Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 2
T. M. Madden:
A Classification of Perfect 4-Solids
Abstract
The notion of regularity of polytopes was generalized to that of perfection
in [9] by Robertson. A polytope $P$ is said to be {\it perfect}\/ if it has
maximal symmetry properties in the sense that $P$ cannot be deformed without
changing its `shape' or its symmetry group. In [3], the study of symmetry of
convex polytopes was extended by Farran and Robertson to convex solids in
general (in other words, compact convex bodies). Thus we have the notion of a
regular solid and a perfect solid. Recently, the family of regular solids
were classified by Madden and Robertson [5]. However, the more general problem
of classifying the perfect solids remains unsolved. In fact, not even the
perfect polytopes have been classified. This paper gives a classification of
the perfect 4-solids, confirming a conjecture of Rostami [12].
In section 0, we explain the terminology and key ideas. For general
background, see Robertson [8]. We introduce the notion of a deformation of a
solid (and in particular of a polytope) in section 1. A polytope $P$ is
perfect if and only if all deformations of $P$ are similar to $P$. The
properties of the fixed point set of a vertex of a perfect polytope are
investigated in section 2. It turns out that this set is one-dimensional.
The conditions for which a perfect polytope is regular are determined in
section 3. In section 4, the orbit vectors of a perfect polytope $P$ provide
useful information about transitivity properties of $GP$, leading to a
classification of perfect 4-polytopes. We conclude with a discussion of the
perfect 4-solids in section 5.
The classification in dimensions $n\geq 5$ remains unresolved. In the cases
$n\geq 6$, there is a problem in that perfect polytopes are known which are
not associated (in terms of their symmetry group) to any regular polytope.
Such polytopes are derived from the root systems $E_6,\, E_7$ and $E_8$.
MSC 1991: 53C35, 52C07, 51F15