Abstract:
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.