I was first inspired to consider this problem while building a set of solids to represent all 32 crystal classes (most commercial model sets omit the less common and less symmetrical classes). I soon realized the answers were different depending on whether I was cutting models out of wood or building them out of cardboard, and whether I was thinking of ease of construction or aesthetic appeal.
Some reasonable possible definitions of "best" include:
Augmented Sphenocorona (J87)
Augmented triangular prism (J49)
|2/m 2/m 2/m
The uniaxial classes which have a single major symmetry axis and additional twofold axes or mirror planes all have certain features in common. For each group, whether trigonal, tetragonal or hexagonal, there are seven possible classes (but some turn out to be degenerate). These can all be derived by taking one of the seven strip space groups and wrapping it around a cylinder. If N is the degree of symmetry, we have:
Elongated trigonal prism
|(Same as 6*)||3/m|
|(Same as 6* 2/m)||3/m m|
Gyroelongated Triangular Bicupola (J44)
Elongated Triangular Gyrobicupola (J36). Eliminating the central prism results in a cuboctahedron.
Elongated square pyramid. Also regular faced.
|4/m 2/m 2/m
Elongated octahedron. Also regular faced.
Gyroelongated square bicupola (J45)
Gyrobifastigium. Also regular faced.
|6/m 2/m 2/m
Hexagonal Prism (equilateral)
Trigonal prism. Also minimal
|4/m 3* 2/m
Cube. Also isohedral, minimal and regular-faced. Other equilateral examples are the octahedron and rhombic dodecahedron, and all Archimedean polyhedra with cubic symmetry.
Tetrahedron. Also isohedral, minimal and regular-faced. The truncated tetrahedron (4 triangles and 4 hexagons) is another example.
Snub cube (6 squares, 32 triangles). Also regular-faced
Created 31 July 2001, Last Update 14 December 2009
Not an official UW Green Bay site