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. Charles-Victor Mauguin modified it a few years later and the Hermann-Mauguin Notation System became widely used. Today it is known, also, as the International System of Notation. Hermann-Mauguin (H-M) 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 H-M symbols may seem a bit confusing at first but as one increasingly develops facility in recognizing the symmetry of crystals, H-M becomes more understandable and more helpful.

The H-M Code

1. The H-M symbols consist of some combination of numerals 1, 2, 3, 4, and 6; numerals with a bar-over 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 3-fold axis of rotational symmetry. Similarly, 2, 4,, and 6 indicate axes of 2-fold, 4-fold, or 6-fold rotation, A 1 symbolizes "no symmetry".

3. The numerals with a bar-over 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 inside-out) 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 2-fold 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 3-fold axis parallel to 3 mirror planes. Given a 3-fold axis parallel to (coincident with) one mirror plane, the 3-fold 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 H-M Symbols

Hermann-Mauguin 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 c-axis. The major exception is in the monoclinic system where it is the b-axis.

For example, the H-M symbol for the Monoclinic Prismatic class is 2/m. It indicates that the b-axis is a 2-fold axis of rotation and that the axis is perpendicular to a mirror plane. The H-M symbol for the Hexagonal pyramidal class is 6 (a single element), meaning that the c-axis is a 6-fold axis of rotation and there is no other symmetry. 32 (same as 3 2) is a 3-fold axis coincident with the c-axis and three 2-fold axes each coincident with the 3 a-axes. 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 4-fold axes of rotation, each perpendicular to a mirror plane. The second element is for the four 3-fold axes that are the body diagonals of the cube (why four of them? The 4-fold axes require them.) And the third element is for the 2-fold axes (edge to edge diagonals) each perpendicular to a mirror plane.

H-M 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.



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.

Herman-Mauguin Symbols: Triclinic Classes

Class Nbr. Class Name H-M Symbol Interpretation 1 Pedial 1 No symmetry. No axes of rotation and no mirror planes. No center of symmetry. Faces on opposite sides are not equivalent. 2 Pinacoidal 1 A center of symmetry. Each face or point is matched on the opposite side by inversion through the center (i.e. Opposite sides are equivalent). No axes of rotation and no mirror planes.

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 2-fold principal axis in the prismatic and sphenoidal classes is, uniquely, the b-axis. The absence of a vertical mirror plane containing both the b-axis and c-axis makes recognition of the prismatic class fairly easy, even when the lower termination is in matrix and the 2-fold axis cannot be observed. The single vertical mirror plane is the key in both the prismatic and domatic classes.
2 is always a 2-fold b-axis.
m is always a vertical mirror plane containing the a-axis and c-axis.

Herman-Mauguin Symbols: Monoclinic Classes

Class Nbr. Class Name H-M Symbol Interpretation 3 Domatic m A single mirror plane perpendicular to the b-axis. 4 Sphenoidal 2 A 2-fold axis coincident with the b-axis. No mirror plane. 5 Prismatic 2/m A 2-fold axis (b-axis) perpendicular to a mirror plane. The mirror plane is visible when viewing the termination; if the crystal is in matrix the 2-fold axis may not be visible.

The Orthorhombic Crystal System
The orthorhombic crystal system is unique in that it has a 2/fold axis coincident with the c-axis. 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 2-fold 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 = a-axis or a/c plane.
2nd position = b-axis or b/c plane.
3rd position = c-axis.

Herman-Mauguin Symbols: Orthorhombic Classes

Class Nbr. Class Name H-M ____Symbol____ Interpretation 6 Orthorhombic pyramidal mm2 Two perpendicular mirror planes the intersection of which is a 2-fold c-axis. There is no horizontal mirror plane and the class is hemimorphic. 7 Orthorhombic disphenoidal 222 Three 2-fold axes mutually perpendicular and coincident with the crystallographic axes. No mirror planes. 8 Orthorhombic dipyramidal 2/m 2/m 2/m Three 2-fold axes each perpendicular to a mirror plane. Each 2-fold axis is coincident with a crystallographic axis.

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

Herman-Mauguin Symbols: Trigonal Classes

Class Nbr. Class Name H-M ___Symbol___ Interpretation 9 Trigonal pyramidal 3 A 3-fold axis of rotation coincident with the c-axis. No mirror planes. Hemimorphic 10 Rhombohedral 3 A 3-fold axis of rotatory inversion coincident with the c-axis. No mirror planes. 11 Ditrigonal pyramidal 3m A 3-fold axis coincident with the c-axis and 3 vertical mirror planes each containing the c-axis and one a-axis. Hemimorphic 12 Trigonal trapezohedral 32 A 3-fold axis coincident with the c-axis and perpendicular to 3 2-fold axes. Each 2-fold axis is coincident with an a-axis. 13 Hexagonal scalenohedral 3 2/m A 3-fold axis of rotatory inversion perpendicular to 3 2-fold axes. Each 2-fold axis is coincident with an a-axis and perpendicular to a mirror plane. Holohedral

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

Herman-Mauguin Symbols: Hexagonal Classes

Class Nbr. Class Name H-M ____Symbol____ Interpretation 14 Trigonal dipyramidal 6 = 3/m A 6-fold inversion axis is equivalent to a 3-fold axis perpendicular to a mirror plane. No minerals have been found as crystals. 15 Hexagonal pyramidal 6 A 6-fold axis of rotation coincident with the c-axis. Hemimorphic. 16 Hexagonal dipyramidal 6/m A 6-fold axis of rotation coincident with the c-axis and perpendicular to a mirror plane. 17 Ditrigonal dipyramidal 6m2 = 3/m m A 6-fold inversion axis coincident with the c-axis and centered in 3 vertical mirror planes each containing an a-axis; 3 2-fold axes coincident with the a-axes. Equivalent to a 3-fold axis perpendicular to a mirror plane and central to 3 vertical mirror planes each containing an a-axis. 18 Dihexagonal pyramidal 6mm A 6-fold axis coincident with and central to 6 vertical mirror planes; 3 coincident with the a-axes and 3 spaced evenly between the a-axes. Hemimorphic 19 Hexagonal Trapezohedral 622 A 6-fold axis coincident with the c-axis and perpendicular to 6 2-fold axes; 3 coincident with the a-axes and 3 spaced evenly between the a-axes. Enantiomorphic 20 Dihexagonal dipyramidal 6/m 2/m 2/m 6-fold axis coincident with the c-axis and perpendicular to 6 2-fold axes; 3 coincident with the a-axes and 3 spaced evenly between the a-axes; each axis perpendicular to a mirror plane. Holohedral

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

Herman-Mauguin Symbols: Tetragonal Classes

Class Nbr. Class Name H-M __Symbol____ Interpretation 21 Tetragonal disphenoidal 4 A 4-fold rotatory inversion axis coincident with the c-axis 22 Tetragonal pyramidal 4 A 4-fold axis coindident with the c-axis. Hemimorphic 23 Tetragonal dipyramidal 4/m A 4-fold axis coincident with the c-axis and perpendicular to a mirror plane. 24 Tetragonal scalenohedral 42m A 4-fold inversion axis coincident with the c-axis that is perpendicular to 2 2-fold axes that are coincident with the a-axes. 2 vertical mirror planes containing the c-axis and intermediate to the a-axes. 25 Ditetragonal pyramidal 4mm A 4-fold axis parallel to and included in 4 vertical mirror planes. 2 mirror planes are coincident with the a-axis; the other two are intermediate. Hemimorphic 26 Tetragonal trapezohedral 422 A 4-fold axis coincident with the c-axis and perpendicular to 4 2-fold axes; two of which are coincident with the a-axes and two of which are intermediate to the a-axes. 27 Ditetragonal dipyramidal 4/m 2/m 2/m A 4-fold axis coincident with the c-axis and perpendicular to a mirror plane. 2 2-fold axes perpendicular to 2 vertical mirror planes and coincident with the a-axes; and 2 2-fold axes intermediate to the a-axes and perpendicular to 2 vertical mirror planes.

The Isometric (Cubic) System
The unique element of symmetry in the isometric system is the presence of 4 3-fold axes. The 4-fold 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 = a-axes.
2nd position = cube body corner diagonals .
3rd position = cube body edge diagonals.

Herman-Mauguin Symbols: Isometric Classes

Class Nbr. Class Name H-M __Symbol____ Interpretation 28 Tetartoidal 23 Three 2-fold axes coincident with the a-axes and four 3-fold diagonal axes. 29 Diploidal 2/m 3 Three 2-fold axes coincident with the a-axes and each perpendicular to a mirror plane; and four diagonal 3-fold inversion axes. 30 Hextetrahedral 43m Three mutually perpendicular 4-fold inversion axes coincident with the a-axes; four axes of 3-fold symmetry; and six diagonal mirror planes. 31 Gyroidal 432 Three 4-fold axes coincident with the a-axes; four diagonal 3-fold axes; and six 2-fold axes. 32 Hexoctahedral 4/m 3 2/m Three 4-fold axes perpendicular to mirror planes; 4 body-diagonal 3-fold inversion axes; 6 edge-diagonal 2-fold axes each perpendicular to a mirror plane .

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Links to the Crystallography Series

1. Determining Symmetry of Crystals: An Introduction
2. Miller Indices
3. Hermann-Mauguin Symmetry Symbols
4. Crystallography: The Monoclinic System

Crystallography: The Orthohombic 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/symmetry-elements-the-32-crystal-classes/

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.