The discipline of crystallography has developed a descriptive terminology which is applied to crystals and crystal features in order to describe their structure, symmetry, and shape. This terminology defines the crystal lattice which provides a mineral with its ordered internal structure. It also describes various types of symmetry. By considering what type of symmetry a mineral species possesses, the species may be categorized as a member of one of six crystal systems and one of thirtytwo crystal classes.
The concept of symmetry
describes the periodic repetition of structural features. Two general
types of symmetry exist. These include translational symmetry and
point symmetry. Translational symmetry describes the periodic
repetition of a motif across a length or through an area or volume. Point
symmetry, on the other hand, describes the periodic repetition of a motif
around a point.
Reflection, rotation, inversion, and
rotoinversion are all point symmetry operations. A reflection
occurs when a motif on one side of a plane passing through the center of a
crystal is the mirror image of a motif which appears on the other side of
the plane. The motif is said to be reflected across the mirror plane
which divides the crystal. Rotational symmetry arises when a
structural element is rotated a fixed number of degrees about a central
point before it is repeated. If a crystal possesses inversion
symmetry, then every line drawn through the center of the crystal will
connect two identical features on opposite sides of the crystal.
Rotoinversion is a compound symmetry operation which is produced by
performing a rotation followed by an inversion.
A specified motif which is translated
linearly and repeated many times will produce a lattice. A lattice
is an array of points which define a repeated spatial entity called a
unit cell. The unit cell of a lattice is the smallest unit which
can be repeated in three dimensions in order to construct the lattice.
The corners of the unit cell serve as points which are repeated to form
the lattice array; these points are termed lattice points.
The number of possible lattices is
limited. In the plane only five different lattices may be produced by
translation. The French crystallographer Auguste Bravais (18111863)
established that in threedimensional space only fourteen different
lattices may be constructed. These fourteen different lattice structures
are thus termed the Bravais lattices.
The reflection, rotation, inversion,
and rotoinversion symmetry operations may be combined in a variety of
different ways. There are thirtytwo possible unique combinations of
symmetry operations. Minerals possessing the different combinations are
therefore categorized as members of thirtytwo crystal classes;
each crystal class corresponds to a unique set of symmetry operations.
Each of the crystal classes is named according to the variant of a
crystal form which it displays. Each crystal class is grouped as
one of the six different crystal systems according to which
characteristic symmetry operation it possesses.
A crystal form is a set of
planar faces which are geometrically equivalent and whose spatial
positions are related to one another by a specified set of symmetry
operations. If one face of a crystal form is defined, the specified set
of point symmetry operations will determine all of the other faces of the
crystal form.
A simple crystal may consist of only a
single crystal form. A more complicated crystal may be a combination of
several different forms. The crystal forms of the five nonisometric
crystal systems are the monohedron or pedion, parallelohedron or pinacoid,
dihedron, or dome and sphenoid, disphenoid, prism, pyramid, dipyramid,
trapezohedron, scalenohedron, rhombohedron and tetrahedron. Fifteen
different forms are possible within the isometric system.
Each crystal class is a member of one
of six crystal systems. These systems include the isometric,
hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic systems.
The hexagonal crystal system is further broken down into the hexagonal and
rhombohedral divisions. Every crystal of a certain crystal system will
share a characteristic symmetry element with the other members of its
system. The crystal system of a mineral species may sometimes be
determined visually by examining a particularly wellformed crystal of the
species.
Crystals possess a regular, repetitive
internal structure. The concept of symmetry describes the
repetition of structural features. Crystals therefore possess symmetry,
and much of the discipline of crystallography is concerned with describing
and cataloging different types of symmetry.
Two general types of symmetry exist.
These consist of translational symmetry and point symmetry.
Translational symmetry describes the periodic repetition of a structural
feature across a length or through an area or volume. Point symmetry, on
the other hand, describes the periodic repetition of a structural feature
around a point. Reflection, rotation, and inversion are all point
symmetries.
The concept of a lattice is
directly related to the idea of translational symmetry. A lattice is a
network or array composed of single motif which has been translated and
repeated at fixed intervals throughout space. For example, a square which
is translated and repeated many times across the plane will produce a
planar square lattice.
The unit cell of a lattice is
the smallest unit which can be repeated in three dimensions in order to
construct the lattice. In a crystal, the unit cell consists of a specific
group of atoms which are bonded to one another in a set geometrical
arrangement. This unit and its constituent atoms are then repeated over
and over in order to construct the crystal lattice. The surroundings in
any given direction of one corner of a unit cell must be identical to the
surroundings in the same direction of all the other corners. The corners
of the unit cell therefore serve as points which are repeated to form a
lattice array; these points are termed lattice points. The vectors
which connect a straight line of equivalent lattice points and delineate
the edges of the unit cell are known as the crystallographic axes.
The number of possible lattices is
limited. In the plane only five different lattices may be produced by
translation. One of these lattices possesses a square unit cell while
another possesses a rectangular unit cell. The third possible planar
lattice possesses a centered rectangular unit cell, which contains a
lattice point in the center as well as lattice points on the corners. The
unit cell of the fourth possible planar lattice is a parallelogram, and
that of the final planar lattice is a hexagonal unit cell which may
alternately be considered a rhombus.
The French crystallographer Auguste Bravais (18111863) established that in threedimensional space only fourteen different lattices may be constructed. The fourteen Bravais lattices may be divided among six crystal systems. These are the isometric or cubic, tetragonal, orthorhombic, monoclinic, triclinic, and hexagonal systems. (The six crystal systems are discussed below.) The Bravais lattices are furthermore of three different types. A primitive lattice has only a lattice point at each corner of the threedimensional unit cell. A bodycentered lattice contains not only lattice points at each corner of the unit cell but also contains a lattice point at the center of the threedimensional unit cell. A facecentered lattice possesses not only lattice points at the corners of the unit cell but also at either the centers of just one pair of faces or else at the centers of all three pairs of faces. The fourteen Bravais lattices are therefore the primitive cubic, bodycentered cubic, facecentered cubic, primitive tetragonal, bodycentered tetragonal, primitive orthorhombic, bodycentered orthorhombic, single facecentered orthorhombic, multiple facecentered orthorhombic, primitive monoclinic, single facecentered monoclinic, primitive triclinic, single facecentered hexagonal, and rhombohedral lattices. (The rhombohedral lattice is a subset of the hexagonal crystal system.)
Point symmetry describes the
repetition of a motif or structural feature around a single reference
point, commonly the center of a unit cell or a crystal. The different
pointsymmetry operations are reflection, rotation, inversion, and the
combined operation rotoinversion.
A reflection occurs when the
structure features on one side of a plane passing through the center of a
crystal are the mirror image of the structural features on the other side.
The plane across which the reflection occurs is then termed a mirror
plane.
Rotational symmetry arises when
a structural element is rotated a fixed number of degrees about a central
point and then repeated. A square, for example, possesses 4fold
rotational symmetry because it may be rotated four times by 90° about
its central point before it is returned to its original position. Each
time it is rotated by 90° the resultant square will be identical in
appearance to the original square.
If a crystal possesses inversion
symmetry, then any line which is drawn through the origin at the
center of the crystal will connect two identical features on opposite
sides of the crystal.
Rotoinversion is a compound
symmetry operation which is produced by performing a rotation followed by
an inversion. 1fold, 2fold, 3fold, 4fold, and 6fold rotoinversion
operations exist. Most of these rotoinversions may alternately be
described by a specified set of rotation, reflection and inversion
operations. A 1fold rotoinversion is equivalent to rotation by 360°
followed by inversion. This procedure is ultimately equivalent to a
single inversion. A 2fold rotoinversion axis is equivalent to reflection
through a mirror plane perpendicular to the rotoinversion axis. A crystal
which possesses a 3fold rotoinversion axis is equivalent to one which
possesses both 3fold rotational symmetry and inversion symmetry. A
6fold rotoinversion is equivalent to 3fold rotation and reflection
across a mirror plane which lies at right angles to the rotation axis.
The only rotoinversion operation which cannot be replaced by a combination
of rotations, reflections and inversions is 4fold rotoinversion.
The reflection, rotation, inversion,
and rotoinversion symmetry operations may be combined in a variety of
different ways. There are thirtytwo different possible combinations of
these symmetry elements. Minerals possessing the different combinations
are therefore categorized as members of 32 possible crystal
classes. According to this schema, each crystal class corresponds to
a unique set of symmetry operations. Each crystal class is then placed
into one of the six different crystal systems so that several different
classes are members of each system.
Every crystal class is a member of one
of the six crystal systems. These systems include the isometric,
hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic crystal
systems. The hexagonal crystal system is further broken down into
hexagonal and rhombohedral divisions.
Every crystal class which belongs to a
certain crystal system will share a characteristic symmetry element with
the other members of its system. For example, all crystals of the
isometric system possess four 3fold axes of symmetry which proceed
diagonally from corner to corner through the center of the cubic unit
cell. In contrast, all crystals of the hexagonal division of the
hexagonal system possess a single sixfold axis of rotation.
In addition to the characteristic
symmetry element, a crystal class may possess other symmetry elements
which are not necessarily present in all members of the same system. The
crystal class which possesses the highest possible symmetry or the highest
number of symmetry elements within each system is termed the
holomorphic class of the system. For example, crystals of the
holomorphic class of the isometric system possess inversion symmetry,
three 4fold axes of rotational symmetry, the characteristic set of four
3fold axes of rotational symmetry which is indicative of the isometric
crystal system, six 2fold axes of rotational symmetry, and nine different
mirror planes. In contrast, a crystal which is not a member of the
holomorphic class yet still belongs to the isometric system may possess
only three 2fold axes of rotational symmetry and the characteristic four
3fold axes of rotational symmetry.
The crystal system of a mineral
species may sometimes be determined in the field by visually examining a
particularly wellformed crystal of the species.
A crystal form is a set of faces
which are geometrically equivalent and whose spatial positions are related
to one another according to the symmetry of the crystal. If one face of a
crystal form is defined, the point symmetry operations which specify the
class to which the crystal belongs also determine the other faces of the
crystal form.
Fifteen different forms are possible
within the isometric or cubic system. These include the hexoctahedron,
gyroid, hextetrahedron, diploid, and tetartoid, among others. The crystal
forms of the remaining five crystal systems are the monohedron or pedion,
parallelohedron or pinacoid, dihedron, or dome and sphenoid, disphenoid,
prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron, and
tetrahedron.
The crystal forms which occur in each
crystal class and system must possess a symmetry complementary to that of
the associated crystal class and system. For example, a monohedron, which
possesses only one face, will never occur in a crystal with inversion
symmetry because the inversion operation requires that an equivalent face
be present on the opposite side of the crystal.
A simple crystal may consist of only a
single crystal form. A more complicated crystal may be a combination of
several different forms. All forms which occur in a crystal of a
particular system must be compatible with that crystal system.
The reflection, rotation, inversion,
and rotoinversion symmetry operations may be combined in thirtytwo
different ways. Thirtytwo different crystal classes are therefore
defined so that each crystal class corresponds to a unique set of symmetry
operations. Each of the crystal classes is named according to the variant
of a crystal form which it displays. For example, the isometric
hexoctahedral class belongs to the isometric crystal system and
demonstrates the hexoctahedral crystal form. The rhombic pyramidal,
tetragonal pyramidal, trigonal pyramidal and hexagonal pyramidal classes
each display a variant of the crystal form which is called a pyramid.
Each crystal class is a member of one
of the six different crystal systems according to which
characteristic symmetry operation it possesses. For example, all crystals
of the isometric system possess four 3fold axes of symmetry, while
minerals of the tetragonal system possess a single 4fold symmetry axis
and crystals of the triclinic class show no symmetry at all. The rhombic
pyramidal crystal class is thus a member of the orthorhombic crystal
system, the tetragonal pyramidal class is a member of the tetragonal
crystal system, and the trigonal and hexagonal pyramidal classes are
members of the rhombohedral (trigonal) and hexagonal divisions of the
hexagonal crystal system respectively.
The following table lists in bold type
the six crystal systems. Included are the isometric, hexagonal,
tetragonal, orthorhombic, monoclinic, and triclinic systems. The
tetragonal crystal system is further separated into the hexagonal and
trigonal or rhombohedral divisions. Under each crystal system the table
lists by name the crystal classes which occur within that system. For
example, the crystal classes which occur within the trigonal crystal
system are the trigonal monohedral and trigonal parallelohedral crystal
classes. Adjacent to the listing of each crystal class is the symmetry of
the class.
When listing the symmetry of each
crystal class an axis of rotational symmetry is represented by the capital
letter A. Whether this axis is a 2fold, 3fold, or 4fold axis is
indicated by a subscript following the letter A. The number of such axes
present is indicated by a numeral preceding the capital A.
1A_{2}, 2A_{3}, and
3A_{4} thus represent one 2fold axis of rotation,
two 3fold axes, and three 4fold axes respectively. A center of
inversion is noted by the lowercase letter 'i' while a mirror plane is
denoted by 'm'. The numeral preceding the m indicates how many mirror
planes are present. Axes of rotary inversion are usually replaced by
the equivalent rotations and reflections. For example, a 2fold
rotoinversion axis is equivalent to reflection through a mirror plane
perpendicular to the rotoinversion axis. A crystal which possesses a
3fold rotoinversion axis is equivalent to one which possesses both 3fold
rotational symmetry and inversion symmetry. A 6fold rotoinversion is
equivalent to 3fold rotation and reflection across a mirror plane at
right angles to the rotation axis. The only rotoinversion operation which
cannot be thus replaced is 4fold rotoinversion, which is indicated by
R_{4}.
The class which possesses the highest
possible symmetry within each crystal system is termed the holomorphic
class of that system. The holomorphic class of each crystal system is
indicated in the table by bold type. For example, the triclinic
parallelohedron is the holomorphic class of the triclinic crystal system
while the isometric hexoctahedron is the holomorphic class of the
isomorphic or cubic crystal system. The characteristic symmetry element
of each crystal system is listed in bold type. It is thus apparent that
the characteristic symmetry element of the isometric crystal system is the
possession of four 3fold axes of rotational symmetry, while the
characteristic symmetry element of the rhombohedral division of the
hexagonal crystal system is the possession of a single 3fold axis of
rotational symmetry.
Crystal System  Crystal Class / Crystal Form  Symmetry of Class  
Isometric System 
hexoctahedron
gyroid hextetrahedron diploid tetartoid 
i, 3A_{4},
4A_{3},
6A_{2}, 9m
3A_{4}, 4A_{3}, 6A_{2} 3A_{2}, 4A_{3}, 6m i, 3A_{2}, 4A_{3}, 3m 3A_{2}, 4A_{3} 

Hexagonal System  Hexagonal Division 
dihexagonal dipyramid
hexagonal trapezohedron dihexagonal pyramid ditrigonal dipyramid hexagonal dipyramid hexagonal pyramid trigonal dipyramid 
i, 1A_{6},
6A_{2}, 7m
1A_{6}, 6A_{2} 1A_{6}, 6m 1R_{6}, 3A_{2}, 3m i, 1A_{6}, 1m 1A_{6} 1R_{6} 
Rhombohedral Division 
hexagonal scalenohedron
trigonal trapezohedron ditrigonal pyramid rhombohedron trigonal pyramid 
i, 1A_{3}, 3A_{2},
3m
1A_{3}, 3A_{2} 1A_{3}, 3m i, 1A_{3} 1A_{3} 

Tetragonal System 
ditetragonal dipyramid
tetragonal trapzohedron ditetragonal pyramid tetragonal scalenohedron tetragonal dipyramid tetragonal pyramid tetragonal disphenoid 
i, 1A_{4}, 4A_{2},
5m
1A_{4}, 4A_{2} 1A_{4}, 4m 1R_{4}, 2A_{2}, 2m i, 1A_{4}, 1m 1A_{4} 1R_{4} 

Orthorhombic System 
rhombic dipyramid
rhombic disphenoid rhombic pyramid 
i, 3A_{2}, 3m
3A_{2} 1A_{2}, 2m 

Monoclinic System 
prism
sphenoid dome 
i, 1A_{2}, 1m
1A_{2} 1m 

Triclinic System 
parallellohedron
monohedron 
i
no symmetry 
