Steven Dutch, Natural and Applied Sciences,
University of Wisconsin  Green Bay
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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:
1  
1* 
2  
m  
2/m 
222
Orthorhombic disphenoid (shown viewed along 2fold axis and as inscribed in rectangular prism.) 4 faces. Also minimal 

mm
Orthorhombic fan: two or more tetragonal disphenoids side by side. 3k + 2 faces for k disphenoids. 

2/m 2/m 2/m
Orthorhombic dipyramid 8 faces 
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:
3  
3m  
(Same as 6*)  3/m 
(Same as 6* 2/m)  3/m m 
32
Skew trigonal trapezohedron 6 faces 

3*  
3*2m
Rhombohedron. Also minimal and equilateral 6 faces 
4  
4/m  
4mm  
4/m 2/m 2/m
Tetragonal dipyramid 8 faces 

422
Skew tetragonal trapezohedron 8 faces 

4*  
4* 2/m
Tetragonal disphenoid. Also minimal 4 faces. 
6  
6/m  
6mm  
6/m 2/m 2/m
Hexagonal dipyramid 12 faces 

622
Skew hexagonal trapezohedron 12 faces 

6*  
6* 2m
Trigonal dipyramid. If faces equilateral (two tetrahedra base to base) then crystal is equilateral and regularfaced. 6 faces 
4/m 3* 2/m
Cube. Also equilateral, minimal and regularfaced. Other isohedral examples are the octahedron and rhombic dodecahedron, and all closed forms of this class 6 faces. 

2/m 3*
Nonregular pentagonal dodecahedron (pyritohedron) 12 faces 

4* 3m
Tetrahedron. Also equilateral, minimal and regularfaced. Other isohedral examples are all closed forms of this class 4 faces 

432
Gyroid 24 faces 

23
Tetartoid 12 faces 
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Created 31 July 2001, Last Update 14 December 2009
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