City Research Online

Abstract

A convenient rectangular cartesian coordinate system is constructed for the regular icosahedron. This allows us to write down the coordinates of every vertex, so enabling the direction-ratios of all the symmetry axes to be obtained. Our results afford a fresh approach to various geometrical features of the icosahedron and of the icosahedral group.

The circumsphere of a regular icosahedron may be converted into possible models for the spherical carbon molecule C60 (the buckyball), by expanding the 12 vertex points into 12 regular spherical pentagons of equal size. We examine quantitatively the changing pentagonal-hexagonal pattern as the pentagon size expands up to the possible maximum, at which stage the circumsphere is entirely covered with pentagons i.e. is the circumsphere of a regular dodecahedron. Arguments are given for a unique choice model which conforms to the chemical data.

Prolate spheroid models for C70 and C80 are also considered.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics Q Science > QD Chemistry
Departments:	Doctoral Theses School of Science & Technology > School of Science & Technology Doctoral Theses School of Science & Technology

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