- Title
- Symmetrical polyhedra
- Creator
- Lurie, Jos
- Subject
- symmetrical polyhedra, crystallography, platonic solids, archimedean polyhedra
- Date
- 2008-09-23T09:53:32Z
- Type
- Book
- Identifier
- uj:1684
- Identifier
- http://hdl.handle.net/10210/1036
- Description
- Much of the material of this book was prepared over a period commencing more than a decade ago and, while a few instances have considered publishing it commercially, the cost in relation to the potential market have been the reason for not implementing this. Over the centuries philosophers and mathematicians have been fascinated by regular polyhedra. Those that have attracted particular attention are essentially isometric1 with high symmetry. It is these and related forms that are largely dealt with in this book. Mathematics necessarily demands a rigid proof of a proposition and a clear distinction between observational evidence and watertight verification. Typical was the proposed solution following three centuries of mathematical endeavour of the close packed spheres problem. Stated simply: what volume is occupied by space in the closest packing of identical solid spheres? Professor Hsiang required one hundred pages of tricky geometry to produce a mathematical solution (apparently not universally accepted) to a problem which the author faced in calculating the theoretical maximum porosity of close-packed equal-sized spheres for an engineering geology text. Doubtless the problem, from different viewpoints, has been faced by others. A practical solution (without mathematical proof was obtained in two hours and using a few lines of simple calculations by converting it into a polyhedral problem! The author was unaware that Kepler had approached the problem originally in this way. Crystallographers are concerned only with those polyhedra whose external form is prescribed by a three dimensional repeating pattern of molecular groups. Excluded is five-fold symmetry and thus consideration of a host of most beautiful polyhedra. Furthermore, only three true stellations are encountered among the crystallographically possible polyhedra. Also, since the development of Xray diffraction, crystallographers have focussed mainly on the internal arrangement patterns of atomic components and interest in external crystal morphology has declined considerably. Through career involvement in mineralogy, chemistry, geology, gemmology and engineering the author was struck by the recurrence in these disciplines of polyhedral phenomena. Perspectives are different but inevitably there is a remarkable convergence when following a particular aspect.
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