Some common minerals like galena, garnet, fluorite, and magnetite are found in cubic or isometric crystal systems. The crystals of this group have three mutually perpendicular axes of equal lengths. These axes are interchangeable and are designated as a1, a2, and a3. The axes of cubic/isometric crystal systems are as follows.

## Classes of Isometric System of Crystals

There are five classes in the cubic or isometric system of crystals, out of which the most important classes are: (i) Galena type, (ii) Pyrite types, and (iii) Tetrahedrite type.

**(1). Galena Type of Cubic Crystal System: **

It is the normal class of cubic system, which possesses the highest degree of symmetry.

**Symmetry Elements:** The symmetry elements of this class are as follows:

- Planes 9
- 3 axial planes,
- 6 diagonal planes)

- Axes 13
- Three crystallographic axes of 4-fold symmetry,
- Four diagonal axes of 3-fold symmetry,
- Six diagonal axes of 2-fold symmetry

**Forms: **There are 7 basic forms in this class.

**Cube (100%):**This form is composed of six square faces that make 90**°**angles with each other. Each face intersects one of the crystallographic axes and is parallel to the other two.**Octahedron (111):**Octahedron is composed of eight equilateral triangular faces. Each face intersects the three crystallographic axes at equal lengths.**Dodecahedron (110):**This is a polyhedron composed of 12 rhomb-shaped faces. Each face intersects the two crystallographic axes equally and is parallel to the third.**Trapezohedron (hll):**This form of the cubic crystal system have 24 trapezium-shaped faces. Each face intersects one crystallographic axis at a unit distance and the other two at equal multiples. The most common trapezohedron is (211).**Tetrahedrahexon (hko):**The tetrahexahedron is made up of 24 faces, each of which is an isosceles triangle. Each face intersects one axis at unity, the second axis at some multiple of unity and is parallel to the third. The most common tetrahexhedron is (210).**Tris-octahedron (hhl):**This crystalline form of cubic system has 24 congruent faces meeting on the edges of a regular octahedron. Each face intersects two crystallographic axes at unity and the third at some multiple of it. The most common trisoctahedron is (221).**Hexoctahedron (hkl):**This form is made up of 48 triangular faces. Each face intersects all three crystallographic axes at unequal distances. A common hexoctahedron is (321).

### (2). Pyrite type of Cubic/Isometric Crystal Systems:

**Symmetry Elements: **The symmetry elements of this class of cubic crystals are as follows.

- Plane 3 (axial)
- Axes 7
- 3 crystallographic axes of 2-fold symmetry
- 4 diagonal axes of 3-fold symmetry

**Forms: **Pyritohedron and diploid are the important forms of the pyrite class of cubic/isometric crystal system.

**Pyritohedron (hko):**This form of pyrite class is composed of 12 pentagonal faces. Each face intersects one axis at unity, the second axis at some multiple of unity, and is parallel to the third. The most common pyritohedron is (210). There are two pyritohedra (i) Positive (210), and (ii) Negative (120). A rotation of 90° about one of the crystallographic axes brings the positive pyritohedron into the negative position, the positive and negative pyritohedron together form 24 faces of the tetrahexahedron of the normal class.**Diploid (hkl):**The second form of the pyrite class of cubic crystals is known as the diploid form, which is composed of 24 faces that correspond to one-half of the faces of the hexoctahedron of the normal class. Each face intersects the axes at unequal distances. A common diploid is Diploid-(321). There are two diploids in this form; one is positive (321), and the other is negative (312). The positive and negative forms together comprise all the faces of the hexoctahedron of the normal class.

In addition to the pyritohedron and diploid, the other forms such as cube, octahedron, dodecahedron, trisoctahedron, and trapezohedron, are also found in the pyrite type class. These forms are geometrically similar to the normal class but they show a lower symmetry due to the presence of striations and etch marks.

### (3). Tetrahedrite Class of Cubic/Isometric Crystal System:

**Symmetry Elements of Tetrahedrites: **The symmetry elements of this class are as follow:

- Planes 6, (diagonal)
- Axes:
- 3 crystallographic axes of 2-fold symmetry
- 4 diagonal axes of 3-fold symmetry

**Forms:** Tetrahedron(lll), Tristetrahedron(hll), deltoid-dodecahedron (hhl), Hexatetrahedron (hkl) are the four typical forms of this crystal class.

**Tetrahedron (lll):**It is a solid bounded by four faces each of which is an equilateral triangle. Each face intersects the crystallographic axes at equal lengths. There are two tetrahedrons: (i)positive (lll) and negative (ll’l).**Tristetrahedron (hll):**This forms has 12 faces. Its faces correspond to one-half of those of the trapezohedron of normal class. Each face intersects one axis at unity and the other two at equal multiples. The most common tristetrahedron is (211). There are two tristetrahedrons; one is positive (hll), and the other is negative (hl’l).**Deltoid Dodecahedron (hhl):**This form has 12 faces has 12 faces, which correspond to one-half of those of the trisoctahedron of the normal class. Each face intersects the two axes at equal lengths and the third at a greater length. The common deltoid form is ”deltoid (221). There are, further, two types of Deltoid Dodecahedrons, one is positive (hhl), and the other is negative (hh’l).**Hextetrahedron (hkl):**This form of tetrahedroid class has 24 triangular faces, which correspond to one-half of the faces of the hexoctahedron of the normal class. Each face intersects all the three crystallographic axes at different lengths. The most common hextetrahedron is (321). Further more, there are two types of hextetrahedrons: (i) positive (hkl), (ii) negative (hk’l).

The other forms found in tetrahedrite class are cube, dodecahedron, and tetrahexahedron. These are geometrically similar to the normal class but have a lower symmetry due to difference in their molecular structure.

**Related Posts:**

- Introduction to Crystallography
- Unit Cells of the Crystal Lattice
- Crystallographic Axes
- Interfacial Angle of Crystals
- Parameters of Crystal faces
- Crystallographic Notation
- Forms of Crystal faces
- Symmetry Elements of Crystals
- Six Types of Crystal Systems