夏建国律师 论文作者吴怡凡韩心怡南京师范大学附属中学指导老师夏建国
Rn中正n-单形有理性的初等方法新探论文作者:吴怡凡韩心怡南京师范大学附属中学指导老师:夏建国南京师范大学2015.8标题:Rn中正n-单形有理性的初等方法新探摘要由于空间的局限性,人们对于高维空间的概念极为模糊,并常常无法想象出高维空间的构造。
然而在一些物理理论中,高维空间的研究有十分重要的应用。本文旨在依靠高中生现有的知识,研究高维空间内正n单形的有理性质,分别探究:(1)在n维空间中,存在至少一个其格点坐标全为有理数的正n-单形时n所满足的条件;(2)在n维空间中,存在至少一个其格点坐标和边长全为有理数的正n-单形时n所满足的条件。
本文在研究过程中使用了解析几何、矩阵、微积分等方法,有效地求出了正n单形的体积,并藉由体积完成了主要探究过程。
在文章的末尾还给出了符合条件顶点坐标和边长都有理的正8-单形的解。通过这个探究过程,本文分别得出(1)当n为偶数时,n维空间存在正n-单形各顶点坐标全为有理数的充要条件;(2)当n为奇数时,n维空间存在正n-单形各顶点坐标全为有理数的必要条件;(3)当n为偶数时,n维空间存在正n-单形各顶点坐标和边长全为有理数的必要条件;(4)当n为奇数时,n维空间存在正n-单形各顶点坐标全为有理数的必要条件;(5)正8-单形各顶点坐标和边长全为有理数的特殊解。
关键词:高维空间,正单形,有理AbstractBecauseofthelimitofspace,peopleareboundtohavelittleconceptofhigherdimensionsandarelikelytofinditdifficulttoimaginewhatdoeshigherdimensionexactlylooklike.
However,the1studyofhigherdimensionalwayshaveveryimportantapplicationsinsomephysicstheories.
Inthepaper,authorsusetheknowledgelearnedinhighschool,dealwiththerationalpropertyofregularn-simplexinhighdimen-sion.
Morespecifically,authorstalkabout:(1)conditionthatnmustsatisfywheninn-dimension,thereexistsatleastoneregularn-simplexwhosevertices’coordinatesareallrationalnumbers;(2)conditionthatnmustsatisfywheninn-dimension,thereexistsatleastoneregularn-simplexwhosevertices’coordinatesandwhoseedges’lengthareallrationalnumbers.
Inthispaper,authorsuseanalyticalgeometry,matrix,calculusetc.
tocalculatethevolumeofn-simplexandgainmostofthecon-clusionsthroughtheformulaofvolume.Attheendofthepaper,authorsgivethegeneralformulaoftheregularn-simplexeswhosevertices’coordinatesandwhoseedges’lengthareallrationalnum-bers.
Throughtheentireresearch,thepapergets:(1)whenniseven,thenecessaryandsufficientconditionthatnmustsatisfywheninn-dimension,thereexistsatleastoneregularn-simplexwhosevertices’coordinatesareallrationalnumbers;(2)whennisodd,theneces-saryconditionthatmustbesatisfiedwheninn-dimension,thereexistsatleastoneregularn-simplexwhosevertices’coordinatesareallrationalnumbers;(3)thenecessaryconditionthatnmustsat-isfywheninn-dimension,thereexistsatleastoneregularn-simplexwhosevertic
