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:
Orthorhombic disphenoid (shown viewed along 2-fold axis and as inscribed in rectangular prism.)
4 faces. Also minimal
Orthorhombic fan: two or more tetragonal disphenoids side by side.
3k + 2 faces for k disphenoids.
|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:
|(Same as 6*)||3/m|
|(Same as 6* 2/m)||3/m m|
Skew trigonal trapezohedron
Rhombohedron. Also minimal and equilateral
|4/m 2/m 2/m
Skew tetragonal trapezohedron
Tetragonal disphenoid. Also minimal
|6/m 2/m 2/m
Skew hexagonal trapezohedron
Trigonal dipyramid. If faces equilateral (two tetrahedra base to base) then crystal is equilateral and regular-faced.
|4/m 3* 2/m
Cube. Also equilateral, minimal and regular-faced. Other isohedral examples are the octahedron and rhombic dodecahedron, and all closed forms of this class
Non-regular pentagonal dodecahedron (pyritohedron)
Tetrahedron. Also equilateral, minimal and regular-faced. Other isohedral examples are all closed forms of this class
Created 31 July 2001, Last Update 14 December 2009
Not an official UW Green Bay site