow the procedure of the arguments, they overcame these difficulties eagerly and recognized two contents of mathematics in the pre master’s course. Through the program in the pre master’s course in a mathematical domain, we noticed that learners with high motivation for mathematics study many advanced contents extensively and extend their interests for themselves only with the help of introducing well−suited orientations and with the proper support to give some necessary concepts

On the Areas of Cyclic and Semicyclic Polygons

We investigate the ``generalized Heron polynomial'' that relates the squared area of an n-gon inscribed in a circle to the squares of its side lengths. For a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a certain variety derived from the variety of binary (2m-1)-forms having m-1 double roots. Thus we obtain explicit formulas for the areas of cyclic heptagons and octagons, and illuminate some mysterious features of Robbins' formulas for the areas of cyclic pentagons and hexagons. We also introduce a companion family of polynomials that relate the squared area of an n-gon inscribed in a circle, one of whose sides is a diameter, to the squared lengths of the other sides. By similar algebraic techniques we obtain explicit formulas for these polynomials for all n <= 7.Comment: 22 page