### Abstract:

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.