Section 3:

Crystal Structure and Crystal System

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  1. Introduction
  2. Symmetry and Lattices
          Bravais Lattices
          Point Symmetry Operations
  3. Crystal Systems
          Isometric
          Hexagonal
          Tetragonal
          Orthorhombic
          Monoclinic
          Triclinic
  4. Crystal Forms
          Monohedron
          Parallelohedron
          Dihedron
          Disphenoid
          Prism
          Pyramid
          Dipyramid
          Trapezohedron
          Scalenohedron
          Rhombohedron
          Tetrahedron
  5. Crystal Classes
          Table of Crystal Classes


1. Introduction

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      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 thirty-two 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 (1811-1863) established that in three-dimensional 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 thirty-two possible unique combinations of symmetry operations. Minerals possessing the different combinations are therefore categorized as members of thirty-two 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 non-isometric 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 well-formed crystal of the species.


2. Symmetry and Lattices

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Symmetry

      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.

Lattices

      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.

Bravais Lattices

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      The French crystallographer Auguste Bravais (1811-1863) established that in three-dimensional 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 three-dimensional unit cell. A body-centered lattice contains not only lattice points at each corner of the unit cell but also contains a lattice point at the center of the three-dimensional unit cell. A face-centered 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, body-centered cubic, face-centered cubic, primitive tetragonal, body-centered tetragonal, primitive orthorhombic, body-centered orthorhombic, single face-centered orthorhombic, multiple face-centered orthorhombic, primitive monoclinic, single face-centered monoclinic, primitive triclinic, single face-centered hexagonal, and rhombohedral lattices. (The rhombohedral lattice is a subset of the hexagonal crystal system.)

Point Symmetry Operations

      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 point-symmetry 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 4-fold 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. 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotoinversion operations exist. Most of these rotoinversions may alternately be described by a specified set of rotation, reflection and inversion operations. A 1-fold rotoinversion is equivalent to rotation by 360° followed by inversion. This procedure is ultimately equivalent to a single inversion. A 2-fold rotoinversion axis is equivalent to reflection through a mirror plane perpendicular to the rotoinversion axis. A crystal which possesses a 3-fold rotoinversion axis is equivalent to one which possesses both 3-fold rotational symmetry and inversion symmetry. A 6-fold rotoinversion is equivalent to 3-fold 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 4-fold rotoinversion.
      The reflection, rotation, inversion, and rotoinversion symmetry operations may be combined in a variety of different ways. There are thirty-two 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.


3. Crystal Systems

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      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 3-fold 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 six-fold 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 4-fold axes of rotational symmetry, the characteristic set of four 3-fold axes of rotational symmetry which is indicative of the isometric crystal system, six 2-fold 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 2-fold axes of rotational symmetry and the characteristic four 3-fold axes of rotational symmetry.
      The crystal system of a mineral species may sometimes be determined in the field by visually examining a particularly well-formed crystal of the species.

Isometric
      The isometric crystal system is also known as the cubic system. The crystallographic axes used in this system are of equal length and are mutually perpendicular, occurring at right angles to one another.
      All crystals of the isometric system possess four 3-fold axes of symmetry, each of which proceeds diagonally from corner to corner through the center of the cubic unit cell. Crystals of the isometric system may also demonstrate up to three separate 4-fold axes of rotational symmetry. These axes, if present, proceed from the center of each face through the origin to the center of the opposite face and correspond to the crystallographic axes. Furthermore crystals of the isometric system may possess six 2-fold axes of symmetry which extend from the center of each edge of the crystal through the origin to the center of the opposite edge. Minerals of this system may demonstrate up to nine different mirror planes.
      Examples of minerals which crystallize in the isometric system are halite, magnetite, and garnet. Minerals of this system tend to produce crystals of equidimensional or equant habit. (Please refer to Section 2 for more information on crystal habit.)

Hexagonal
      Minerals of the hexagonal crystal system are referred to three crystallographic axes which intersect at 120° and a fourth which is perpendicular to the other three. This fourth axis is usually depicted vertically.
      The hexagonal crystal system is divided into the hexagonal and rhombohedral or trigonal divisions. All crystals of the hexagonal division possess a single 6-fold axis of rotation. In addition to the single 6-fold axis of rotation, crystals of the hexagonal division may possess up to six 2-fold axes of rotation. They may demonstrate a center of inversion symmetry and up to seven mirror planes. Crystals of the trigonal division all possess a single 3-fold axis of rotation rather than the 6-fold axis of the hexagonal division. Crystals of this division may possess up to three 2-fold axes of rotation and may demonstrate a center of inversion and up to three mirror planes.
      Minerals species which crystallize in the hexagonal division are apatite, beryl, and high quartz. Minerals of this division tend to produce hexagonal prisms and pyramids. Example species which crystallize in the rhombohedral division are calcite, dolomite, low quartz, and tourmaline. Such minerals tend to produce rhombohedra and triangular prisms.

Tetragonal
      Minerals of the tetragonal crystal system are referred to three mutually perpendicular axes. The two horizontal axes are of equal length, while the vertical axis is of different length and may be either shorter or longer than the other two. Minerals of this system all possess a single 4-fold symmetry axis. They may possess up to four 2-fold axes of rotation, a center of inversion, and up to five mirror planes.
      Mineral species which crystallize in the tetragonal crystal system are zircon and cassiterite. These minerals tend to produce short crystals of prismatic habit.

Orthorhombic
      Minerals of the orthorhombic crystal system are referred to three mutually perpendicular axes, each of which is of a different length than the others.
      Crystals of this system uniformly possess three 2-fold rotation axes and/or three mirror planes. The holomorphic class demonstrates three 2-fold symmetry axes and three mirror planes as well as a center of inversion. Other classes may demonstrate three 2-fold axes of rotation or one 2-fold rotation axis and two mirror planes.
      Species which belong to the orthorhombic system are olivine and barite. Crystals of this system tend to be of prismatic, tabular, or acicular habit.

Monoclinic
      Crystals of the monoclinic system are referred to three unequal axes. Two of these axes are inclined toward each other at an oblique angle; these are usually depicted vertically. The third axis is perpendicular to the other two. The two vertical axes therefore do not intersect one another at right angles, although both are perpendicular to the horizontal axis.
      Monoclinic crystals demonstrate a single 2-fold rotation axis and/or a single mirror plane. The holomorphic class possesses the single 2-fold rotation axis, a mirror plane, and a center of symmetry. Other classes display just the 2-fold rotation axis or just the mirror plane.
      Mineral species which adhere to the monoclinic crystal system include pyroxene, amphibole, orthoclase, azurite, and malachite, among many others. The minerals of the monoclinic system tend to produce long prisms.

Triclinic
      Crystals of the triclinic system are referred to three unequal axes, all of which intersect at oblique angles. None of the axes are perpendicular to any other axis.
      Crystals of the triclinic system may be said to possess only a 1-fold symmetry axis, which is equivalent to possessing no symmetry at all. Crystals of this system possess no mirror planes. The holomorphic class demonstrates a center of inversion symmetry.
      Mineral species of the triclinic class include plagioclase and axinite; these species tend to be of tabular habit.


4. Crystal Forms

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      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.

Monohedron
The monohedral crystal form is also called a pedion. It consists of a single face which is geometrically unique for the crystal and is not repeated by any set of symmetry operations. Members of the triclinic crystal system produce monohedral crystal forms.

Parallelohedron
The parallelohedral crystal form is also called a pinacoid. It consists of two and only two geometrically equivalent faces which occupy opposite sides of a crystal. The two faces are parallel and are related to one another only by a reflection or an inversion. Members of the triclinic crystal system produce parallelohedral crystal forms.

Dihedron
The dihedron consists of two and only two nonparallel geometrically equivalent faces. The two faces may be related by a reflection or by a rotation. The dihedron is termed a dome if the two faces are related only by reflection across a mirror plane. If the two faces are related instead by a 2-fold rotation axis then the dihedron is termed a sphenoid. Members of the monoclinic crystal system produce dihedral crystal forms.

Disphenoid
Members of the orthorhombic and tetragonal crystal systems produce rhombic and tetragonal disphenoids, which possess two sets of nonparallel geometrically equivalent faces, each of which is related by a 2-fold rotation. The faces of the upper sphenoid alternate with the faces of the lower sphenoid in such forms.

Prism
A prism is composed of a set of 3, 4, 6, 8, or 12 geometrically equivalent faces which are all parallel to the same axis. Each of these faces intersects with the two faces adjacent to it to produce a set of parallel edges. The mutually parallel edges of all intersections of the prism sides then form a tube. Prisms are given names based on the shape of their cross section. Variants of the prism form include the rhombic prism, tetragonal prism, trigonal prism, and hexagonal prism. A prism in which the large faces are divided into two mirror-image faces which intersect with one another at an oblique angle is called a ditetragonal prism, a ditrigonal prism, or a dihexagonal prism. Prisms are associated with the members of the monoclinic crystal system.

Pyramid
A pyramid is composed of a set of 3, 4, 6, 8, or 12 faces which are not parallel but instead intersect at a point. The orthorhombic, tetragonal and hexagonal crystal systems all produce pyramids. These pyramids are named according to the shape of their cross-section in the same way that prisms are. Thus are produced the rhombic pyramid, tetragonal pyramid, trigonal pyramid, and hexagonal pyramid. Each large face of the ditetragonal pyramid, ditrigonal pyramid, and dihexagonal pyramids is divided into two mirror-image faces which occupy an oblique angle with respect to one another.

Dipyramid
The dipyramidal crystal form is composed of two pyramids placed base-to-base and related by reflection across a mirror plane which runs parallel to and adjacent to the pyramid bases. The upper and lower pyramids may each have 3, 4, 6, 8, or 12 faces; the dipyramidal form therefore possesses a total of 6, 8, 12, 16, or 24 faces. The orthorhombic, tetragonal and hexagonal crystal systems all produce dipyramids. These dipyramids are named for the shape of their cross-section just as prisms and pyramids are, resulting in the rhombic dipyramid, trigonal dipyramid, tetragonal dipyramid, and hexagonal dipyramid. The large faces of the ditetragonal, ditrigonal and dihexagonal dipyramids are divided into two mirror-image faces which intersect one another at an oblique angle.

Trapezohedron
A trapezohedron is a crystal form possessing 6, 8, or 12 trapezoidal faces. The tetragonal crystal system and both the trigonal and hexagonal divisions of the hexagonal crystal system produce trapezohedral crystal forms. Trigonal trapezohedra possess three trapezoidal faces on the top and three on the bottom for a total of six faces; tetragonal trapezohedra have four faces on top and four on the bottom for a total of eight faces; and hexagonal trapezohedra have six faces on top and six on the bottom, resulting in twelve faces total.

Scalenohedron
A scalenohedron consists of 8 or 12 faces, each of which is a scalene triangle. The faces appear to be grouped into symmetric pairs. The tetragonal and hexagonal crystal systems produce the scalenohedral crystal form, of which examples may be further described as trigonal, tetragonal and hexagonal scalenohedra.

Rhombohedron
The rhombohedral crystal form possesses six rhombus-shaped faces. A rhombohedron resembles in appearance a cube which is poised upright upon one corner and has been either flattened or elongated along an axis which runs diagonally from corner to corner through the center. The rhombohedral crystal form is produced only by members of the trigonal and rhombohedral divisions of the hexagonal crystal system.

Tetrahedron
A tetrahedron is composed of four triangular faces. In crystals of the isometric system each face is an identical equilateral triangle. In crystals of the tetragonal system each face is an identical isoceles triangle; this variant of the tetrahedron is called a tetragonal tetrahedron. In crystals of the orthorhombic system the faces consist of two pairs of different isoceles triangles; the crystal is then termed a rhombic tetrahedron.


5. Crystal Classes

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      The reflection, rotation, inversion, and rotoinversion symmetry operations may be combined in thirty-two different ways. Thirty-two 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 3-fold axes of symmetry, while minerals of the tetragonal system possess a single 4-fold 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.

Table of the 32 Crystal Classes

      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 2-fold, 3-fold, or 4-fold 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. 1A2, 2A3, and 3A4 thus represent one 2-fold axis of rotation, two 3-fold axes, and three 4-fold 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 2-fold rotoinversion axis is equivalent to reflection through a mirror plane perpendicular to the rotoinversion axis. A crystal which possesses a 3-fold rotoinversion axis is equivalent to one which possesses both 3-fold rotational symmetry and inversion symmetry. A 6-fold rotoinversion is equivalent to 3-fold rotation and reflection across a mirror plane at right angles to the rotation axis. The only rotoinversion operation which cannot be thus replaced is 4-fold rotoinversion, which is indicated by R4.
      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 3-fold axes of rotational symmetry, while the characteristic symmetry element of the rhombohedral division of the hexagonal crystal system is the possession of a single 3-fold axis of rotational symmetry.






Crystal System Crystal Class / Crystal Form    Symmetry of Class




Isometric System hexoctahedron
gyroid
hextetrahedron
diploid
tetartoid
i, 3A4, 4A3, 6A2, 9m
3A4, 4A3, 6A2
3A2, 4A3, 6m
i, 3A2, 4A3, 3m
3A2, 4A3




Hexagonal System Hexagonal Division dihexagonal dipyramid
hexagonal trapezohedron
dihexagonal pyramid
ditrigonal dipyramid
hexagonal dipyramid
hexagonal pyramid
trigonal dipyramid
i, 1A6, 6A2, 7m
1A6, 6A2
1A6, 6m
1R6, 3A2, 3m
i, 1A6, 1m
1A6
1R6
Rhombohedral Division      hexagonal scalenohedron
trigonal trapezohedron
ditrigonal pyramid
rhombohedron
trigonal pyramid
i, 1A3, 3A2, 3m
1A3, 3A2
1A3, 3m
i, 1A3
1A3




Tetragonal System ditetragonal dipyramid
tetragonal trapzohedron
ditetragonal pyramid
tetragonal scalenohedron
tetragonal dipyramid
tetragonal pyramid
tetragonal disphenoid
i, 1A4, 4A2, 5m
1A4, 4A2
1A4, 4m
1R4, 2A2, 2m
i, 1A4, 1m
1A4
1R4




Orthorhombic System     rhombic dipyramid
rhombic disphenoid
rhombic pyramid
i, 3A2, 3m
3A2
1A2, 2m




Monoclinic System prism
sphenoid
dome
i, 1A2, 1m
1A2
1m




Triclinic System parallellohedron
monohedron
i
no symmetry






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