Steven Dutch, Natural and Applied Sciences,
University of Wisconsin  Green Bay
Firsttime Visitors: Please visit Site Map and Disclaimer. Use "Back" to return here.
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
General tetrahedron. It is a remarkable fact that fully eight of the 32 crystal classes can be represented by tetrahedra. 4 faces 

1*
General oblique parallelepiped 6 faces 
2
Tetrahedron with 2fold axis 4 faces 

m
Tetrahedron with mirror plane 4 faces 

2/m
General monoclinic prism 6 faces 
222
Orthorhombic disphenoid 4 faces. Also isohedral 

mm
Tetrahedron with two mirror planes 4 faces 

2/m 2/m 2/m
General rectangular parallelepiped 6 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
Two skew dissimilar pyramids 6 faces. 

3m
Trigonal pyramid with flat base 4 faces 

(Same as 6*)  3/m 
(Same as 6* 2/m)  3/m m 
32
Trigonal trapezohedron 6 faces. Also isohedral 

3*
Trapezohedron with skew prism faces 12 faces 

3*2m
Rhombohedron. Also isohedral and equilateral. 6 faces. 
4
Two skew dissimilar pyramids 8 faces. 

4/m
Tetragonal prism with second set of skew faces. Angle A not equal to B and sets of faces not congruent. 10 faces 

4mm
Tetragonal pyramid 5 faces 

4/m 2/m 2/m
Square prism 6 faces 

422
Tetragonal trapezohedron 8 faces. Also isohedral 

4*
Skew truncated prism 8 faces 

4* 2/m
Tetragonal disphenoid 4 faces. Also isohedral 
6
Two skew dissimilar pyramids 12 faces. 

6/m
Hexagonal prism with second set of skew faces. Angle A not equal to B and sets of faces not congruent. 14 faces 

6mm
Hexagonal pyramid 7 faces 

6/m 2/m 2/m
Hexagonal prism 8 faces. 

622
Hexagonal trapezohedron 12 faces. Also isohedral 

6* (same as 3/m)
Triangular prism with second set of skew faces. Angle A not equal to B and sets of faces not congruent. 8 faces 

6* 2/m
Trigonal prism. Also equilateral and regularfaced. 5 faces 
4/m 3* 2/m
Cube. Also isohedral, equilateral and regularfaced 6 faces 

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

4* 3m
Tetrahedron. Also isohedral, equilateral and regularfaced. 4 faces 

432
Gyroid. Also isohedral. 24 faces 

23
Tetartoid. Also isohedral. 12 faces 
Return to Symmetry Index
Return to Professor Dutch's Home page
Created 31 July 2001, Last Update 14 December 2009
Not an official UW Green Bay site