a point group: the closure of the generators its Hermann-Mauguin symbol names, written in given axes

Syntax

pg = symmetry('m-3m')

By its international symbol.

pg = symmetry('D2')

By its Schoenflies symbol or an alias.

pg = symmetry(spaceId=225)

The point group of a space group.

pg = symmetry(pointId=45)

By its number in symmetry.pointGroups.

pg = symmetry('2', abg=[90*degree, 100*degree, 90*degree])

The monoclinic setting from the lattice angles.

pg = symmetry(rotation.byAxisAngle(zvector, 90*degree))

The group the elements generate.

Description

A group says nothing about where it acts: whether it describes a crystal or a specimen, and in which coordinate system, is the business of the frame carrying it, crystalFrame or specimenFrame, which builds its group from the same symbol. The elements are built from the symbol in the order of its generators, as orthogonal transforms, proper rotations and rotoinversions; nothing refers to an element by its position.

A symbol of a monoclinic group, ‘2’, ‘m’ or ‘2/m’, names no unique axis; with the lattice angles abg the unique axis is the one whose angle is not 90 degree, as MATLAB.

Input

`name`	str, the Hermann-Mauguin or Schoenflies symbol, a lattice name or an alias such as ‘orthotropic’; or rotation or orthogonalTransform, elements that generate the group; or a symmetry to copy; the triclinic group ‘1’ when omitted
`axes`	vector3d or an array (3, 3), the axes a, b, c the elements are written along; x, y, z when omitted

Keywords

keyword	default	description
`pointId`		the number of the group in symmetry.pointGroups, 1 to 47
`laueId`		the number of a Laue class, 1 to 11
`spaceId`		the number of a space group, 1 to 230, which gives its point group
`abg`		the lattice angles alpha, beta, gamma in radians, which choose the unique axis of a monoclinic group

Output

`pg`	symmetry, with pg.pointGroup its symbol, pg.rot its elements, pg.id its number

Members

calcAngleDistribution	the distribution of the misorientation angle: of a misorientation density, of a list of misorientations, or of the uniform density of two crystal frames
calcAxisDistribution	the axis distribution of an SO(3) function, an S2 function of the rotation axes symmetric under the rotations the two frames share
char	the symbol with a bar over every rotoinversion axis under utf8 output
conventionalSector	the conventional sector of the group, the one MTEX draws, with up and east choosing its placement
disjoint	return the intersection of two point groups in their current coordinate axes
elements	the rotational axes and their multiplicities, both signs, symmetry/elements
eq	Return self==value
factor	factor two groups using their common symmetry operations
fundamentalRegion	the fundamental region of orientations of this group and another one, the Dirichlet domain of the identity
fundamentalRegionEuler	the box in Euler angles that carries every orientation at least once (fundamentalRegionEuler): the two-folds perpendicular to z halve Phi, and one of the reflections they make on phi1 or phi2 halves that angle when it folds its turn about zero fundamentalRegionEuler(cs, ss): MATLAB’s name of the Euler plotting box
fundamentalSector	the fundamental sector of the group on the sphere, the region of the directions one symmetric copy of each falls into
id	the number of the group in the table of the point groups
isLaue	whether the group contains the inversion
isProper	whether the group holds proper rotations only
Laue	the smallest Laue group containing this one: every element once proper and once improper
LaueName	the Hermann-Mauguin symbol of the Laue group, the smallest one containing this group
maxAngle	maxAngle(cs), maxAngle(cs1, cs2): the largest rotation angle of the fundamental region of the frames, MATLAB’s maxAngle of a symmetry
multiplicityPerpZ	the highest order of a rotation axis of the group perpendicular to z, 1 if there is none
multiplicityZ	the highest order of a rotation axis of the group along z, 1 if there is none
name	the Hermann-Mauguin symbol of the group, as pointGroup
ne	Return self!=value
nfold	the maximal n-fold of the group, and about given axes, symmetry/nfold
numProper	the number of proper rotations of the group
numSym	the number of elements of the group
plot	plot(x, …) draws x as its kind is drawn
pointGroup	the Hermann-Mauguin symbol of the group, ‘unknown’ when its elements name no group
pointGroups	the table of the 32 point groups, the Hermann-Mauguin symbols with their Schoenflies names
properGroup	form a proper group from the rotational parts of all symmetry operations
properSubGroup	retain the symmetry operations whose determinant is +1
rotations	return the proper rotations already contained in the point group
Schoenflies	the Schoenflies symbol of the group, ‘unknown’ when its elements name no group
symmetry	a point group: the closure of the generators its Hermann-Mauguin symbol names, written in given axes
union	generate the smallest point group containing the operations of both inputs
WignerD	the Wigner-D matrices of rotations, or the Wigner-d matrices of a second Euler angle, of one degree or of all degrees up to a bandwidth

MATLAB’s symmetry_index

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/symmetry_index_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

The class symmetry