Symmetry groups — symmetry

Frequently orientations are subject to some symmetry group \(\mathbb{G}\), which is a finite set of rotations (satisfying certain properties). For any rotation \(Q\) in \(\mathbb{G}\), the rotation matrix \(Q \cdot R\) represents the same orientation as \(R\) does.

Usage

symmetry_group(
  group = c("triclinic", "ray_in_plane", "line_in_plane", "monoclinic",
    "trigonal-trapezohedral", "hexagonal-trapezohedral", "trivial")
)

Arguments

group: character (symmetry class) or integer number of the enantiomorphic point group. See table below for details. Also accepts ""ray_in_plane" (equivalent to triclinic symmetry) and "line_in_plane" (monoclinic).

Value

list of rotation parameters

Details

Symmetry	Enantiomorphic Point Group	Example
`triclinic`	1	Ray in plane, plagioclase
`monoclinic`	2	Line in plane, orthoclase, gypsum, muscovite, clinopyroxene, clinoamphibole
`orthorhombic`	222	olivine, aragonite, marcasite, orthopyroxenes
`tetragonal`	4	Pyramidal: zircon
	422	Trapezohedral
`trigonal`	3	Pyramidal, Rhombohedral
	32	Trapezohedral: \(\alpha\)-Quartz
`hexagonal`	6	Pyramidal, Rhombohedral
	622	Trapezohedral: \(\beta\)-Quartz
`cubic`	23	Tetartoidal
	432	Hexoctahedral: galena, pyrite, fluorite

Examples

symmetry_group("triclinic")
#> [[1]]
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> [[2]]
#>      [,1] [,2] [,3]
#> [1,]   -1    0    0
#> [2,]    0    1    0
#> [3,]    0    0   -1
#> 

symmetry_group(2)
#> [[1]]
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> [[2]]
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0   -1    0
#> [3,]    0    0   -1
#> 
#> [[3]]
#>      [,1] [,2] [,3]
#> [1,]   -1    0    0
#> [2,]    0    1    0
#> [3,]    0    0   -1
#> 
#> [[4]]
#>      [,1] [,2] [,3]
#> [1,]   -1    0    0
#> [2,]    0   -1    0
#> [3,]    0    0    1
#>