HermannMauguin Symmetry Symbols
Last Updated: 11th Aug 2020By Don Peck & Erin Delventhal
You may want to review the Mindat Article, The Symmetry of Crystals: An Introduction, before studying this article.
In 1830, Johann Hessel proved mathematically that, since crystals exhibited only 2, 3, 4, and 6 fold symmetry and as a result of Rene Hauy's Law of Rational Intercepts, there are only 32 possible combinations of symmetry elements, and therefore only 32 possible classes of crystals. In 1928 Carl Hermann presented his notation for relating symmetry elements to crystals. CharlesVictor Mauguin modified it a few years later and the HermannMauguin Notation System became widely used. Today it is known, also, as the International System of Notation. HermannMauguin (HM) Symmetry Symbols are used to convey in a concise manner the symmetry of the 32 crystal classes.
The symbols for each class are known, also, as a Point Group. The HM symbols may seem a bit confusing at first but as one increasingly develops facility in recognizing the symmetry of crystals, HM becomes more understandable and more helpful.
The HM Code
1. The HM symbols consist of some combination of numerals 1, 2, 3, 4, and 6; numerals with a barover 1, 3, 4, and 6;; the letter m; and a slash, /.
2. The numerals 1, 2, 3, 4, and 6 indicate an axis of rotational symmetry (usually referred to as an axis of rotation). For example, 3 indicates that a rotation of 120^{o} brings the crystal into coincidence with its original position in space and this occurs 3 times in a rotation of 360^{o}. It is a 3fold axis of rotational symmetry. Similarly, 2, 4,, and 6 indicate axes of 2fold, 4fold, or 6fold rotation, A 1 symbolizes "no symmetry".
3. The numerals with a barover show an axis of rotatory inversion, usually called an axis of inversion. A 4 means that a rotation of 90^{o} followed by an inversion through the center brings the crystal to occupy the same space as at the start. Four such operations brings the crystal to its original position (Operationally, turn crystal 90^{o} to the right, then invert top to bottom clockwise. Four times returns the crystal to its original position). In reality, all points on the surface of a crystal rotate n degrees and invert through the center (somewhat like turning the crystal insideout) but this is physically impossible. Thus operationally, we follow a single point, such as a corner. 1 is one complete rotation plus an inversion through the center and merely indicates a center of symmetry.
4. The letter m, indicates a plane of symmetry, usually referred to as a mirror plane. A mirror plane divides a body (crystal) such that the half on one side of the plane is the mirror image of the half on the opposite side.
5. A slash, /, means "perpendicular to". Thus, 2/m means "a 2fold axis of rotation perpendicular to a mirror plane".
6. Two symbols in succession means "parallel to". 3m (two elements: same as 3 m) denotes a 3fold axis parallel to 3 mirror planes. Given a 3fold axis parallel to (coincident with) one mirror plane, the 3fold symmetry requires that there be three mirror planes at 120^{o} angles to each other and with each of the three planes divided by the axis. The axis of rotation lies in the three mirror planes
The HM Symbols
HermannMauguin Symbols have 1, 2 , or 3 elements. The first element usually refers to the principle axis of rotation. The second element is usually for secondary axes of rotation and/or mirror planes; and the third element is for remaining symmetry. The principle axis of rotation is most often coincident with the caxis. The major exception is in the monoclinic system where it is the baxis.
For example, the HM symbol for the Monoclinic Prismatic class is 2/m. It indicates that the baxis is a 2fold axis of rotation and that the axis is perpendicular to a mirror plane. The HM symbol for the Hexagonal pyramidal class is 6 (a single element), meaning that the caxis is a 6fold axis of rotation and there is no other symmetry. 32 (same as 3 2) is a 3fold axis coincident with the caxis and three 2fold axes each coincident with the 3 aaxes. The three fold axis requires that given one horizontal axis or plane there must be three of them. 4/m 3 2/m) is for the hexoctahedral class in the isometric system. The first element is for the three 4fold axes of rotation, each perpendicular to a mirror plane. The second element is for the four 3fold axes that are the body diagonals of the cube (why four of them? The 4fold axes require them.) And the third element is for the 2fold axes (edge to edge diagonals) each perpendicular to a mirror plane.
HM Symbols for Each Crystal Class
Each Crystal Class has its own unique symbols, or point group. We will consider them individually. In the following tables the crystal systems and the classes are in ascending order of symmetry.
Clicking the thumbnail photo with each system will enlarge it. Clicking the name of the mineral on the enlarged photo will take you to the Mindat mineral page where, in each case, there is a rotatable crystal model.
The Triclinic Crystal System
The symmetry of the triclinic system is unique in that it exhibits only a center of symmetry or no symmetry at all. Only 8% of all minerals crystallize in the triclinic system, and nearly all do so in the pinacoidal class with similar faces on opposite sides of the crystal.
HermanMauguin Symbols: Triclinic Classes  

The Monoclinic Crystal System
Twenty seven percent of all known minerals crystallize in the monoclinic system and the vast majority are in the prismatic class. The 2fold principal axis in the prismatic and sphenoidal classes is, uniquely, the baxis. The absence of a vertical mirror plane containing both the baxis and caxis makes recognition of the prismatic class fairly easy, even when the lower termination is in matrix and the 2fold axis cannot be observed. The single vertical mirror plane is the key in both the prismatic and domatic classes.
2 is always a 2fold baxis.
m is always a vertical mirror plane containing the aaxis and caxis.
HermanMauguin Symbols: Monoclinic Classes  

The Orthorhombic Crystal System
The orthorhombic crystal system is unique in that it has a 2/fold axis coincident with the caxis. That axis can be seen in all three classes when observing the termination of the crystal, making orthorhombic crystals fairly easy to identify. Two vertical mirror planes at 90^{o} angles to each other and containing the 2fold axis, in the Orthorhombic dipyramidal class are quite obvious, also. A major problem is that many triclinic and monoclinic minerals have axial angles very close to 90^{o}.
1st position = aaxis or a/c plane.
2nd position = baxis or b/c plane.
3rd position = caxis.
HermanMauguin Symbols: Orthorhombic Classes  

The Trigonal Crystal System
The unique feature of trigonal crystal system symmetry is a single 3fold or 3fold inversion axis. If more than one 3fold or 3fold inversion axis is found, the crystal is in the isometric (cubic) system. Trigonal crystals are often recognizable from a termination view. The 3fold symmetry is apparent in the distribution of faces. Further, the tripartite termination is often set above a trigonal or hexagonal crosssection or prism when viewed from above.
1st position = caxis.
2nd position = aaxes and/or a/c planes.
HermanMauguin Symbols: Trigonal Classes  

The Hexagonal Crystal System
The unique feature of symmetry in the hexagonal system is a 6fold axis of rotation or a 6fold inversion axis. In the dihexagonal pyramidal and hexagonal trapezohedral classes the planes or 2fold axes are located so three of the planes or 2fold axes, respectively, include the aaxes. The other three, in each case, are evenly spaced between the axes.
1st position = caxis.
2nd position = aaxes or a/c plane.
3rd position = alternate (30^{o} from aaxes).
HermanMauguin Symbols: Hexagonal Classes  

The Tetragonal System
A single 4fold axis of rotational symmetry or rotatory inversion is the unique feature of the tetragonal crystal system. if there is more than one 4fold axis, the crystal belongs in the isometric (cubic) system.
1st position = caxis
2nd position = aaxes and/or a/c planes.
3rd position = alternate (45^{o} from aaxes)
HermanMauguin Symbols: Tetragonal Classes  

The Isometric (Cubic) System
The unique element of symmetry in the isometric system is the presence of 4 3fold axes. The 4fold axes are not present in all classes and as such are not the unique feature. Minerals in the isometric system comprise approximately 7% of all known minerals. About 2/3 of them are in the hexoctahedral class.
1st position = aaxes.
2nd position = cube body corner diagonals .
3rd position = cube body edge diagonals.
HermanMauguin Symbols: Isometric Classes  

Links to the Crystallography Series
 Determining Symmetry of Crystals: An Introduction
 Miller Indices
 HermannMauguin Symmetry Symbols
 Crystallography: The Monoclinic System
Crystallography: The Trigonal System
Crystallography: The Hexagonal System
Crystallography: The Tetragonal System
Crystallography: The Isometric System
References
Mason, Brian and Berry, L.G. (1968) Elements of Mineralogy. W. H. Freeman and Company, San Francisco.
Peck, Donald B. (2007) Mineral Identification: A Practical Guide for the Amateur Mineralogist. Mineralogical Record, Tucson, Arizona.
https://en.wikipedia.org/wiki/Crystallographic_point_group
https://kaushikmitra5.wordpress.com/2013/09/12/symmetryelementsthe32crystalclasses/
Acknowledgements
The photos used in this article are all from Mindat Archives. We are indebted to A. Bleeker, John Betts, Joseph Freillich, Rob Levinski, Doug Merson, Michael Roarke, and Dominik Schlafi for sharing them with us.
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