二十面体壳层结构──纪念C_（60）发现10周年

Icosahedral Shell Structures──Geodesic Domes,Virus,C60 and Fullerenes

圆顶建筑、病毒衣壳和Ｃ＿（６０）分子笼的生成机制和尺寸差别很大，但都属二十面体壳层，遵循同一几何规律。圆顶建筑和病毒衣壳由近似全等的三角面构成，有１２个五键连３￣５顶，１０（Ｔ－１）个六键连３￣６顶，其中三角面数Ｔ＝ｈ￣２＋ｈｋ＋ｋ２是二十面体的一个三角面中的小三角面数，ｈ，ｋ是六角坐标系中３￣５顶的坐标。另一方面，全碳分子笼Ｃ＿ｎ（包括其原型Ｃ６０）由１２个五角面与１０（Ｔ－１）个六角面围成，全是三键连顶（ｈ，ｋ是一个五角面中心的坐标）。显然，这两类二十面体壳层是对偶的，也就是它们的面、顶有互换关系。本文讨论这些二十面体壳层的几何特征及富勒烯的多样性，强调不同学科间的联系和科学研究的继承与发展的关系。

Though the dimensions and formation mechanisms of geodesic domes,theprotein capsids of virus,and the C60 molecular cage vary significantly,they all belong tothe same category of the icosahedral shells and consequently follow the same geometricalrules,Geodesic domes and virus capsids consist of triangular faces of almost the same size(quasi-equivalence)forming 12 penta-connected 3 5 and 10(T-1)hexa-connected 3 6 ver-tices,where T=h2+hk+k2 is the triangular number or the number of small triangles in anicosahedral face,and h,k are the hexagonal coordinates between two 3 5 vertices. On theother hand,the fullerene Cn,including the archetype C60,consists of 12 pentagonal facesand 10(T-1)hexagonal faces forming only tri-connected vertices,now h,k become thehexagonal coordinates between the centers of two pentagonal faces,Obviously,these twokinds of icosahedral shells are duals with each other,or,in other words,their faces andvertices are interchanged.The geometrical characteristics of these icosahedral shells arediscussed and the great variety of fullerenes is emphasized.