Best Representations of 3-Dimensional Symmetry: Regular polygon Faces

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

What Do We Mean By "Best?"

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:

Minimum number of faces
Isohedral solids: faces all the same shape (obviously desirable for building cardboard models)
Equilateral solids: all edges the same length
Regular polygon faces: best for aesthetic appeal but it is known that not all crystal classes can be represented this way, at least not by convex models.

Triclinic

	1
	1*

Monoclinic

Augmented Sphenocorona (J87)

17 faces

2/m

Orthorhombic

222

Augmented triangular prism (J49)

8 faces

2/m 2/m 2/m

Bilunabirotunda (J91)

14 faces

Uniaxial Classes

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:

N: A single major symmetry axis alone
N/m: Symmetry axis perpendicular to a mirror plane
Nm: Symmetry axis with mirror planes intersecting along the axis
N/m m: Symmetry axis perpendicular to a mirror plane, and mirror planes intersecting along the axis. A combination of N/m and Nm. This is the holosymmetric class: it contains all the other symmetries as subsets
N2: Symmetry axis with 2-fold axes perpendicular to it
N*: N-fold rotoinversion axis
N*2m: N-fold rotoinversion axis with 2-fold axes perpendicular to it and with mirror planes intersecting along the N-fold axis

Trigonal - Rhombohedral

	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

Tetragonal

	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

Hexagonal

	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

Isometric

	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. 6 faces.
	2/m 3*
	4* 3m Tetrahedron. Also isohedral, minimal and regular-faced. The truncated tetrahedron (4 triangles and 4 hexagons) is another example. 4 faces
	432 Snub cube (6 squares, 32 triangles). Also regular-faced
	23

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Created 31 July 2001, Last Update 11 June 2020