Best Representations of 3-Dimensional Symmetry: Equilateral

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

Triclinic

1

Triangular prism with equilateral ends, cut obliquely to base and  prism axis. Prism edges equal triangle edges.

5 faces

1*

Rhombic-faced parallelepiped with all angles unequal and not equal to 90 degrees

6 faces

Monoclinic

2

Triangular prism with equilateral ends, cut normal to base and obliquely to prism axis. Prism edges equal triangle edges.

5 faces

m

Triangular prism with equilateral ends, cut obliquely with base of triangle normal to prism axis. Prism edges equal triangle edges.

5 faces

2/m

Rhombic prism obliquely truncated.

6 faces.

Orthorhombic

  222
mm

Roof prism (compound of cube and triangular prism)

7 faces

2/m 2/m 2/m

Rhombic prism with length equal to rhomb edges

6 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

Highly symmetric classes are fairly easy to represent as equilateral solids but classes lacking mirror planes are quite a bit harder. One (small) class can be called skew cupolas.

At left is a regular triangular cupola. Cupolas can also be square or pentagonal. However, for hexagons the construction simply yields a flat tesselation, so  only 3-, 4-, and 5-fold cupolas exist.
We can construct a cupola by starting with a tetrahedron and pulling the faces apart, maintaining 3-fold symmetry. If we pull perpendicular to the edges we get the regular cupola above. If we pull at some other angle, we get a skew cupola as at left. The altitude of the solid is the same whether the cupola is regular or skew. We can construct 4- and 5- fold cupolas the same way starting with square or pentagonal pyramids. Again, the altitude of the solid is the same whether the cupola is regular or skew. 

However, there is a class of equilateral solids, the zonohedra, that can be a basis for finding equilateral solids of desired uniaxial symmetry, using methods analogous to those for building skew cupolas.

Zonohedra have the property that all edges belong to one of a few sets of parallel lines. If the zonohedron has polar symmetry, it follows that the edges are all equal and the faces are all rhombi.

At left is a zonohedron with 9-fold symmetry. We can see it has mirror planes and equatorial 2-fold axes, so it has symmetry 9*2m.

We can truncate the zonohedron across any plane of vertices normal to the axis. We get a solid with a regular polygon base. 

Changing the axial proportions of the solid does not affect the parallellism or equality of the edges. Thus we can rescale the solid so that the triangular faces become equilateral. We thus get a solid with 9m symmetry. We can combine two solids base to base to get a solid with 9/m m symmetry. For large N we may have to include several bands of rhombuses. 

We can then separate faces as we did in creating skew cupolas. At left is a polar view of the polyhedron above. At right is the same solid expanded along the red lines, which are equal in length to all the other edges.

The polar vertex becomes a regular 9-gon and the base becomes a non-regular 18-gon. The overall solid has 9-fold symmetry.

Thus we can create equilateral n-fold shapes. We can combine two base to base to obtain n/m symmetry or rotate one with respect to the other to obtain n22 symmetry. 

Trigonal - Rhombohedral

3

Skew triangular cupola. Start with a tetrahedron and pull the faces apart, maintaining 3-fold symmetry. The altitude of the solid is the same whether the cupola is regular or skew.

8 faces.

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

General rhombohedron. Also minimal and isohedral.

6 faces

Tetragonal 

4

Skew square cupola. Start with a square pyramid and pull the faces apart, maintaining 4-fold symmetry. The altitude of the solid is the same whether the cupola is regular or skew.

10 faces.

4/m

Paired skew square cupolas. 

18 faces

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*

Paired skew triangular cupolas

14 faces

6* 2/m

Trigonal prism. Also minimal.

6 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*

Octahedron with vertices truncated by pairs of equilateral triangles such that the truncated octahedron faces become equilateral (but non-regular) hexagons. Although this solid looks flattened, its dimensions along symmetry axes are identical.

20 faces

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 14 December 2009

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