The geometric constructions obtained with only straightedge and compass are famous and play a special role in the development of geometry. On the one hand, the constructibility of ?gures is a key ingredient in Euclid ...The geometric constructions obtained with only straightedge and compass are famous and play a special role in the development of geometry. On the one hand, the constructibility of ?gures is a key ingredient in Euclid geometry and, on the other hand, unconstructibility gave birth to famous open problems of the ancient Greece which were unlocked only in the nineteenth century using discoveries in algebra. This paper discusses the mechanization of straightedge and compass constructions. It focuses on the algebraic approaches and presents two methods which are implemented; one is due to Lebesgue and the other one was jointly designed by Gao and Chou. Some links between the algebraic approach of constructions and synthetic geometry are described.展开更多
It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of hep...It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of heptadecagon has been discussing all the time. Many different construction methods are proposed although they are based on the same theory. Simplification of the construction is still a sensible problem. Here, we propose a simple method for constructing regular heptadecagon with the fewest steps. The accumulation of construction errors is also avoided. This method is more applicable than the previous construction.展开更多
基金supported by Strasbourg University and French CNRS
文摘The geometric constructions obtained with only straightedge and compass are famous and play a special role in the development of geometry. On the one hand, the constructibility of ?gures is a key ingredient in Euclid geometry and, on the other hand, unconstructibility gave birth to famous open problems of the ancient Greece which were unlocked only in the nineteenth century using discoveries in algebra. This paper discusses the mechanization of straightedge and compass constructions. It focuses on the algebraic approaches and presents two methods which are implemented; one is due to Lebesgue and the other one was jointly designed by Gao and Chou. Some links between the algebraic approach of constructions and synthetic geometry are described.
文摘It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of heptadecagon has been discussing all the time. Many different construction methods are proposed although they are based on the same theory. Simplification of the construction is still a sensible problem. Here, we propose a simple method for constructing regular heptadecagon with the fewest steps. The accumulation of construction errors is also avoided. This method is more applicable than the previous construction.
