All 32 Crystal classes including triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic or isometric system.

1. ISOMETRIC SYSTEM
In this system, all the three crystallographic axes are equal in length, and are
perpendicular to each other i.e.
a1 = a2 = a3;
a1 | a2 | a3
1. NORMAL CLASS
SYMMETRY ELEMENTS:
1. AXIS OF SYMMETRY: It is an imaginary line passing through the crystal, which
when rotated through an angle of 360 ̊, similar faces/ edges/ solid angles will come to
the space (similar configuration) twice, thrice, four or six times.
Crystals belonging to this class are characterized by three axes of 4-fold symmetry (tetrad),
which are coincident with the crystallographic axes. There are four axes of 3-fold symmetry
(triad), which emerge in the middle of each of the octants formed by intersection of
crystallographic axes.
In addition, there are six axes of 2-fold rotation (diad), each of which bisects one of the
angles between two crystallographic axes. In totality, there are 13 axes of symmetry; 34
, 43
,
and 62
.

2. 2. PLANE OF SYMMETRY: It is an imaginary plane passing through the crystal,
dividing it into two parts, such that one part is the mirror image of the other.
There are 9 planes of symmetry in this class of cubic system: 3 axial planes (along 2
crystallographic axes), and 6 diagonal planes.
3. CENTRE OF SYMMETRY: It is an imaginary point in the crystal that any line drawn
through it intersects the surface of the crystal at equal distance on either side.

3. FORMS:
We have 7 forms in the normal class of isometric system, also known as Galena-type.
CUBE
6 faces
OCTAHEDRON
8 faces

4. RHOMBIC-DODECAHEDRON
12 faces

5. TRIS OCTAHEDRON
24 faces
Front Back
TETRAHEXAHEDRON
Front
Back
24 faces

6. TRAPEZOHEDRON
Front
Back
24 faces

7. T
e
t
r
a
h
e
d
r
a
T
HEXOCTAHEDRON
Back
48 faces
Front

8. 2. TETRAHEDRAL CLASS
The symmetry of this class is as follows: There are three axes of binary symmetry, which
coincide with the crystallographic axes. There are also four diagonal axes or trigonal
symmetry, which coincide with the octahedral axes. There are six diagonal planes of
symmetry. There is no centre of symmetry.
FORMS:
Tetrahedron
The tetrahedron, as its name indicates, is a four-faced solid, bounded by planes meeting the
axes at equal distances. Its general symbol is (111). Each of the four faces of a tetrahedron is
an equilateral triangle. The tetrahedron is the regular triangular pyramid of geometry, but
crystallographically it must be so placed that the axes join the middle points of opposite
edges, and one axis is vertical.
There are two possible tetrahedrons:
The positive tetrahedron (111), having the symbols (1 1 1), (1̅ 1̅ 1), (1 1̅ 1̅ ), (1̅ 1 1̅) and the
negative tetrahedron, having the same geometrical form and symmetry, but the indices of its
four faces are(1̅ 1 1), (1 1̅ 1), (1 1 1̅ ),( 1̅ 1̅ 1̅).

9. Tetragonal Tristetrahedron
The general symbol is (221). There are twelve faces, each a quadrilateral, belonging to this
form, distributed as determined by the tetrahedral type of symmetry. They correspond to
twelve of the faces of the trisoctahedron, namely, all those falling in, alternate octants. This
type of solid is sometimes called a tetragonal tristetrahedron, or a deltoid dodecahedron. It
does not occur alone among crystals, but its faces are observed modifying other form. There
is also a complementary negative form, corresponding to the positive form, related to it in
precisely the same way as the negative to the positive tetrahedron. Its twelve faces are those
of the trisoctahedron, which belong to the other set of alternate octants.
Trigonal Tristetrahedron
The general symbol is (hll), here (211); it is bounded by twelve like triangular faces,
distributed after the type demanded by tetrahedral symmetry, and corresponding
consequently to the faces of the alternate octants of the form (hll) - the trapezohedron - of the

10. normal class. This type of solid is sometimes called a trigonal tristetrahedron or
trigondodecahedron.
Hextetrahedron
It has the general symbol (hkl), here (321), and is bounded by twenty-four faces distributed
according to tetrahedral symmetry, that is, embracing all the faces of the alternate octants of
the forty-eight-faced hexoctahedron. This form is sometimes called a hextetrahedron or
hexakistetrahedron.

11. 3. PYRITOHEDRAL CLASS (DIPLOIDAL CLASS)
A. Hermann-Maugin Symbol- 2/m 3
B. Center Of Symmetry- Present
C. Plane Of Symmetry- 3 plane of symmetry
D. Axis Of Symmetry- 3 axis of 2 fold symmetry and 4 axis of 3 fold symmetry
E. Minerals – Pyrite,Sperrylite
F. Forms-There are 7 forms present in the pyritohedral class of isometric system.
 Cube-In this form there are 6 faces each being a square. It has The miller
indices of the form is {100}
 Dodecahedron(rhombic)-In this form there are 12 faces each being a
rhombic. It has 24 edges and 14 vertices. The miller indice of the form is
{110}.
 Octahedron-In this form there are 8 faces each being an equilateral
triangle.There are 12 edges and 6 vertices. The
miller indices of this form is {111}.

12.  Pyritohedron-It is a type of dodechahedron with 12 faces each being a
pentagon. The miller indices of the form is {210} or {hko}.The negative
pyritohedron has the miller indices of {120} or {kho}.
 Trisoctahedron- In this form there are 24 faces which resembles three
triangular faces for every face of an octahedron.The miller indices of this form
is {221} or {hhk}.
 Trapezohedron-In this form there are 24 faces. The miller indices of this
form is {211} or {hkk}.

13.  Diploid- In this form there are 24 faces. The miller indices of this form is
{321} or {hkl} and the negative diploid has the miller indices of {231} or
{khl}.
4. PLAGIOHEDRAL CLASS
Axis of Symmetry:
3 axes of four-fold symmetry
4 axes of three-fold symmetry
6 axes of two-fold symmetry
Total – 13
Plane of Symmetry:
Absent
Centre of Symmetry:
Absent
FORMS
1. Deltoidal Icositetrahedron:
It is a 24-faced dual polyhedron with each face made up of 4 edges, also known as a
trapezoidal icositetrahedron.

14. Pentagonal Icositetrahedron:
It is a 24-faced dual polyhedron with each face made up of 5 edges. This form is known as
the Cuprite Type.
5. TETARTOHEDRA LCLASS
Symmetry and Typical Forms:
The fifth remaining possible class under the isometric system is illustrated, which represents
the twelve-faced solid corresponding to the general symbol (hkl). The distribution of its faces
is 18. This form is sometimes called a tetrahedral-pentagonuldodecahedron. It is seen to have
one-fourth as many faces as the form in the normal class. Hence there are four similar solids
which together embrace all the faces of the hexoctahedron .These four solids, which are
distinguished as right-handed (positive and negative) and left-handed (positive and negative),
are enantiomorphic, and hence the salts crystallizing here may be expected to also how
circular polarization .There main forms of the class are (besides the cube and rhombic
dodecahedron) the tetrahedrons, the pyritohedrons. The tetragonal and trigonal tris-
tetrahedron geometrically they are like the solids of the same names already described. This
class has no plane of symmetry and no center of symmetry.
There are three axes of binary symmetry normal to the cubic faces, and four axes of trigonal
symmetry normal to the face, so the tetrahedron

15. This group is illustrated by artificial crystals of barium nitrate, strontium nitrate, sodium
chlorate etc. Further, the species ullmanmte, which shows some time pyritohedron and again
tetrahedral forms, both having the same composition, must be regarded in this class.
Symmetry Elements
3A2
, 4A3
Three 2-fold axes and four 3-fold axes of rotational symmetry.
Three 2-fold axes of rotational symmetry are coincident with the crystallographic axes
whereas four 3-fold axes through the corners of the crystal.
General Form
The general form, thetetartohedron, sometimes known as the tetrahedral-pentagonal
dodecahedron, has 12 faces. The form misanalogoustoone-fourth the faces of the
hexoctahedron, making possible four similar solids: left positive and negative, and right
positive and negative. The class is enantiomorphic .The general symbol for the form is {hkl}:
{231}.

16. Miler Indices:

17. TETRAGONAL SYSTEM
The Tetragonal System includes all the forms which are referred to three axes at right angles
to each other of which the two horizontal axes are equal to each other in length and
interchangeable and the third, the vertical axis, is either shorter or longer. The horizontal axes
are designated by the letter a; the vertical axis by c. The length of the vertical axis expresses
properly the axial ratio of a:c, a being uniformly taken as equal to unity. The axes are
orientated and their opposite ends designated by plus and minus signs exactly as in the case
of the Isometric System.
Seven classes are embraced in this system. Of these the normal class is common and
important among minerals; two others have several representatives, and another a single one
only. It may be noted that in four of the classes the vertical axis is an axis of tetragonal
symmetry; in the remaining three it is an axis of binary symmetry only.
1. NORMAL CLASS ZIRCON TYPE
SYMMETRY
The forms belonging to the normal class of the tetragonal system have one principal axis of
tetragonal symmetry which coincides with the vertical crystallographic axis, c. There are also
four horizontal axes of binary symmetry, two of which coincide with the horizontal
crystallographic axes while the other two are diagonal axes bisecting the angles between the
first two.
Further they have one principal plane of symmetry, the plane of the horizontal
crystallographic axes. There are also four vertical planes of symmetry which pass through the

18. vertical crystallographic axis c and make angles of 45" with each other. Two of these latter
planes include the horizontal crystallographic axes and are known as axial planes of
symmetry. The other two are known as diagonal planes of symmetry.
Plane of Symmetry-5 (Vertical – 4, Horizontal – 1)
Axis of Symmetry-5 (14
– vertical, 42
– horizontal)
Centre of Symmetry Is Present
The various possible forms under the normal class of this system are as follows:
Symbols
1. Base or basal pinacoid 2 Face (001)
2. Prism of the first order 4 Face (110)
3. Prism of the second order 4 Face (100)
4. Ditetragonal prism 8 Face (hk0), (210)
5. Pyramid of the first order 8 Face (hhl), (223)
6. Pyramid of the second order 8 Face (h01), (203)
7. Ditetragonal pyramid 16 Face (hkl), (321)
Base or basal pinacoid
The base is the form which includes the two similar faces which are parallel to the plane of
horizontal axes, The faces have the indices (001) and (001) respectively. The general symbol
is (001). It is open form as they do not enclose a space, consequently these forms can occur
only in combination with the other forms.

19. Prisms
Prisms have been defined to the forms whose faces are parallel to the vertical axis of crystal,
while they meet the two horizontal axis. In tetragonal system, the four faced solid form
whose planes are parallel both to the vertical axis and horizontal axis is also called prism.
There are three types of prisms.
Prism of First order:
It is bounded four similar faces which are parallel to vertical axis; meet the
horizontal axes at equal distances. Its general symbol is (110). It is a
prism with interfacial angle 90°. The indices of the four faces are (110),
(110), (110), (110)
Prism of Second order:
It includes four faces which are parallel to
vertical axis and to a horizontal
axis. Therefore it has the general symbol is
(100). The angle between any two
adjacent face is 90 ° They have indices
(100) (010) (100) (010).
Ditetragonal Prism:
It is bounded eight similar faces, each of which is parallel to the vertical
axis, while meeting the two horizontal axis at unequal distances, It has
general symbol is (hk0). The common forms of the general type have
indices is. (310), (210), (320) etc.

20. Pyramids.
There are three types of pyramids corresponding to the prisms, lt. Is a closed form having
eight faces, each face of which cuts the vertical axis and cuts one or more horizontal axes, at
unequal or equal distances.
Pyramid of First order.
It is a form whose similar eight faces are intersecting the two horizontal axes at unit distance
and also intersect the vertical axis. Hence the general symbol is (hhl) or (221)
Pyramid of Second order:
It is a form whose similar eight faces intersect the one of the horizontal axes and vertical axis
and parallel to the other horizontal axis. The general symbol is (h0l) or (201)

21. Ditetragonal Pyramid.
Ditetragonal pyramid or double eight sided pyramid is a closed form, of each faces intersect
three crystallographic axes at unequal distances. Hence lts general symbol is (hkl) as (321)
etc.
Mineral crystallizing in this class
Important minerals crystallizing in this class are Zircon, Cassiterite, Rutile,
Appophyllite,Vesuvianite, Octahedrite etc.
HEMIMORPHIC CLASS

22. 2. SPHENOIDAL CLASS (Aka chalcopyrite type symmetry)
Two vertical diagonal plane of symmetry
Three crystallographic axis of two fold symmetry
FORMS
The typical forms present in this class of the system are sphenoid and
tetragonal scalenohedron
1.Sphenoid
It is a four faced solid resembling a tetrahedron but each face is isosceles triangle. . It
may be considered as derived from the first order pyramid of the normal class by the
development of only the alternate faces of the latter. There are therefore possible two
complementary forms known as the positive and negative sphenoid .The general symbol of
the positive unit sphenoid is (111) and its faces have the indices: 111, 1ത1ത1, 1ത11ത, 11ത1ത.
Negative sphenoid has the number (11ത1).
Sphenoid
2.Tetragonal scalenohedron
It is bounded by eight similar faces, each face is scalene triangle, hence it is called
tetragonal scalenohedron The general symbol is (hkl) It may be considered as derived from
the ditetragonal pyramid of the normal class by taking the alternate pairs of faces of latter
form.

23. Tetragonal scalenohedron
HEMIMORPHIC FORM:
It is also derived from a holohedral form and has only half the number of faces as in
hemihedral form. In this case, all the faces of the form are developed only on one extremity
of the crystal , being absent from the other extremity . In other words , such a crystal will not
be symmetrical with reference to centre of symmetry.
So, hemimorphism is the property wherein the two ends of a crystallographic axis are not
related by symmetry ; thus , the faces that terminate the axis on opposing ends are not
symmetrically equivalent . One of the minerals that best exemplifies hemimorphism in its
morphology is tourmaline. By virtue of the lack of a centre of symmetry, all hemimorphic
crystals also exhibit the property of piezoelectricity. Ultimately , it is the atomic arrangement
of the crystals that dictates it is hemimorphic or not. Of the 32 classes, 9 are hemimorphic. In
these the two ends of crystallographic axis are not equivalent by symmetry. This requires the
absence of the following symmetry elements: a centre of symmetry, a mirror plane oriented
perpendicular to the unique crystallographic axis, and a two fold axis of rotation oriented
perpendicular to unique crystallographic axis.
3. DITETRAGONAL-PYRAMIDAL CLASS, 4MM
Symmetry content - 1A4, 4m
This class has a single 4-fold axis and 4 mirror planes. The mirror planes are
not shown in the diagram, but would cut through the edges and center of the
faces shown. Note that the ditetragonal pyramid is a set of 8 faces that form a
pyramid on the top of the crystal. Only one rare mineral forms in the crystal
class.
The Ditetragonal-pyramidal Class

24. (= Hemimorphy of Holohedric ) 4 m m
Because the Tetragonal Crystal System recognizes a main axis (that is) to be
distinguished from the other crystallographic axes, a special type of hemihedric, namely
hemimorphy is possible in all cases where there initially is a mirror plane perpendicular to
this main axis. To suppress this mirror plane is to apply a hemimorphy. Each bipyramid
and each basic pinacoid decays into an upper and lower half which are independent of
each other. Consequently we can find, for example, a combination in which the upper
faces are the faces of a hemimorphous pyramid, which is closed at the bottom by a
hemimorphous (basic) pinacoid, which is called a pedion.
The symmetry content of this Class is :
Four (2 + 2) vertical mirror planes.
One 4-fold polar rotation axis.
The different forms of Ditetragonal pyramidal (hemimorphic) forms are as follows-
1. Pedion
2. Pedion
3. Tetragonal prism
4. Ditetragonal prism
5. Tetragonal pyramid
6. Tetragonal pyramid
7. Ditetragonal monopyramid
A Pedion or Monohedron is derived by hemimorphy from. It is a
Basic Pinacoid.
Miller indices: {001}
A second Pedion, also derived from a Basic Pinacoid by
hemimorphy.
Miller indices: {001}
Miller indices: (100) and (110)

25. A Protoprism does not change its outer shape when subjected to hemimorphy, but despite its
outer appearance it does not possess a main mirror plane anymore which can become evident
by physical features.
Tetragonal prism ii yields tetragonal prism ii but of lower symmetry.
Ditetragonal prism yields Ditetragonal prism of lower symmetry.
The tetragonal pyramid of type I derived from coloured faces of protopyramid.
Symbol: (hol)
Tetragonal pyramid of type I, derived from the white faces of protopyramid
Symbol : (hol)

26. Tetragonal Pyramid of II type, derived from the Deuteropyramid ( = bipyramid of type
II))
Symbol: (hh݈)
Symbol: (hk݈)
A Ditetragonal Monopyramid, derived from a holohedric Ditetragonal Bipyramid by
applying a hemimorphy.

27. 4. TRIGONAL-TRAPEZOHEDRAL CLASS
SYMMETRY ELEMENTS :
AXIS OF SYMMETRY - 4 AXIS OF 2-FOLD SYMMETRY , 1 AXIS OF 4-FOLD
SYMMETRY
PLANE OF SYMMETRY - ABSENT
CENTRE OF SYMMETRY - ABSENT
FORMS:
The trapezohedral class is analogous to the plagiohedral class of the isometric system, it is
characterized by the absence of any plane or centre of symmetry
Vertical axis is an axis of tetragonal symmetry and perpendicular to this there are four axes of
binary symmetry
GENERAL FORM – (hkl)

28. There are two complementary forms called right and left handed which embraces all the faces
of the ditetragonal pyramidof the normal class
5. TRIPYRAMIDAL CLASS (Scheelite type)
Symmetry
1 vertical axis is an axis of 4 fold symmetry
One horizontal plane of symmetry
Centre of symmetry present
Forms
The distinctive forms of this class are
1. Tetragonal prism of third order
(4faces (hko))
2. Tetragonal pyramid of third order
(8 faces (hkl))
1. Prism of third order
This itself is a square prism identical in appearance to the prism of 1st
and
2nd
order, represented by 4 faces, each meeting 2 horizontal axis at different length
and being parallel to the c axis. Hence general symbol is (hko). The form is
considered to be derived from the ditetragonal prism by the development of half of
its faces. Hence these are two complimentary forms designated as left and right
handed which together account for all the faces of the ditetragonal prism of the
normal class.
2. Pyramid of third order
It is a square pyramid bounded by 8 similar faces each a triangle. It is
similar in appearance to the pyramid of 1st
order and 2nd
order. Each face meets the 3
crystallographic axes at unequal distances giving the general index (hkl). The form
may be considered to be derived from the ditetragonal pyramid by the development
of half of its faces. Hence there will be 2 complimentary forms-right and left handed
whose faces together will account for the faces of the holohedral form.
Other forms of this class- base(001), prism 1st
order & 2nd
order,
p
y
r
a
m
i

29. ds 1st
& 2nd
order. These are geometrically like the corresponding forms in the
normal class but with the symmetry of tripyramidal class. This class shows,
therefore, three types of square pyramid , and hence is called the tripyramidal class.
The indices of faces of 2 complementary prisms as (210) are;
Left: 210, ̄120, ̄2̄10, 1̄20
Right: 120, ̄210, ̄1̄20, 2̄10
The indices of faces of corresponding pyramids as (212) are;
Left: above 212, ̄122, ̄2̄12, 1̄22;
below 21̄2, ̄12̄2, ̄2̄1̄2, 1̄2̄2
Right: above 122, ̄212, ̄1̄22, 2̄12
below 12̄2, ̄21̄2, ̄1̄2̄2, 2̄1̄2
Minerals
1. Scheelite (CaWO4 )
2. Wulfrenite (PbMoO4 )
3. Powellite (CaMoO4 )
4. Meionite (a variety of scapolite)
5. Fergusonite
6. Pinnoite

30. 6. TETARTOHEDRAL CLASS
SYMMETRY
Vertical axis of binary symmetry
No plane of symmetry
No center of symmetry
The symmetry and the distribution of the faces of the general form(hkl)is shown in the
stereographic projection
Forms
1)Third order sphenoid
Third order sphenoid is the only one form present in tetartohedral class.
PYRAMIDAL-HEMIMORPHIC CLASS. WULFENITE TYPE
(Tetragonal Pyramidal or Hemihedral Hemimorphic Class)
Symmetry -The fourth class of the tetragonal system is closely related to the class just
described. It has the same vertical axis of tetragonal symmetry, but there is no horizontal
plane of symmetry. The forms are, therefore, hemimorphic in the distribution of the faces.

31. The species wulfenite of the Scheelite Group among mineral species probably belongs here,
although the crystals do not always show the difference between the pyramidal faces, above
and below, which would characterize distinct complementary forms. In the following figure
a characteristic distinction is exhibited.
In these figures the forms are u(102),e(101), n(ll1) ; also f (230), k(210). z(432),x(311).

32. HEXAGONAL SYSTEM
The Hexagonal system contains the crystals that can be referred to four axes,- three equal
horizontal axes making angles of 1200
with each other, & a vertical axis perpendicular
to the plane containing the horizontal axes. The three horizontal axes are lettered a1, a2, a3
and the vertical axis is c. Hence, the symbols of faces in the Hexagonal System have four
numbers and the axial order is a1 a2 a3 c.
The Normal or highest symmetry class belonging to the Hexagonal system is called The
Dihexagonal-bipyramidal Class.
1. Dihexagonal-bipyramidal class or Normal class
SYMMETRY:
PLANES OF SYMMETRY: There are seven planes of symmetry with one horizontal
plane of symmetry & six vertical planes of symmetry.
Planes, 7 {4 Axial (1 horizontal, 3 vertical)}
{3 diagonal, vertical}
AXES OF SYMMETRY: There are seven axes of symmetry. One axis i.e., the vertical
crystallographic axis is an axis of six-fold symmetry. Other than that we have six axes
showing two- fold symmetry. They are 3 horizontal crystallographic axes and 3 diagonal
axes.
Axes, 7{6II
(horizontal, 3 crystallographic axes, 3 diagonal)}
{1VI
(vertical crystallographic axis)}
CENTRE OF SYMMETRY: There is one centre of symmetry.
FORMS:
We have seven forms in the normal class including the open forms and the closed
forms.

33. 1. Basal pinacoid- This is an open form consisting of two faces, each cutting the
vertical axis & being parallel to the three horizontal axes. Symbols of the faces are
(0001) & (0001).
2. Hexagonal Prism (11૛0) or of Second Order- This open form is a prism of
6 faces, the horizontal crystallographic axes joining the centres of opposite parallel faces.
Each face is parallel to the vertical axis, and cuts all three horizontal axes, one at the unit
distance & the other two at twice this distance. The miller indices of the 6 faces are
(1120), (2110), (1210), (2110), (1120), (1210).
3. Hexagonal Prism (10૚0) or of First Order- This open form is a prism of 6
faces, each face parallel to the vertical axis & to one horizontal axis and meeting the
other two horizontal axes at equal distances. The miller indices of the 6 faces are (0110),
(1100), (1010), (0110), (1100), (1010).
4. Dihexagonal prism- This open form has 12 faces each parallel to the vertical axis
and meeting the three horizontal axes at three different length. Its general form is (hi݇0).
The miller indices for this form are (3120), (2130), (1230), (1320), (2310), (3210),

34. (3120), (2130), (1230), (1320), (2310), (3210).
5. Hexagonal Bipyramid (second order) (hh૛ࢎl) - This is a closed form
analogous with the Second Order Prism (1120); each of its 12 faces cuts vertical axis and
all three horizontal, one at the unit distance and the other two at twice this distance. The
miller indices of the faces are (1121), (2111), (1211), (2111), (1121), (1211), (1121),
(2111), (1211), (2111), (1121), (1211).
.
6. Hexagonal Bipyramid (first order) (h0ࢎl)- This is a closed form analogous
with the First Order Prism (1010); It consists of 12 faces each cutting two horizontal
axes at equal distances and being parallel to the other horizontal axis, & cutting the
vertical axis.The miller indices of the faces are (1011), (1101), (1011), (0111), (1101),
(1011), (1011), (1101), (1011), (0111), (1101), (1011).

35. 7. Dihexagonal Bipyramid- This is a double 12 sided pyramid with all total 24
faces. Each face cuts the horizontal axes at unequal distances and also the vertical axis.
The symbol for the general form is (hk݅l). The miller indices of the faces are (3121),
(2131), (1231), (1321), (2311), (3211), (3121), (2131), (1231), (1321), (2311),
(3211), (3121), (2131), (1231), (1321), (231ሬԦ1), (3211), (3121), (2131), (1231), (1321),
(2311), (3211).

36. 2. TRIGONAL CLASS

37. 3. HEMIMORPHIC CLASS (ZINCITE)
Axis of Symmetry:
One Vertical axis of Six fold symmetry.
Plane of Symmetry:
Six vertical planes.
Centre of Symmetry:
Centre of symmetry is Absent.
Forms:
The forms has a basal plane (or pedions).
Positive pyramid of three face and negative pyramid of three face.
Positive prism of three face and negative prism of three face.
.
Zincite

38. 4. TRAPEZOHEDRAL CLASS:
β-quarts type (hexagonal trapezohedral, trapezohedral hemihedral, or hexagonal holoaxial)
SYMMETRY ELEMENTS
AXIS OF SYMMETRY:
1 VERTICAL AXIS OF 6 FOLD SYMMETRY
6 HORIZONTAL AXIS OF 2 FOLD SYMMETRY
PLANE OF SYMMETRY: NOT PRESENT
But the vertical axis is an axis of hexagonal symmetry, and there are, further, six horizontal axes of
binary symmetry.
CENTRE OF SYMMETRY: NOT PERESENT
The symmetry and the distribution of the faces of the typical form (hki᷈l) is shown in the stereographic
projection. The typical forms may be derived from the dihexagonal pyramid by the omission of the
alternate faces of the latter. There are two possible types known as the right and the left hexagonal
trapezohedrons, which are enantiomorphous, and the few crystallised salts falling in this class show
circular polarization. A modification of quarts known as β-quartz is also described as belonging here.
The indices of the right form (213᷈1) are as follows:
5. TRIPYRAMIDAL CLASS
This class is important because it includes the common species of the Apatite Group,
apatite, pyromorphite, mimetite, vanadinite.

39. The typical form is the hexagonal prism (hk0) and the hexagonal prism (hkl), each
designated as third order.
These forms ( Third Order Prism and Third Order Pyramid ) may be considered as derived
from the corresponding dihexagonal forms of the normal class by the omission of one half of
the faces of the latter.
Prism and Pyramid of the Third Order
The prism of the third order has six like faces and the form is a regular hexagonal prism with
angles of 60°.
Fig :- Third Order Prism
The six faces of the right-handed form (203̅0) have the indices :-
213̅0, 1̅32̅0, 3̅210, 2̅1̅30, 13̅20, 32̅1̅0.
The faces of the complementary left-handed form have the indices :-
123̅0, 2̅31̅0, 3̅210, 1̅2̅30, 23̅10, 31̅2̅0
These two forms together embrace all the faces of the dihexagonal prism.
The pyramid is also a regular double hexagonal pyramid of the third order, and in its
relations to the other hexagonal pyramids of the class it is analogous to the square pyramid of
the third order met with in the corresponding class of the tetragonal system.

40. Fig (a) :- Third 0rder Pyramid
The faces of the right-handed form ( 213̅1 ) are :-
Above - 213̅1, 1̅32̅1, 3̅211, 2̅1̅31, 13̅21, 32̅1̅1.
Below - 213̅1̅, 1̅32̅1̅, 3̅211̅, 2̅1̅31̅, 13̅21̅, 32̅1̅1̅.
There is also a complementary left-handed form, which with this embraces all the faces of the
dihexagonal pyramid.
Fig (b) :- Third Order Pyramid
The cross section of Fig.(b) shows in outline the position of the first order prism, and also
that of the right-handed prism of the third order.

41. The prism and pyramid just do not often appear on crystals as predominating forms, though
this is sometimes the case, but commonly these faces are present modifying other
fundamental forms.
Other Forms
The remaining forms of the class are geometrically like those of the normal class , the base
(0001), the first order prism (101̅0), the second order prism (112̅0), the first order pyramids
(h̅ 0h̅ l), and the second order pyramids (h.h.2h̅ .l).
This class is given its name of Tripyramidal because its forms include three distinct types of
pyramids.
6. TRIGONAL TETAROHEDRAL
Trigonal Tetarohedral class is belongs to Hexagonal System of crystal.
This class is important because Phosphate minerals show Trigonal Tetarohrdral crystal
structure.
This class is divided into two forms-
1. Rhombohedron
2. Scalenohedron
1. Rhombohedron- Rombohedron is bounded by six faces, each a rhomb. It has six like
lateral edges forming a zigzag line about the crystal, and six like terminal edges, three
above and three is alternate position bellow. The vertical axis joins the two trihedral solid
angels, and the horizontal axis joint the middle points of the opposite sides.
The general symbol of rhombohedron is (h0hl) and the successive faces of the
unit form (1 0 ̅1 1)
Indices of rhomohedron-

42. Fig- Positive Rhomohedron Fig- Negative Rhombohedron
Above - 1 0 ̅1 1, ̅1 1 0 1, 0 ̅1 1 1
Bellow- 01 ̅1 ̅1, ̅1 0 1 ̅1, 1 ̅1 0 ̅1
Positive and negative Rhomohedron-
To every positive rhomohedron there may be an inverse and complimentary form, identical
geometrically, but bounded by face falling in the alternative sectants. Thus the negative form
of the unit rhomohedron (0 1 ̅1 1) the following faces-
Above- 0 1 ̅1 1, ̅1 0 1 1, 1 ̅1 0 1
Bellow- ̅1 1 0 ̅1, 0 ̅1 1 1, 1 0 1 ̅1
2. Scalenohedron- The general symbol of Scalenohedron is h k ̅l l. It is the solid bounded
by twelve faces, each a scalene triangle. It has roughly the shape of a double six sided
pyramid, but there are two sets of terminal edges, one more obtuse than the outer, and the
lateral edges form a zigzag edge around the form like that the rhombohedrum.
Indices of scalenohedron-
Above :- 2 1 ̅3 1, ̅2 3 ̅1 1, ̅3 2 1 1, ̅1 ̅2 3 1, 1 ̅3 2 1, 3 ̅1 ̅2 1
Bellow :- 1 2 ̅3 ̅1, ̅1 3 ̅2 ̅1, ̅3 1 2 ̅1, ̅2 ̅1 3 ̅1, 2 ̅3 1 ̅1, 3 ̅2 ̅1 ̅1
For negative scalenohedron –
Above – 1 2 ̅3 1, ̅1 3 2 ̅1, ̅3 1 2 1, ̅2 ̅1 3 1, 2 ̅3 1 1 , 3 ̅2 ̅1 1

43. Bellow – ̅2 3 ̅1 ̅1, ̅3 2 1 ̅1, ̅1 ̅2 3 ̅1, 1 ̅3 2 ̅1, 3 1 ̅2 ̅1, 2 1 ̅3 ̅1
Fig – Scalenohedron
7. PYRAMIDAL-HEMIMORPHIC CLASS.
NEPHELITE TYPE.
It is also known as Hexagonal pyramidal or Pyramidal Hemihedral
Hemimorphic Class
This is the fourth class under hexagonal division, as the name suggest the forms
are hemimorphic
PLANES OF SYMMETRY:
Single horizontal plane of symmetry is absent
AXES OF SYMMETRY:
Vertical axes of 6 fold symmetry

44. Figure 1: Symmetry Of Pyramidal-Hemimorphic Class
FORMS:
The form would be like the upper half of pyramid of third order
The species nephelite belong here

45. TRIGONAL SYSTEM
1. RHOMBOHEDRAL CLASS. CALCITE TYPE
The typical forms of the rhombohedral class are the rhombohedron and the scalenohedron.
There are three planes of symmetry only; these are diagonal to the horizontal crystallographic
axes and intersect at angles of 60° in the vertical crystallographic axis. This axis is with these
forms an axis of trigonal symmetry; there are, further, three horizontal axes diagonal to the
crystallographic axes of binary symmetry. This group is hence analogous to the tetrahedral
class of the isometric system, and the sphenoidal class of the tetragon 111 system.
Rhombohedron (fig.244)- Geometrically described, the rhombohedron is a solid bounded by
six like faces, each a rhomb. It has six like lateral edges forming a zigzag line about the
crystal, and six like terminal edges, three above and three in alternate position below. The
vertical axis joins the two trihedral solid angles, and the horizontal axes join the middle
points of the opposite sides.
The general symbol the rhombohedron is (h0h̅ l), and the successive faces of the unit form
(101̅1) have the indices:
Above, 101̅1, 1̅101, 01̅11; below, O11̅1̅, 1̅011̅, 11̅01̅.
The geometrical shape of the rombohedron varies widely as the angles change, and
conseqently the relatIve length of the vertical axis c (expressed in terms of the horizontal
axes, a = 1). As the vertical axis diminishes, the rhombohedrons become more and more
obtuse or flattened; and as it increases they become more and more acute. A cube placed with
an octahedral axis vertical is obviously the limiting case between the obtuse and acute forms
where the interfacial angle is 90°. In Fig. 244 of calcite the normal rhombohedral angle is
74°55' and c = 0·854, while for Fig. 246 of hematite this angle is 94° and c = 1·366.

46. Positive and Negative Rhombohedrons. - To every positive rhombohedron there may be an
inverse and complementary form, identical geometrically, but bounded by faces falling in the
alternate sectants. Thus the negative form of the unit rhombohedron (011̅1) shown in Fig. 245
has the faces:
Above, 011̅1, 1̅011, 11̅01; below, 1̅101̅, 01̅11̅, 101̅1̅.
Of the figures already referred to, Figs. 244, 246, 250 are positive, and Figs. 245, 247, 248,
249 negatIve, rhombohedrons; Fig. 251 shows both forms.
It will be seen that the two complementary positive and negative rhombohedrons of given
axial length together embrace all the like faces of the double six-sided hexagonal pyramid of
the first order. In each case the form which is geometrically a double hexagonal pyramid (in
Fig. 254 with c and m), is in fact a combination of the two unit rhombohedrons, positive and
negative.
Scalenohedron (fig.259) - The scalenohedron, shown in Fig. 259, is the general form for this
class corresponding to the symbol h k ı̅ l. It is a solid, bounded by twelve faces, each a
scalene triangle. It has roughly the shape of a double six-sided pyramid, but there are two sets
of terminal edges, one more obtuse than the other, and the lateral edges form a zigzag edge
around the form like that of the rhombohedron. It may be considered as derived from the
dihexagonal pyramid by taking the alternating pairs of faces of that form. It is to be noted that
the faces in the lower half of the form do not fall in vertical zones with those of the upper

47. half. Like the rhombohedrons, the scalenohedrons may be either positive or negative.
The positive forms correspond in position to the positive rhombohedrons and conversely.
The positive scalenohedron (213̅1), Fig. 259 has the following indices for the several
faces: Above 213̅1, 2̅31̅1, 3̅211, 1̅2̅31, 13̅21 , 31̅2̅1
Below 123̅1̅, 1̅32̅1, 3̅121̅, 2̅1̅31̅, 23̅11, 321̅1̅.
For the complementary negative scalenohedron (1231) the indices of the faces are:
Above 123̅1, 1̅32̅1, 3̅121, 2̅1̅31, 23̅11, 32̅1̅1.
Below 2̅31̅1̅, 3̅2̅11̅, 1̅2̅31̅, 13̅21̅, 31̅2̅1̅, 213̅1̅.
2. RHOMBOHEDRAL HEMIMORPHIC CLASS (tourmaline type)
Axes of symmetry :
Vertical axis of 3 fold symmetry
3 Vertical diagonal planes of symmetry
Forms of symmetry :
Forms Miller indices
Pedions (0001)-1 face , (0001)-1 face
Trigonal prisms ( 1St
order ) (101ത0)-3 faces , (011ത0)-3 faces
2nd
order prism (112ത0)-6 faces
Ditrigonal prism (hkଓ̅0)- Right [6 faces]
Left [6 faces]
1st
order hemipyramid (h0݄l)-Upper [+ve-3 faces]
Upper [-ve-3 faces]
Lower [+ve-3 faces]
Lower [-ve-3 faces]
2nd
order hemipyramid (hh2݄തl)-Upper [6 faces]
Lower [6 faces]
Ditrigonal hemipyramid (hkଓ̅l)- Upper [+ve-3 faces]
Upper [-ve-3 faces]
Lower [+ve-6 faces]
Lower [-ve-6 faces]

48. 3. TRIGONAL – TRAPEZOHEDRAL
QUARTZ TYPE
SYMMETRY
3 Horizontal axes of twofold symmetry
Vertical axes of three fold symmetry
Plane of symmetry absent
Centre of symmetry absent
FORMS
The characteristic form of the Trapezohedral class is Trigonal Trapezohedron.
TRIGONAL TRAPEZOHEDRON
Hexagonal Trapezohedron is bounded by six similar faces, each face intersects three
crystallographic axes at unequal distances. Hence the general symbol is (h k i l). it may be
derived from the dihexagonal pyramid of the normal class by taking one quarter of the faces
of the latter. There are therefore four such Trapezohedrons, two positive, called respectively
right handed and left handed, and two similar negative forms, also right and left handed, and
two similar negative forms, also right and left handed. These forms are enantiomorphs.
Circular polarization is striking character of the species belonging to this class.
Other forms are geometrically like those of corresponding class of rhombohedral
class – calcite type. They are base (0001), prism of first order (1010) and negative and
positive as (1011) and (0111). This class includes minerals of the species quartz and cinnabar.

49. 4. TRIRHOMBOHEDRON CLASS
Symmetry elements:
i. Axis of symmetry: 13
, the vertical axis shows trigonal symmetry.
ii. Plane of symmetry: This class is characterised by the absence of all planes of
symmetry.
iii. Centre of symmetry: Centre of symmetry is present in this class.
Forms:
The distinctive forms of this class are the rhombohedron of the second order, and the
hexagonal prism and rhombohedron, each of the third order. The class is thus characterised
by three rhomboherdons of distinct types each + and –), and hence the name given to it.
The second order of rhombohedron may be derived by taking one half the faces of the normal
hexagonal pyramid of the second order. There will be two complimentary forms known as
positive and negative.
The rhombohedron of the third order has a general symbol of (hkଓ̅k) and may be derived from
the normal dihexagonal pyramid by taking one quarter of the faces of the latter.
The hexagonal prism of the third order may be derived from the normal dihexagonal prism by
taking one half of the faces of the latter.

50. 5. TRIGONAL TETARTOHEDRAL HEMIMORPHIC CLASS
 It is also known as Trigonal Pyramidal class.
 The Herman Mogen symbol is 3.
 The mineral Simpsonite is an example for this class.
 The miller indices are hkīl.
Axis of symmetry: 13
, the vertical axis shows trigonal symmetry.
Plane of symmetry: This class is characterized by the absence of all planes of symmetry.
Centre of symmetry: Centre of symmetry is Absent in this class.

52. ORTHORHOMBIC SYSTEM
1. Hemimorphic class
Symmetry
Two vertical plane of symmetry
One vertical axis of two fold symmetry
The forms belonging to hemimorphic class characterized by two unlike plane of symmetry
and one axis of binary symmetry. This class doesn’t posses the center of symmetry. The
forms are therefore hemimorphic. The two planes of symmetry are parallel to the pinacoids
‘a’ (100) and ‘b’ (010). The prisms geometrically like those of the normal class as the macro
pinacoid and brachy pinacoid. But the two basal planes become independent form (001) ,
(001ത). There are also two macrodomes (101) and (101ത) .
2. SPHENOIDAL CLASS (Epsomite Type)
The forms of the remaining class of the orthorhombic-sphenoidal class are characterized by
three unlike rectangular axis of binary symmetry which coincide with the crystallographic
axes.
Axis of symmetry: 32
. This class is characterized by three axes of two fold symmetry.
Plane of Symmetry: No planes of symmetry.
Centre of Symmetry: No centre of symmetry.

53. The general form is hkl here and has four faces only, and the corresponding solid is a
rhombic sphenoid, analogous to the sphenoid of the tetragonal system.
MONOCLINIC SYSTEM
The monoclinic system includes all the forms which are referred to three unequal axes,
having one of their axial inclinations oblique.
The axis are designated as follows: the inclined axis or the clino axis is a , the ortho axis is b
and the vertical axis is c.
1. NORMAL CLASS (Gypsum Type)
Axis of symmetry : 12
. This class is characterised by one axis of two fold symmetry.
Plane of Symmetry : There is one plane of symmetry.
Centre of Symmetry : Centre of symmetry is present.
Forms:
Orthopinacoid or a-pinacoid (100)
Clinopinacoid or b- pinacoid (010)
Base or c- pinacoid (001)
Prisms (hk0)
Orthodomes (h0l) (݄ത0݈ሻ
Clinodomes (0kl)
Pyramids (hkl) (݄ത݈݇ሻ

54. 1. Pinacoids:
There are three pinacoids in this class, orthopinacoid, clinopinacoid and basal
pinacoid.
a. Orthopinacoid (100): includes two faces parallel to the ortho axis and the vertical
axis.
b. Clinopinacoid (010): includes two faces parallel to the plane of symmetry, that is
the plane of the clino axis and the c axis.
c. Basal pinacoid (001): includes two terminal faces above and below, parallel to the
plane of axis a and b.
2. Prisms:
The prisms are all of one type, the oblique rhombic prisms. There are there types of
prisms:
a. Orthoprisms (hk0): also called macroprisms
b. Clinoprisms (hk0): also called brachyprisms
c. Unit prisms (110)
3. Orthodomes: the four faces parallel to the ortho-axis b,and meeting the other two axis,
fall into two sets of two each, having the general symbol (h0l) and (݄ത0݈ሻ.
4. Clinodomes: these are the forms whose faces are parallel to the inclined axis a, while
intersecting the other two axes.

55. 5. Pyramids: the pyramids in the the monoclinic system are are all hemipyramids,
embracing 4 faces only in each form, corresponding to the general form (hkl).
The figure above shows various mineral groups that crystallize in this system.
2. HEMIMORPHIC CLASS
Axis of symmetry : 12
. This class is characterised by one axis of two fold symmetry
Plane of Symmetry : There is no plane of symmetry.
Centre of Symmetry : Centre of symmetry is absent.

56. The hemimorphic character is distinctly show in the distribution of the clinodomes and
pyramids; corresponding to this the artificial salts belonging here often exhibit arked
pyroelectric phenomena.
3. CLINOHEDRAL CLASS OR DOMATIC CLASS
This class is also known as Clinohedral Class, Clinohedrite Type, Hemihedral, or Planar Class.
SYMMETRY ELEMENTS:
AXIS/CENTER OF SYMMETRY:
Minerals belonging to this class have no axes or center of symmetry; H-M symbol - m.
PLANE OF SYMMETRY
In this class there is a single plane of symmetry. Faces related by a mirror plane are called
domes.
MILLER INDICES:
(10 FACES)
TRICLINIC SYSTEM
The triclinic system includes all the forms which are referred to three unequal axes
with all their intersections oblique. When oriented one axis has a vertical position and

57. is called the c axis a second axis lies in the front-to-back plane, sloping toward the
observer, and is called the a axis. The remaining axis is designated as b. the a axis is
shorter as in the orthorhombic system and is called the bachy-axis.
1. NORMAL CLASS
Axis of symmetry : There is no axis of symmetry
Plane of Symmetry : There is no plane of symmetry.
Centre of Symmetry : Centre of symmetry is present.
Forms:
The various types of forms are given in the table below:
Macropinacoid or a-pinacoid (100)
Brachypinacoid or b- pinacoid (010)
Base or c- pinacoid (001)
Prisms (hk0) (h݇0ሻ
Macrodomes (h0l) (݄ത0݈ሻ
Brachydomes (0kl) (0݈݇ሻ
Pyramids (hkl) (݄ത݈݇ሻ (݄݇ത݈ሻ ሺ݄݇ത݈ሻ

58. The figure show various minerals that crystallize in the triclinic system.
2. Triclinic Asymmetric Class
 It is also known as Hemihedral or Pedial Class.
 It has no Planes of Symmetry and Axis of Symmetry.
 The form has one face only.
 The Herman Morgen symbol is 1.
 Calcium Thiosulphate (CaS2O3.6H2O) is an example for this class.
 The Miller indices are (100).