The Area of Cyclic Polygons: Recent Progress on Robbins’ Conjectures

On the Robbins problem for cyclic polygons

Cyclic polygons with rational sides and area

We generalise the notion of Heron triangles to rational-sided, cyclic n-gons with rational area using Brahmagupta’s formula for the area of a cyclic quadrilateral and Robbins’ formulæ for the area of cyclic pentagons and hexagons. We use approximate techniques to explore rational area n-gons for n greater than six. Finally, we produce a method of generating non-Eulerian rational area cyclic n-g...

Geometry of pentagons: from Gauss to Robbins

An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give several proofs and extensions of this beautiful formula to hexagon etc. and consider special cases of affine–regular polygons. The Gauss pentagon formula is, in...

Mirror Symmetry and the Classification of Orbifold Del Pezzo Surfaces

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces. We explore mirror symmetry for del Pezzo surfaces with c...

Deriving Geometry Theorems by Automated Tools

Derivation of geometry theorems belongs to mighty tools of automated geometry theorem proving. By elimination of suitable variables in the system of algebraic equations describing a geometric situation we get required formulas. The power of derivation is presented on computation of the area of planar polygons given by their lengths of sides and diagonals. This part we conclude with derivation o...