This paper delves into the historical background,precise division drawing methods,and approximate division drawing methods of the circumference of a circle,and provides a comprehensive summary.The problem of dividing ...This paper delves into the historical background,precise division drawing methods,and approximate division drawing methods of the circumference of a circle,and provides a comprehensive summary.The problem of dividing the circumference of a circle,as an important milestone in the history of mathematical development,not only demonstrates the progress of geometry but also has a profound impact on the formation of algebra,modern mathematics,and even contemporary abstract mathematics.The history of the problem of dividing the circumference of a circle can be traced back to ancient Greece,where mathematicians explored the division of the circle through geometric drawing using a straightedge and compass.One of the most famous methods is the approach proposed by Hippasus to approximate the circumference of a circle by increasing the number of sides of a regular polygon.However,mathematical proofs in the 19th century showed that certain division problems,such as the trisection of an angle,cannot be solved with just a straightedge and compass.With the development of algebra,the problem of dividing the circumference of a circle gradually transformed into solving radical equations.The thesis introduces in detail the precise drawing methods for dividing the circumference into three,six,four,eight,five,ten,and fifteen equal parts.For example,the trisection is achieved by constructing an inscribed equilateral triangle,while the hexasection is based on the trisection to further construct a regular hexagon.The drawing methods for the division into four and eight parts involve the perpendicular bisector of a circle and the bisection of the circumference.The drawing methods for the division into five and ten parts are based on the construction of an inscribed regular decagon,while the division into fifteen parts is achieved by constructing an inscribed regular pentadecagon.The paper further explores the approximate division of the circumference into seven and nine parts,as well as how to construct an approximate regular n-gon,such as a regular undecagon,using Tobi's method.Although these methods cannot achieve completely precise division,they provide a way to divide the circumference of a circle without advanced mathematical tools.The paper also provides error analysis,showing the range of errors for approximate division methods with different numbers of sides.The article summarizes the importance of the problem of dividing the circumference of a circle,emphasizing the subtlety of compass and straightedge drawing and the expansion of thinking.At the same time,the paper points out that there are still many methods waiting to be explored in the approximate division of the circumference,and proposes future research directions,that is,how to optimize and expand existing approximate division methods to achieve higher precision in dividing the circumference of a circle.展开更多
文摘This paper delves into the historical background,precise division drawing methods,and approximate division drawing methods of the circumference of a circle,and provides a comprehensive summary.The problem of dividing the circumference of a circle,as an important milestone in the history of mathematical development,not only demonstrates the progress of geometry but also has a profound impact on the formation of algebra,modern mathematics,and even contemporary abstract mathematics.The history of the problem of dividing the circumference of a circle can be traced back to ancient Greece,where mathematicians explored the division of the circle through geometric drawing using a straightedge and compass.One of the most famous methods is the approach proposed by Hippasus to approximate the circumference of a circle by increasing the number of sides of a regular polygon.However,mathematical proofs in the 19th century showed that certain division problems,such as the trisection of an angle,cannot be solved with just a straightedge and compass.With the development of algebra,the problem of dividing the circumference of a circle gradually transformed into solving radical equations.The thesis introduces in detail the precise drawing methods for dividing the circumference into three,six,four,eight,five,ten,and fifteen equal parts.For example,the trisection is achieved by constructing an inscribed equilateral triangle,while the hexasection is based on the trisection to further construct a regular hexagon.The drawing methods for the division into four and eight parts involve the perpendicular bisector of a circle and the bisection of the circumference.The drawing methods for the division into five and ten parts are based on the construction of an inscribed regular decagon,while the division into fifteen parts is achieved by constructing an inscribed regular pentadecagon.The paper further explores the approximate division of the circumference into seven and nine parts,as well as how to construct an approximate regular n-gon,such as a regular undecagon,using Tobi's method.Although these methods cannot achieve completely precise division,they provide a way to divide the circumference of a circle without advanced mathematical tools.The paper also provides error analysis,showing the range of errors for approximate division methods with different numbers of sides.The article summarizes the importance of the problem of dividing the circumference of a circle,emphasizing the subtlety of compass and straightedge drawing and the expansion of thinking.At the same time,the paper points out that there are still many methods waiting to be explored in the approximate division of the circumference,and proposes future research directions,that is,how to optimize and expand existing approximate division methods to achieve higher precision in dividing the circumference of a circle.
