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
Augmented Sphenocorona (J87) 17 faces 

2/m 
222  
mm
Augmented triangular prism (J49) 8 faces 

2/m 2/m 2/m
Bilunabirotunda (J91) 14 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
Elongated trigonal prism 7 faces 

(Same as 6*)  3/m 
(Same as 6* 2/m)  3/m m 
32
Gyroelongated Triangular Bicupola (J44) 26 faces 

3*  
3*2m
Elongated Triangular Gyrobicupola (J36). Eliminating the central prism results in a cuboctahedron. 20 faces 
4  
4/m  
4mm
Elongated square pyramid. Also regular faced. 9 faces 

4/m 2/m 2/m
Elongated octahedron. Also regular faced. 12 faces 

422
Gyroelongated square bicupola (J45) 34 faces 

4*  
4* 2/m
Gyrobifastigium. Also regular faced. 8 faces 
6  
6/m  
6mm  
6/m 2/m 2/m
Hexagonal Prism (equilateral) 8 faces 

622  
6*  
6* 2/m
Trigonal prism. Also minimal 5 faces 
4/m 3* 2/m
Cube. Also isohedral, minimal and regularfaced. Other equilateral examples are the octahedron and rhombic dodecahedron, and all Archimedean polyhedra with cubic symmetry. 6 faces. 

2/m 3*  
4* 3m
Tetrahedron. Also isohedral, minimal and regularfaced. The truncated tetrahedron (4 triangles and 4 hexagons) is another example. 4 faces 

432
Snub cube (6 squares, 32 triangles). Also regularfaced 

23 
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Created 31 July 2001, Last Update 14 December 2009
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