Symmetrical polyhedra

**Authors:**Lurie, Jos**Date:**2008-09-23T09:53:32Z**Subjects:**symmetrical polyhedra , crystallography , platonic solids , archimedean polyhedra**Type:**Book**Identifier:**uj:1684 , 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.**Full Text:**false

**Authors:**Lurie, Jos**Date:**2008-09-23T09:53:32Z**Subjects:**symmetrical polyhedra , crystallography , platonic solids , archimedean polyhedra**Type:**Book**Identifier:**uj:1684 , 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.**Full Text:**false

Lurie and Ponelat's catalogue of symmetrical polyhedra

**Authors:**Lurie, Jos**Date:**2008-09-23T09:53:05Z**Subjects:**symmetrical polyhedra , platonic solids , archimedean polyhedra , crystallography**Type:**Book**Identifier:**uj:1683 , http://hdl.handle.net/10210/1035**Description:**The scope of this catalogue is more-or-less confined to the most symmetrical polyhedra exemplified by the socalled Platonic solids (the five convex forms each of which consists ofaset of identical regular polygon faces) and their symmetry associates including the Archimedean polyhedra. The five solids are the tetrahedron, the hexahedron (cube), the octahedron, the dodecahedron and the icosahedron. These fall into three symmetry groups: tetrahedral, octahedral and icosahedral. The seven members of the last two groups, together with a combination of all, are given on page iv. Because of its relatively low symmetry the tetrahedral group receives somewhat cursory attention. The symmetrical polyhedra described are by no means exhaustive - even with the constraint of considering only the most symmetrical ones there are, in fact infinite possibilities. However, examples produced using several techniques are presented here and these processes (especially producing successive generations) can be employed to produce ever more obscure but highly symmetrical polyhedra. The first contributor to this catalogue had been trained as a draughtsman and had studied crystallography and, having encountered a regular pentagonal dodecahedron for the first time managed, without prior knowledge of them, to produce drawings, applying basic crystallographic principles, of all the Archimedean solids (except the two "snub" forms). The seven forms ofthe icosahedral symmetry group were also produced. Many other symmetrical polyhedra were also "discovered" before being introduced to the Cundy and Rollett classic "Mathematical Models". The logo on the cover of this catalogue was produced by using a stereogram and following Penfield's description but manual draughting of the more complex forms is hugely problematic and the second contributor's role in producing these by computer became indispensable. The computerised portion of the material ofthis catalogue was implemented by Sven Ponelat between October 1993 and April 1997 with the use of an Autocad programme. It largely involved techniques that at the time, had not been used before and, as far as can be established, are little known at present. Fundamentally, it involved utilising the symmetry of a given polyhedron to generate further positions of the polyhedron which can be unioned together. Provided all the components of a given symmetry element are utilised, the resulting compound retains the full symmetry of the starting polyhedron. Thus, partial utilisation of a symmetry element which produces lower symmetry forms is largely omitted. The analysis of the intersections of the compounds generated in terms of their combined convex forms is a new technique apparently. The first author has continued to produce forms up to the present (2007) such as the duals of some forms which have been executed, largely manually, and to systematise the study. Besides utilising a fixed orientation, all combinations and compounds have been rendered in colour to simplify interpretation and comparisons. The analysing of intersections in terms of the components of the combinations so produced apparently has notbeen attempted before.**Full Text:**false

**Authors:**Lurie, Jos**Date:**2008-09-23T09:53:05Z**Subjects:**symmetrical polyhedra , platonic solids , archimedean polyhedra , crystallography**Type:**Book**Identifier:**uj:1683 , http://hdl.handle.net/10210/1035**Description:**The scope of this catalogue is more-or-less confined to the most symmetrical polyhedra exemplified by the socalled Platonic solids (the five convex forms each of which consists ofaset of identical regular polygon faces) and their symmetry associates including the Archimedean polyhedra. The five solids are the tetrahedron, the hexahedron (cube), the octahedron, the dodecahedron and the icosahedron. These fall into three symmetry groups: tetrahedral, octahedral and icosahedral. The seven members of the last two groups, together with a combination of all, are given on page iv. Because of its relatively low symmetry the tetrahedral group receives somewhat cursory attention. The symmetrical polyhedra described are by no means exhaustive - even with the constraint of considering only the most symmetrical ones there are, in fact infinite possibilities. However, examples produced using several techniques are presented here and these processes (especially producing successive generations) can be employed to produce ever more obscure but highly symmetrical polyhedra. The first contributor to this catalogue had been trained as a draughtsman and had studied crystallography and, having encountered a regular pentagonal dodecahedron for the first time managed, without prior knowledge of them, to produce drawings, applying basic crystallographic principles, of all the Archimedean solids (except the two "snub" forms). The seven forms ofthe icosahedral symmetry group were also produced. Many other symmetrical polyhedra were also "discovered" before being introduced to the Cundy and Rollett classic "Mathematical Models". The logo on the cover of this catalogue was produced by using a stereogram and following Penfield's description but manual draughting of the more complex forms is hugely problematic and the second contributor's role in producing these by computer became indispensable. The computerised portion of the material ofthis catalogue was implemented by Sven Ponelat between October 1993 and April 1997 with the use of an Autocad programme. It largely involved techniques that at the time, had not been used before and, as far as can be established, are little known at present. Fundamentally, it involved utilising the symmetry of a given polyhedron to generate further positions of the polyhedron which can be unioned together. Provided all the components of a given symmetry element are utilised, the resulting compound retains the full symmetry of the starting polyhedron. Thus, partial utilisation of a symmetry element which produces lower symmetry forms is largely omitted. The analysis of the intersections of the compounds generated in terms of their combined convex forms is a new technique apparently. The first author has continued to produce forms up to the present (2007) such as the duals of some forms which have been executed, largely manually, and to systematise the study. Besides utilising a fixed orientation, all combinations and compounds have been rendered in colour to simplify interpretation and comparisons. The analysing of intersections in terms of the components of the combinations so produced apparently has notbeen attempted before.**Full Text:**false

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