正多面体的一类定值  

A Class of Fixed Values for Regular Polyhedron

作　　者：李凤清[1] 徐志军[2] LI Fengqing;XU Zhijun(School of Teacher Education,Sichuan Vocational and Technical College,Suining Sichuan,629000,China;School of Digital Culture and Tourism,Sichuan Vocational and Technical College,Suining Sichuan,629000,China)

机构地区：[1]四川职业技术学院教师教育学院,四川遂宁629000 [2]四川职业技术学院数字文旅学院,四川遂宁629000

出　　处：《四川职业技术学院学报》2024年第6期163-168,共6页Journal of Sichuan Vocational and Technical College

摘　　要：本文给出正十二面体与正二十面体的一个性质:“设正多面体A_(1)A_(2)A_(3)…A_(V)的中心为O,顶点数为V,P为异于O的任一点,记θ_(i)=∠POA_(i)(i=1,2,3,…,V),那么i=1∑^(V)cosθ_(i)=0,i=1∑^(V)cos^(2)θ_(i)=V/3.且当V≥6时有i=1∑^(V)cos^(3)θ_(i)=0,当V≥12时有i=1∑^(V)cos^(4)θ_(i)=V/5.”并得到一个推论:中心位于空间直角坐标系原点且外接球半径为的正十二面体或正二十面体A_(1)-A_(2)A_(3)…A_(v-1)-A_(v)的顶点数为V,A_(i)(x_(i),y_(i),z_(i)…,V),m为正奇数.则i=1∑^(V)x_(i)^(2)=i=1∑^(V)y_(i)^(2)=i=1∑^(V)z_(i)^(2)=VRi=1∑^(V)x_(i)^(2)/3;i=1∑^(V)x_(i)^(4)=i=1∑^(V)y_(i)^(4)=i=1∑^(V)z_(i)^(4)=VR^(4)/5;i=1∑^(V)x_(i)^(m)=i=1∑^(V)y_(i)^(m)=i=1∑^(V)z_(i)^(m)=0.This article presents a property of regular dodecahedron and regular icosahedron:"Suppose the center of a regular polyhedron A_(1)A_(2)A_(3)…A_(V)is O,the number of vertices is V,and P is an any point that is different from O,ifθ_(i)=∠POA_(i)(i=1,2,3,…,V),then,i=1∑^(V)cosθ_(i)=0,i=1∑^(V)cos^(2)θ_(i)=V/3,and when V≥6,i=1∑^(V)cos^(3)θ_(i)=0,when V≥12,i=1∑^(V)cos^(4)θ_(i)=V/5.”Then an inference is drawn:a regular dodecahedron or icosahedron A_(1)-A_(2)A_(3)…A_(v-1)-A_(v)whose center is at the origin of a rectangular coordinate system in space and whose outer sphere radius is R,has the number of vertices,A_(i)(x_(i),y_(i),z_(i)…,V),and is positive odd,then i=1∑^(V)x_(i)^(2)=i=1∑^(V)y_(i)^(2)=i=1∑^(V)z_(i)^(2)=VRi=1∑^(V)x_(i)^(2)/3;i=1∑^(V)x_(i)^(4)=i=1∑^(V)y_(i)^(4)=i=1∑^(V)z_(i)^(4)=VR^(4)/5;i=1∑^(V)x_(i)^(m)=i=1∑^(V)y_(i)^(m)=i=1∑^(V)z_(i)^(m)=0.

关 键 词：正多面体 定值 正五棱锥 

分 类 号：O182[理学—数学]