A Historical Study Related To Euler’s Polyhedron Formula

Euler’s polyhedron formula can be understood as the topological properties of polyhedron,is a relational formula for the number of vertices,edges and faces of a polyhedron.The first topological property of polyhedron was discovered by Descartes around 1630.The same property was rediscovered by Euler in 1750 and is now known as Euler’s polyhedron formula.This paper focuses on the work of these mathematicians and their influence on each other,as well as the historical clues to the definition and counterexamples of the polyhedron,through a study of the original literature of Descartes,Euler,Legendre,Cauchy,Staudt,Poinsot,Lhuilier,Hessel and Poincaré,and other related work,with the following main findings.1.The early history of Euler’s polyhedron formula is sorted out,focusing on the differences and connections between the work of Descartes and Euler on Euler’s polyhedron formula.The polyhedron formula was mentioned by Euler in his correspondence with Goldbach in 1750.However,according to historical records,Descartes had also discovered a similar theorem around 1630.This project is a separate study of Descartes’ manuscripts on polyhedron,together with a review of Euler’s papers on polyhedron,to sort out the thinking behind Descartes’ and Euler’s discovery of polyhedral formulas and to compare their work in detail.2.The history of the proof of Euler’s polyhedron formula is examined,interpreting the work of mathematicians such as Euler,Legendre,Cauchy and Staudt,and comparing the ideas of these mathematicians in their proofs and their connections.Early proofs of Euler’s polyhedron formula relied on measurements of angles and ordinary geometric quantities.The first rigorous proof was given by Legendre in 1794.A new proof of Euler’s polyhedron formula was given by Cauchy in 1811,and in 1847 Staudt gave a definition of Euler’s polyhedron and proved it.In addition,other mathematicians have tried to give proofs of Euler’s polyhedron formula,and this topic compares the ideas and connections between the proofs of these mathematicians.3.Based on “Proofs and Refutations”,a history of polyhedron defined and examples that do not satisfy the polyhedral formula is compiled for the period 1750-1850.Since some polyhedral formulas only apply to convex polyhedron,mathematicians wanted to determine which polyhedron satisfied Euler’s polyhedral formula.in 1810 Poinsot gave examples that did not satisfy the polyhedral formula,the four-star polyhedron.in 1811 Lhuilier proposed three types of examples that did not satisfy the polyhedral formula,being polyhedron with internal polygons,tunnels and cavities respectively.In 1832 Hessel proposed five examples of polyhedron that do not satisfy the polyhedral formula.in 1847 Staudt gave a set of guidelines for describing polyhedron.By comparing these definitions with examples that do not satisfy the polyhedral formula,an attempt is made to answer the background and reasons why polyhedron have not been clearly defined for a long time.