Crystallography: The Isometric System
Last Updated: 7th Nov 2018By Donald Peck & Alfred Ostrander
The Isometric Crysal System, also known as the Cubic Crystal System, is in terms of symmetry the most complex of the seven systems. That is the bad news. The good news is that it is probably the most easily recognized.
The system has five crystal classes. There are approximately 500 minerals that crystallize in the system (2018): close to half in the hexoctahedral class, fewer than 100 in each of the hextetrahedral and the diploidal classes, and very small numbers in the tetartoidal and gyroidal classes.
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 Isometric Crystal System
The unique element in the symmetry of the isometric system is the four 3fold axes of rotational symmetry. The three 4fold axes, while definitive when they are found, are absent in three of the five isometric classes. Unlike the isometric, the trigonal system has only one 3fold axis, where it is the principal axis, and differentiates the two systems.
Isometric Crystallographic Axes
The forms of the Isometric Crystal System are related to three equal and mutually perpendicular axes by the symmetry of the system. The three axes, labelled a_{1} (front to back). a_{2} (side to side), and a_{3} (top to bottom), are the geometric axes of a cube and intersect at 90^{o} angles to each other.
Note the convention for designating the positive and negative ends of the axes. The positive end of a_{1} faces the viewer, positive a_{2} is to the right, and positive a_{3} is at the top.
The three axes are equivalent and indistinguishable from each other in actual crystals.
General & 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, and by definition each at a different distance. The general symbol for the general form is {hkl}. Since the three isometric axes are the same length, the intercepts must be at different fractions of the axial unit length. Therefore, in the isometric system, h, k, and l must have different values. This requires forms with Miller indices such as {123},{325},{431}, etc. A Miller Index may not be 0 (zero/zed) for the general form.
Forms of the Isometric Crystal System: Click any figure to enlarge it  

All forms in the Isometric Crystal System are closed forms. That is, the faces of the form completely enclose space.
Hexahedron: The hexahedron, or cube, is a special form that can be found in any crystal class of the isometric system. The ideal form has 6 square faces. Each face is perpendicular to a single crystallographic axis. The general symbol is {h00}: (100). (010}, (001). {100), etc.
Octahedron: is a special form found in any class of the system. The ideal form has 8 equilateral triangular faces. Each face intercepts all three axes at equal distances. The octahedron replaces the corners of the cube. Its general symbol is {hhh}: (111), (111). (111}, etc.
Dodecahedron: The dodecahedron [dodec = 12] is a special form with 12 rhomboid faces, each of which intercepts two crystallographic axes and is parallel to the third. Faces are parallel to diagonal mirror planes. The form bevels the edges of a cube. Its general symbol is {h0l}: (101), (011), (110), etc. Sometimes known as a rhombohedral or rhombic dodecahedron, the rhombic is added to differentiate the form from the pentagonal dodecahedron (pyritohedron), tristetrahedral dodecahedron (aka trigional tristetrahedron or simply, tristetrahedron), and deltahedral dodecahedron (aka tetragonal tristetrahedron, or deltahedron).
Tetrahexahedron: The tetrahexahedron [tetra (4) x hex (6) = 24] is a special form with 24 faces that are isosceles triangles. Four faces of the form replace each face of the cube, making a low pyramid. Each face intercepts two crystallographic axes at unequal distances, and is parallel to the third. The general form is {hk0}: (210), {310}, {320), etc. There may be more than one tetrahexahedron on the same crystal.
Trisoctahedron: The trisoctahedron [tris (3) x Oct (8) = 24] is a special form with 24 faces that are isosceles triangles. Each face intercepts three axes. One axial intercept is at unit length, the other two are equal fractional lengths. A group of three kite shaped faces can replace an octahedral face on the corner of a cube Each face points to an edge of the cube. For the upperright hand corner of the cube, it is two faces above one (contrast with the trapezohedron). The general symbol is {hhl}: (221), (331), etc.
Trapezohedron: The trapezohedron has 24 faces each of which is a trapezoid, but may be truncated to other geometric shapes. Each face intercepts three axes. Each face intercepts two axes at a multiple of unit length and the third axis at the same multiple fractional lengths. Like the trisoctahedron three faces can replace an octahedral face on the corner of a cube,but they have not been found as triangles. For the upperright corner of the cube, it is one face over two (contrast with the Trisoctahedron). The general symbol is {hll}: (211), (311), (322), etc. It is often found as a single bevel surrounding the faces on garnets.
Hexoctahedron: The hexoctahedron [6 x 8] is the general form for the isometric hexoctahedral class. It has 48 faces, each of which is a scalene triangle and intercepts all three axes. The axial intercepts are at different lengths on each axis. The general symbol for the form is {hkl}: {321}. The hexoctahedron, when present, is often seen as a group of 6 small triangles on each corner of a cube (most often on fluorite), or as two parallel faces surrounding the rhombic face of a rhombic dodecahedron (on garnet).
Tetrahedron: Tetrahedron (tetra = 4) has 4 faces, each an equilateral triangle and each of which intercepts all three axes at equal distances. There is a positive {111} and a negative form {111}, distinguishable when both are on the same crystal where the positive form is usually larger. The negative form is rotated on the crystal axes 90^{o} from the positive form. Each form represents development of half the faces of the octahedron If both forms are developed equally, the crystal will appear as a hexoctahedral octahedron and can be differentiated from it only by changes in luster or the appearance of striations on alternate crystal faces. Geometrically the two forms are an octahedron, but symmetrically they are different.
Tristetrahedron: The tristetrahedron has 12 faces (3 x 4), each an isosceles triangle. Three raised and inclined faces replace each tetrahedral face. There are both positive, (211), and negative, (211) forms. The general symbol for the form is {hll}: {211}, {311}. The faces are equivalent to half the faces of the trisoctahedron. The tristetrahedron is also known as the trigonal tristetrahedron and the tristetrahedral dodecahedron.
Deltahedron: The deltahedron has 12 faces each of which is a geometric kite (note the four edges of the kite as opposed to the three on the trisoctohedron). There are both positive, {221}, and negative, {221}, forms. The general symbol for the form is {hhl}: {221}. The deltahedron is also known as the tetragonal tristetrahedron and the deltoid dodecahedron.
Hextetrahedron: The hextetrahedron has 24 faces (6 x 4), each of which is a scalene triangle. The six faces are raised to form a shield that replaces the tetrahedral face. There are both positive, {321}, and negative, {321}, forms. The general symbol for the form is {hkl}: {321}. The hextetrahedron is the general form for the Hextetrahedral Class.
Pyritohedron: The pyritohedron is an irregular pentagonal dodecahedron with 12 pentagonal faces, sometimes known as the pentagonal dodecahedron. The faces are not regular pentagons, having one slightly longer edge that is parallel to a crystallographic axis. There are both positive and negative forms. The pyritohedron is derived from the tetrahexahedron, but has lower symmetry. The general symbol for the form is {hk0}: 210, 310, 410, 320, etc. In the figures, above, compare the directions of the edges, pierced by the axes, on the positive pyritohedron to the direction of the striations on the cube. Note that they are parallel. The symmetry is of the pyritohedron and not the cube, as shown by the striations.
Diploid: The diploid has 24 faces, twice the number for the pyritohedron. Two faces, divided by a mirror plane, replace the pyritohedral face. The general symbol for the form is {hkl}: {321}, {421}. etc. It is the general form for the Diploid Class. The diploid is sometimes known as the dyaskisdodecahedron.
Tetartohedron: The tetartohedron has 12 faces, each of which is an irregular pentagon. They occur in groups of three replacing a tetrahedral face. The tetartohedron shows one fourth the number of faces on the hexoctahedron and as such displays four separate forms: the right positive {123} and negative {123} forms and the left positive {132} and negative {132) forms. With left and right handed forms, it is enantiomorphic. The general symbol for the form is {hkl}. The form is sometimes known as the tetrahedralpentagonal dodecahedron.
General Morphology
Isometric means "equal measure". Crystals in the isometric system generally are equant (having the same dimension in all directions). They resemble a geometric cube, octahedron, or tetrahedron; but often with modifying forms that at times approach making them spherical. While individual forms may be difficult to recognize, the general shape of crystals in the system is not. This is particularly true of minerals that crystallize in the hexoctahedral class, and to a slightly lesser extent the diploidal and hextetrahedral classes. The remaining two classes (the gyroidal and tetartoidal classes) have few minerals one is likely to encounter.
Isometric Crystal Classes
There are five crystal classes in the isometric crystal system. Only the hexoctahedral class (244 minerals), the hextetrahedral class (66 minerals), and the diploidal class (59 minerals) are of much interest to mineral collectors. The remaining two classes contain relatively few and/or rare minerals.
The Hexoctohedral Class
The holohedral class (the class with the highest symmetry in this crystal system)HermanMauguin Symmetry Symbols
4/m 3 2/m 4fold axis perpendicular to a mirror plane, 3fold inversion axis, 2fold axis perpendicular to a mirror plane
Symmetry Elements
3 A_{4}, 4 A_{3}, 6 A_{2}, 9M, C
The 3 4fold axes are coincident with the crystallographic axis, normal to the cubic faces. The 4 3fold axes are the corner to corner diagonals of the cubic crystal, normal to the octahedral faces, and are the unique feature of all isometric crystals. The six 2fold axes are edge to edge diagonals of the cubic crystal and normal to dodecahedral faces. There is a mirror plane perpendiular to every axis of symmetry.
General Form
The general form is the hexoctahedron. 48 faces (6 x 8) It has 6 scalene triangles raised from each octahedral face. It can modify the cube with 6 triangles on each corner, most often on fluorite. It also may show as a pair of perallel faces separating dodecahedral faces, typically on garnets.
Special Forms
Hexahedron: (cube) {100}; 6 faces.
Octahedron: {111}; 8 faces.
Dodecahedron: {101}; 12 faces.
Tetrahexahedron: {210}; 24 faces (12 edges with 2 faces. UNFINISHED
Trapezohedron: (211}; 24 faces.
To Look For
1.Look for the equant habit, and the 4fold axis of symmetry together with the 3fold axis of symmetry.
2. Look for any of the standard hexoctahedral forms (cube, octahedron, dodecahedron), slightly modified at edges or corners.
Hexoctahedral Combined Forms: Click any figure to enlarge it.  
Problems
1. There is possible confusion if a crystal in the tetragonal system has a caxis very close in length to the 2 aaxes (as does chalcopyrite).
2. When viewing the corner of a cube or dodecahedron that is buried in matrix, it is easy to view the 3fold axis as being trigonal when it is not.
3. The octahedral form modifying other forms can be replaced by the trisoctahedron, hexoctahedron, deltohedron, or the trapezohedron. They are easily confused.
Model
Symmetry
...3 4fold axes of rotation: coincident with the crystallographic axes.
...4 3fold axes of rotation: through each corner of the cube to the diagonally opposite corner.
...6 2fold axes of rotation: through each edge (normal to {110} ) diagonally to the opposite edge.
...9 mirror planes: 3 in the planes of the 4fold axes; 6 perpendicular to the 2fold axes.
...1 center of symmetry
Forms
...Hexahedron (cube): {100} 6 faces normal to crystallographic axes.
...Dodecahedron: {110} 12 faces, 1 on each edge of the cube (center face of five).
...Deltahedron: {211} 12 faces (3 x 4 corners)Holding the cube to face (100), the three similar faces at the upper right corner have two at the top, and one underneath. If it were a trisoctahedron it would be the reverse, and each face would be "kite" shaped.
...Tetrahexahedron: {210} and {310} 2 forms of 24 faces each (Of five on each edge, 1 and 5 are {310), 2 and 4 are {210}.)
Representative Minerals
Almandine, Cuprite, Diamond, Fluorite, Galena, Magnetite
The Gyroidal Class
A hemihedral class (Forms have half the number of faces as do those in the holohedral class) Also, enantiomorphic (having left and right handed crystals).HermanMauguin Symmetry Symbols
432 4fold axes coincident with the principal axes; 3fold axes through body diagonals, 2fold axes bisecting angles between principal axe.
Symmetry Elements
3A_{4}, 4A_{3}, 6A_{2} 3 4fold axes of rotational symmetry, 4 3fold axes; 6 2fold axes. No mirror planes and no center of symmetery.
Problems:
There are only four minerals in this class, all of them rare. At one time cuprite was considered to be in this class, but it has been shown to be in the Hexoctahedral Class.
The Hextetrahedral Class
Hemihedral (Forms have half the number of faces as do those in the holohedral class)HermanMauguin Symmetry Symbols43m 4fold inversion axes appear to be 2fold axes, coincident with the crystallographic axes; 3fold axes through corners of the ideal cube; mirror planes each containing 2 of the crystallographic axes.
Symmetry Elements3A_{2}, 4A_{3}, 3P The three principle axes are 2fold axes; the 4 3fold axes are normal to the tetrahedral faces; the mirror planes are at right angles to each other and contain the principal axes.
General Form
The general form is the hextetrahedron, there is a positive [hkl} and a negative {hkl} form, 24 faces;
The hextetrahedron is the only form unique to this crystal class. It is derived as half of the hexoctahedron.
Special Forms
Tetrahedron: (Positive and Negative); both forms may be located on the same crystal, one is usually larger than the other.
Delthedron: Positive and negative forms; 12 faces. Does not form a crystal, but the faces modify forms of other crystals.
Cube: 6 faces {100}
Dodecahedron: 12 faces {101}
Tetrahexahedron: 24 faces (201}
Combined Forms: Click any figure to enlarge it.  
To Look For
Look for equant wedge shaped crystals; equilateral triangular faces on alternate octants of the crystal.
Problems
Chalcopyrite is not in this crystal class. It is a tetragonal mineral.
Model
Symmetry
...3 4fold inversion axes; coincident with the principle axes.
...4 3fold axes; through each point of the crystal and the center of the opposite large face
...4 mirror planes; each mirror plane contains one 4fold inversion axis and two 3fold axes.
Forms
...cube (hexahedron): {100}, 6 faces
...2 tetrahedrons: {111} positive, {111} negative; 4 faces each
Representative Minerals
Sodalite, Sphalerite, Tennantite, Tetrahedrite,
The Diploidal Class
HermanMauguin Symmetry Symbols2/m 3 3 2fold axes perpendicular to a mirror plane; 4 3fold axes of rotational inversionSymmetry Elements
3 A^2, 4 A*3 C: 3 mirror planes The 2/fold axes are coincident with the crystallographic axes; each mirror plane is in the plane of 2 crystallographic axes; each 3fold inversion axis is diagonally through the crystal corners from where 3 pyritohedral faces meet; a center of symmetry.
General Form
The General Form is the Diploid. It is not often encountered.
Special Forms
Cube: 6 faces
Octahedron: 8 faces
Dodecahedron: 12 faces
Trapezohedron: 24 faces
Trisoctahedron: 24 faces
Pyritohedron: 12 faces; There are both positive and negative forms.
To Look For
Characteristic pentagonal faces (usually on pyrite);
2fold symmetry on the principal axes;
the absence of a 4fold axis;
striations on pyrite cubes (not usually found on those from Spain)
Problems
A major problem in this class is that the pyritohedron and even more so, the diploid, are not commonly seen, except on pyrite. Most of the 59 minerals in this class are relatively to extremely rare. Pyrite and skutterudite are the most common.
Model
Symmetry
...3 2fold axes of rotation: coincident with crystallographic axes.
...4 3fold axes of rotational inversion: through the 8 corners of the crystal and along the body diagonal.
...3 mirror planes: each is the plane of 2 of the crystallographic axes.
...center of symmetry
Forms
Cube: {100} 6 faces
...Dodecahedron: (210) 12 faces
...Diploid: {321} 24 faces; the general form
Representative MineralsPyrite, Skutterudite
The Tetartoidal Class
HermanMauguin Symmetry Symbols23 Three 2fold axes of rotational symmetry coincident with the crystallographic axes; four 3fold axes through the corners of the crystal.
Symmetry Elements
3 A^2, 4 A^3 Three 2fold axes and four 3fold axes of rotational symmetry.
General Form
The general form, the tetartohedron, sometimes known as the tetrahedralpentagonal dodecahedron, has 12 faces. The form is analogous to onefourth the faces of the hexoctahedron, making possible four similar solids: left positive and negative, and right positive and negative. The class is enantiomorphic. Th general symbol for the form is {hkl}: {231}.
Special Forms
Cube: 6 faces
Dodecahedron: 12 faces
Tetrahedron: 4 faces
Pyritohedron: 12 faces
Tristetrahedron: 12 faces
To Look For
Look for the 2fold principal axes of symmetry. The only other isometric 2fold principal axis is the pyritohedral class. Search, also, for evidence of enantiomorphism (left or right handedness).
Problems
There are approximately 25 minerals in this crystal class and most of them are relatively rare.
Model 2383
Symmetry
...3 2fold axes of rotational symmetry (coincident with crystallographic axes)
...4 3 fold axes of rotational symmetry (through the cube corners and along the body diagonal)
Forms
...Cube: {100} 6 faces
...Dodecahedron: {110} 12 faces
...Pyritohedron: {120} 12 faces
,,,The general form is not present
Representative Minerals
Corderoite, Gersdorffite, Langbeinite, Ullmannite
Links to the Crystallography of each of the systems.
The the pages for the Tetragonal and Triclinic, Systems are under construction.
 Determining Symmetry of Crystals: An Introduction
Miller Indices
HermannMauguin Symmetry Symbols
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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