Crystallography: The Hexagonal System
Last Updated: 14th Nov 2018By Donald Peck & Alfred Ostrander
The crystals that are in the hexagonal crystal system are among the easiest to recognize. Their cross section, viewed from the termination, is usually hexagonal. However, many trigonal crystals are also hexagonal in cross section leading to possible confusion. The key is to look for the principal axis of rotation. It is 6fold in the hexagonal system and 3fold in the trigonal system.
There are some 300 minerals that crystallize in the hexagonal system. Nearly half are found in the dihexagonal dipyramidal class, 25 or 30 in each of the other classes, except for the trigonal dipyramidal class, which holds only three. There are more than 50 minerals not assigned to any class.
You may find it helpful to review the following articles before attempting this one:
Determining Symmetry of Crystals: An Introduction
Miller Indices
HermannMauguin Symmetry Symbols
Determining Symmetry of Crystals: An Introduction
Miller Indices
HermannMauguin Symmetry Symbols
The Unique Symmetry Element of the Hexagonal Crystal System
The 6fold axis of rotational symmetry is unique to the hexagonal crystal system. In two of the classes (ditrigonal dipyramidal and trigonal pyramidal) the principal axis is 6fold axis of rotatory inversion. This is easily confused with a 3fold axis perpendicular to a mirror plane, and in each case the two symmetries are often listed as equivalent. It is not a large problem, since these two classes are not often encountered.
Crystallographic Axes
The hexagonal system, like the trigonal system, is usually thought of as a 4axis system. Three of the axes; commonly labeled a_{1}, a_{2}, and a_{3}; are equal in length and lie in a single horizontal plane that is perpendicular to the principal c axis. The angle between the positive ends of the three a axes, is 120^{o}. The c axis may be either longer or shorter than an a axis.
Take careful notice of the manner in which the a axes are designated. Positive and negative ends of the axes alternate around the c axis. The angle between the positive end of the a_{1}+ and the positive end of the a_{2}+ is 120^{o}, not 60^{o}.
Crystals are normally oriented so the longest (or shortest) axis is vertical and designated as the c axis. If there is a dominant pyramidal or dipyramidal face it is turned toward the viewer. The positive end of the a_{1} axis then emerges usually below the lower left corner of that face.
Miller Indices
MillerBravais Indices for the trigonal and hexagonal systems customarily have four digits indicating intercepts on the a_{1}, a_{2}, a_{3}, and c axes, respectively. In the four axis system crystallographers increasingly are using a three digit index, dropping the third, a_{3}, index. In any case, the a_{3} index is redundant. It is the negative sum of the a_{1} and a_{2} indices. Given four digit indices such as (1210), the resulting 3 digit indices are (120) [i.e. ( 1+ (2) ) = 1 that is the third digit which is dropped ]. Both give us the same information but for beginning crystallographers the four digit system is easier with which to work.
General and Special Forms
There are two types of forms, general forms and special forms.
Any form which is not a general form is a special form. Most often, the general form is the form for which the crystal class is named. It usually appears only in that crystallographic class or in classes of higher symmetry. A general form has the maximum number of faces of any form in its crystal class. Special forms may appear in any crystal class of the system.
The general form intercepts all axes, each at a different distance. Since all three a axes are the same length, a given face of the general form must intercept them at different fractions of the unit length (i.e. the Miller Index for each of the first three intercepts must have different values). The c axis, having a different unit length, may have an index of any value other than zero (0). The form is not always expressed on the crystal.
Forms in the Hexagonal Crystallographic System: Click any image to enlarge it.  

There are 1st, 2nd, and 3rd order prisms, pyramids, and dipyramids in the hexagonal system.
..1st order prism: each face intercepts two of the aaxes at unit distances and is parallel to the third aaxis. {1010}
..2nd order prism: each face intercepts all three aaxes and is perpendicular to one of them. {1120}
..3rd order prism: each face intercepts all three aaxes at different distances, {2130}, and is neither perpendicular nor parallel to any of them.
..The corresponding dipyramidal or pyramidal faces are directly above or below the prism faces (i.e. pyramid face (1011) would be directly above prism face {1010).)
Pyramids & Dipyramids:
Pyramids in the hexagonal crystal system may have 3, 6 or 12 faces, all converging on and meeting at a single end of the c axis. Dipyramids have 6, 12, or 24 faces, half meeting on the positive end and half on the negative end of the c axis. Pyramids are analogous to half a dipyramid and are open forms (the form requires other forms in combination to enclose space). Dipyramids are closed forms (the form encloses space). Pyramids occur only in hemimorphic classes (a horizontal mirror plane and/or a center of symmetry will not allow the dipyramidal form).
^{. }Hexagonal dipyramid: 12 faces. Six faces intercept the +c axis and six the c axis. the general symbol for the 1st Order dipyramid is {h0hl). The 1st Order dipyramid is oriented with the a_{1}+ axis emerging below the lower left of the (1011). The 2nd Order dipyramid general symbol is {hh2hl)/{1121} and is oriented with the a_{1}+ axis emerging below the center of the face. The 2nd order form is rotated 30^{o} on the axes to the left of the 1st Order form. Usually, the most prominent pyramidal form is designated as 1st Order. (See figure below.)
^{. }Hexagonal pyramid: 6 faces. An open form. Found only on hemimorphic crystals as 1st or 2nd order and either positive (upper half) or negative (lower half). The general symbol is {h0hl}/{1011}, 1st order positive; {
^{. }Dihexagonal dipyramid: 24 faces. 12 faces on the upper half of the crystal and 12 on the lower half. The general symbol is {hkil}/{2131}. It is the general form of the holohedral class.
^{. }Hexagonal Dipyramid, 3rd Order: 6 faces. There are two of them, rotated Right and rotated Left. The symbol for the Right dipyramid is {2131}; the Left hand form is {1231}. Each dipyramid face is directly above, or below, the corresponding prism face (q.v. below). Historically these were known as 3rd order hexagonal dipyramics, positive or negative. Today each form is generally known by its Miller indices.
^{. }Trigonal dipyramid: 6 faces. A closed form which modifies hexagonal crystals. 3 faces on the upper half of the crystal and 3 on the lower half. There is both a 1st order and a 2nd order form. The general symbol is {h0hl}/{1011} 1st order and (hh2hl}/(11_21} 2nd order.
Prisms & Diprisms:
Prisms in the hexagonal system have 3, 6, or 12 faces; faces of a given prism are always parallel to and surround the c axis. All prisms are open forms, requiring other forms to close their ends.
^{. }Hexagonal prism: 6 faces. There are two hexagonal prisms, 1st Order and 2nd Order. The 1st Order prism is oriented with an end of each a axis emerging at the edge between two prism faces. The 2nd Order prism is rotated on the c axis (see below) and an end of each a axis emerge from the center of a face. The indices for the 1st Order form are {h0h0}/{1010} and for the 2nd Order form, {hh2h0}{1120}. In each case, if there is a pyramidal face, it is directly above the prism face that has similar indices ( i.e. pyramidal face (1121) would be above prism face (1120).)
^{. }Dihexagonal prism:< 12 faces. Faces are in pairs and the set of alternate interfacial angles are equal, but different from the other set of alternate angles, which also are equal. The general symbol for the form is {hki0}/{2130), where h, k, and i each has a different value.
^{. }Hexagonal prism, 3rd Order: 12 faces; intercepts all three aaxes at different distances {hk_i0}/{2130}. Rarely observed.
^{. }Trigonal prism: 3 faces. There are four of them. 1st Order positive and negative and 2nd Order positive and negative forms. The 1st Order symbol is {1120} ) and the 2nd order symbol is {2110}.
^{. }Ditrigonal prism: 6 faces. The Miller indices are {hki0}/{2130}.
Trapezohedrons: 12 faces. There is both a Left {3121} and a Right {2131} form, thus they are enantiomorphous (left and right handed). A trapezium has four sides with none of them parallel to any other.
Pinacoids & Pedions:^{. }Pinacoid: 2 faces. The faces are parallel and in the hexagonal system at opposite ends of the caxis. The faces are perpendicular to the axis. The form symbol is {0001}.
^{. }Pedion: 1 basal face. In the hexagonal system the face is perpendicular to the caxis. It is found only in hemimorphic classes, where there is no center of symmetry and no horizontal mirror plane. A basal pedion is said to be negative {0001} if it intercepts the negative end of the caxis, positive {0001} it intercepts the positive end.
General Morphology
Approximately one third of all hexagonal minerals are prismatic and/or pyramidal. As such they are easily recognized. Another significant number of them is acicular and often radiating. Since the cross section of acicular crystals is difficult to determine, the crystals of this group are often problematic. Microscopy is often helpful. A third large group is tabular, sometimes foliated, but again, the hexagonal outline is usually easy. Care must be taken, however, to differentiate the hexagonal forms from the very similar outlines of some of the micas.
Crystal Classes
Dihexagonal Dipyramidal Class
There are approximately 135 minerals in this class (2018). That is slightly less than half of all minerals that crystallize in the hexagonal system. It is the holohedral class (highest symmetry) of the hexagonal system .
HermannMauguin Symbol
6/m 2/m 2/m . . . Principal axis is 6fold perpendicular to a mirror plane, secondary axes are 2fold each perpendicular to a mirror plane, alternate axes (30^{o} from 2ndry axes) are 2fold and each perpendicular to a mirror plane.
Symmetry Elements1A_{6} 6A_{2} 7P C . . . 1 6fold axis (the caxis), 6 2fold axes (3 aaxis and 3 bisecting angles between the aaxes) 7 planes (3 in the planes of the aaxes and caxis, 3 bisecting the angles of the aaxes, 1 horizontal, 1 center)
General Form
The general form is the dihexagonal dipyramid: 24 faces, 12 on upper termination and 12 on the lower. The general symbol is {hkil}. It is almost never seen.
Special Forms
..1st order dipyramid
..2nd order dipyramid
..dihexagonal prism
..1st order prism
..2nd order prism
..basal pinacoid
Look For
The hexagonal cross section is often easy. Look carefully at pinacoidal terminations for often very thin faces beveling the edges and/or clipping the corners. In the case of thin acicular crystals, it is often fruitful to examine them under a microscope, both to observe the hexagonal character and the termination for pyramids.
Problems
The "pseudohexagonal" shape of several of the micas. The apparent lack of pyramidal faces on crystals that terminate with a pinacoid.
Model
..Symmetry
....6fold axis: termination view, along the caxis; or rotate the model around the caxis
....2fold axes: 6: the aaxis and the intermediate axes (bisecting the angles between the aaxes)
....mirror planes: 7: 1 plane perpendicular to each aaxis, caxis, and each alternate axis.
....center of symmetry
..Forms
....1st order prism: {100} or {1010)
....1st order dipyramid: {101} or (1011}
....2nd order dipyramid: (111} or (1121}
Representative Minerals
Beryl, Covellite, Gmelinite, Graphite, Molybdenite, Nickeline
Hexagonal Trapezohedral Class
Mineralogically this is a relatively unimportant class. There may be up to 26 minerals, mostly uncommon, quartzbeta is the most important. Enantiomorphic Class.
HermannMauguin Symbol
622 . . . 6fold axis (caxis) perpendicular to 6 2fold axes (the 3 aaxes and the 3 alternate axes)
Symmetry Elements1A_{6} 6A_{2} . . . The 6fold axis, when there is a horizontal 2fold axis, requires six of them.
General Form
The Hexagonal Trapezohedron: 12 faces. 6 on the upper half and 6 on the lower half; {2131} (right hand crystal).
Special Forms
..1st and 2nd order dipyramids
..1st and 2nd order prisms
..dihexagonal prism
..pinacoids
Problems
Minerals in this class are uncommon and, as crystals (which are even more uncommon) are seen usually as microscopic and acicular.
Representative Minerals:
Currierite,Faheyite, Mallestigite, Rhabdophane(Ce),
Dihexagonal Pyramidal Class
This class holds approximately 35 minerals. Zincite is the best known. Other species are either uncommon or rare. The class is hemimorphic ("half shaped" Opposite ends of the caxis are different forms).
HermannMauguin Symbol:
6 m m . . . A 6fold principal axis with 3 vertical mirror planes containing the aaxes and 3 vertical mirror planes midway between the aaxes (the alternate positions).
Symmetry Elements:
1A_{6} 6m . . . 1 6fold principal axis. 3 mirror planes in the planes of the aaxes and caxis, and 3 mirror planes bisecting the angles between the aaxes (alternate positions).
General Form:
The general form is the Dihexagonal Pyramid: 12 faces, either positive of negative
Special Forms
..Hexagonal pyramid: 6 faces,1st and 2nd order, positive and negative
..Dihexagonal prism: 12 faces
..Hexagonal prisms: 6 faces, 1st and 2nd order
..Pedion: 1 basal face; positive {0001) and negative {0001}
Look For
Look for the 6fold axis of symmetry, the hexagonal cross section, and hemimorphism.
Model
..Symmetry:
....1 6fold axis of rotational symmetry (the caxis)
....6 mirror planes: all vertical (3 coincident with the aaxes, 3 in the alternate positions between the axes)
..Forms:
....Hexagonal pyramid 1st Order: form {101} or {1011
....Hexagonal prism 1st Order: form {100} or {1010}
Representative Minerals:
Greenockite, Iodargyrite, Magnesiotaaffeite2N2S, Zincite
Ditrigonal Dipyramidal Class
There are about 25 minerals (2018) in this class. Benitoite and offretite are the most important with others being uncommon to rare.
HermannMauguin Symbol:6 m 2 = 3/m m . . . 6fold axis of inversion coincident with 3 mirror planes and perpendicular to a 2fold axis of rotation within each mirror plane. Equivalent to a 3fold axis perpendicular to a mirror plane and 3 mirror planes coincident with the axis (the horizontal mirror plane requires and implies the 3 2fold axes.)
Symmetry Elements:1A_{_6} 4P C . . . A 6fold axis coincident with 3 mirror planes, each of which contains a 2fold horizontal axis, and a horizontal mirror plane. Alternatively: 1A_{3} 4P . . . A 3fold principal axis coincident with 3 vertical mirror planes and perpendicular to a horizontal mirror plane.
General Form:
The general form is the Ditrigonal dipyramid: 12 faces, {hk_il}
Special Forms
..Trigonal dipyramid: 1st order positive and negative
..Hexagonal dipyramid 2nd order
..Trigonal prism: 1st order, positive and negative
..Ditrigonal prism: 6 faces
..Hexagonal prism: 6 faces, 2nd order
..Pinacoid
Look For:
The benitoite crystal is unmistakable.
Problems:
Although the structure in this class is hexagonal, the morphology appears to be trigonal.
Model
The model of benitoite certainly appears to have trigonal rather than hexagonal symmetry. The ditrigonal dipyramidal class is one of two in the Hexagonal system that has a 6fold rotatory inversion axis and in both cases the 6fold inversion axis is equivalent to a 3fold axis perpendicular to a mirror plane. The 6fold inversion axis is difficult to see; and the model is not easily manipulated to observe it. Give it a try, anyway.
....Symmetry:
....1 3fold axis of rotation (or 6_fold axis of rotatory inversion)
....3 vertical mirror planes
....1 horizontal mirror plane
....1 center of symmetry
These symmetry elements, taken together, are equivalent to 6, a 6fold rotatory inversion axis. The general form is a 3rd order trigonal dipyramid, not shown on the model and rarely observed.
..Forms:
....1 trigonal dipyramid: {102}={1012}; 6 faces; 1st order positive
....1 trigonal dipyramid: {012}={0112}; 6 faces; 1st order negative
....1 trigonal prism: {100}={1010}; 3 faces; 1st order positive
....1 trigonal prism: {010}={0110}; 3 faces; 1st order negative
....1 pinacoid: {001}={0001}; 2 faces; basal
Representative Minerals:
Benitoite, Connellite, Offretite, Schaurteite
Hexagonal Dipyramidal Class
There are about 62 minerals in this class (2018).
HermannMauguin Symbol
6/m . . . A 6fold principal axis perpendicular to a mirror plane.
Symmetry Elements
1A^6 1P C . . . 1 6fold axis of rotational symmetry (the caxis), 1 mirror plane (horizontal), and a center of symmetry.
General Form
The general form is the hexagonal dipyramid {hkil} & {hkil}, 12 faces; Often called the 3rd order hexagonal dipyramid but now generally known by the Miller indices of the form. The form is very rarely seen, and then (almost?) always on apatite as small faces modifying other pyramids.
Special Forms
.. Hexagonal dipyramid: each 12 faces; 6 on the upper termination and 6 on the lower; 1st and 2nd order
.. Prisms: 6 faces, 1st and 2nd order
.. prisms: {hk_i0} and {kh_i0):each 12 faces. (sometimes labeled a 3rd order diprism)
.. pinacoid:
Look For
6fold axis or symmetry; hexagonal cross section, the dipyramid.
Problems
The dipyramid often appears as a very thin face, beveling the edge between the hexagonal prism and the basal pinacoids. The lower half of the dipyramid, if it exists, may be in the matrix and not developed.
Model
.. Symmetry
....1 6fold axis of rotational symmetry: The caxis
....1 mirror horizontal plane: The plane of the aaxes
....Center of symmetry; Use the transparency of the model to observe this.
.. Forms
....1st order hexagonal dipyramid: {101} or {1011}
....1st order hexagonal dipyramid: {201} or {20_21}
....2nd order hexagonal dipyramid: {112} or {1122)
....2nd order hexagonal dipyramid: {111} or {11_21}
....1st order hexagonal prism: {100} or {1010}
....2nd order hexagonal prism: {110} or {1120}
Representative Minerals
Fluorapatite, Mimetite, Pyromorphite, Vanadinite
Hexagonal Pyramidal Class
This hemimorphic class has approximately 30 minerals (2018). The symmetry of the class is the same as the hexagonal dipyramidal class, except there is no horizontal plane of symmetry nor a center of symmetry {See the model for Nepheline, below, to view the problem.
HermannMauguin Symbol:
6 . . . . The principal axis is a 6fold axis of rotational symmetry.
Symmetry Elements:
1A_{6} . . . 1 6fold axis of rotation; the caxis.
General Form:
The general form is the hexagonal pyramid 3rd order: 6 faces {hkil}; almost never developed.
Special Forms:
..Hexagonal pyramid: 6 faces; positive and negative; 1st and 2nd order.
..Hexagonal prisms: 6 faces; 1st, 2nd, and 3rd order.
..Pedion: 1 basal face; positive and negative
Look For:
The 6fold axis of rotational symmetry; no horizontal mirror plane.
Problems:
With the 3rd order pyramid or prism not present, the crystal will appear to have vertical mirror planes; and perhaps a horizontal mirror plane, as well.
Model:
This crystal appears to be hexagonal dipyramidal. It is not. Nepheline is hemimorphic. The upper hexagonal pyramid (positive) and the lower hexagonal pyramid (negative) are different forms and there are two pedions rather than a basal pinacoid. Determining the crystal class, or even the hemimorphism, is impossible from only the forms shown.
....Symmetry
.... 6fold axis or rotational symmetry
.... 6 vertical mirror planes (absence of the general form makes the crystal appear to have higher symmetry)
.... 1 horizontal mirror plane: ditto
.... 1 center of symmetry: ditto
.. ..Forms
.... 1 Hexagonal pyramid 1st order positive: 6 faces {101}=(1011}; (This and the negative form appear to be a hexagonal dipyramid)
.... 1 Hexagonal pyramid 1st order negative: 6 faces {101}={1011}
.... 1 Hexagonal prism: 6 faces {100}={1010}: 1st order
.... 1 Hexagonal prism: 6 faces (110}= {1120}; 2nd order
.... 1 Pedion: 1 face (001)=(0001), basal positive (A hemimorphic class cannot have a basal pinacoid; thus positive and negative pedions).
.... 1 Pedion: 1 face (001)=(0001), basal negative
Representative Minerals:Cancrinite, Nepheline, Zincenite
Trigonal Dipyramidal Class
This is an unimportant class for minerals. Currently (2018) only three rare minerals have been identified in this class.
HermannMauguin Symbol:
6 . . . 6fold rotatory inversion axis (the caxis)
Symmetry Elements:
1A_{6} = 1A_{3} 1P . . . 1 6fold axis of rotatory inversion; which is equivalent to 1 3fold axis perpendicular to a mirror plane.
Links to the Crystallography of each of the systems.
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.
Dana,Edward Salisbury; Foord, William E. (editor); A Textbook of Mineralogy. John Wiley & Sons, Inc., New York
Smith, Jennie R. (1991) Understanding Crystallography. The Rochester Mineralogical Symposium.
Sinkankas, John: Mineralogy: A First Course. A great book with which to start.
Peck, Donald B. (2007) Mineral Identification: A Practical Guide for the Amateur Mineralogist. Mineralogical Record, Tucson, Arizona.
A.E.H Tutton,Crystallography and Practical Crystal Measurement Volume 1 Form and Structure; 2018. An incredibly thorough text. Not for the beginner.
Klein, Cornelis & Hurlbut, Cornelius S., Jr.: Manual of Mineralogy after J. D. Dana;20th Edition
http://www.minsocam.org/ammin/AM20/AM20_838.pdf: Rogers, Austin F. (1935) A historical discussion of the names of crystal forms.
http://www.tulane.edu/~sanelson/eens211/forms_zones_habit.htm: A good explanation and depiction of crytallographic forms.
https://en.wikipedia.org/wiki/Monoclinic_crystal_system , Short explanation of lattices, space groups, hemimorphic & enantiomorphic structure.
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Comments
Thanks Don
Another great ready reference.
It's the little things I often forget (like Benitoite being in this class)
Cheers
Keith Compton
29th Oct 2018 9:37am
Another great ready reference.
It's the little things I often forget (like Benitoite being in this class)
Cheers
Keith Compton
29th Oct 2018 9:37am
Keith, I really wrestled with that rotating model of Benitoite. I think it took me almost two weeks before I had it figured out. Had a terrible time matching the forms to the axes. The symmetry was easy.
Thanks,
Don
Donald B Peck
14th Nov 2018 2:52am
Thanks,
Don
Donald B Peck
14th Nov 2018 2:52am
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