#### ** 1. Introduction**

**symmetry**describes the periodic repetition of structural features. Two full 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 expanse or volume. Point symmetry, on the other hand, describes the periodic repetition of a motif unopen to a point.

**Reflection, rotation, inversion, together with rotoinversion are all betoken symmetry operations.**

A **reflection** occurs when a motif on 1 side of a bird passing through the oculus of a crystal is the mirror picture of a motif which appears on the other side of the plane. The motif is said to live on reflected across the mirror bird which divides the crystal. **Rotational symmetry** arises when a structural chemical element is rotated a fixed publish of degrees nearly a fundamental betoken earlier it is repeated. If a crystal possesses **inversion symmetry**, together with so every line of piece of job drawn through the oculus of the crystal volition connect 2 identical features on reverse sides of the crystal. **Rotoinversion** is a chemical compound symmetry functioning which is produced yesteryear performing a rotation followed yesteryear an inversion.

A specified motif which is translated linearly together with repeated many times volition hit a **lattice**. Influenza A virus subtype H5N1 lattice is an array of points which define a repeated spatial entity called a **unit cell**. The unit of measurement jail cellular telephone of a lattice is the smallest unit of measurement which tin live on repeated inward iii dimensions inward guild to create the lattice. The corners of the unit of measurement jail cellular telephone serve every bit points which are repeated to shape the lattice array; these points are termed **lattice points**.

The publish of possible lattices is limited. In the bird exclusively v dissimilar lattices may live on produced yesteryear translation. The French crystallographer Auguste Bravais (1811-1863) established that inward three-dimensional infinite exclusively xiv dissimilar lattices may live on constructed. These xiv dissimilar lattice structures are thus termed the **Bravais lattices**.

**crystal classes**; each crystal course of written report corresponds to a unique laid of symmetry operations. Each of the crystal classes is named according to the variant of a

**crystal form**which it displays. Each crystal course of written report is grouped every bit 1 of the half-dozen different

**crystal systems**according to which characteristic symmetry functioning it possesses.

A

**crystal form**is a laid of planar faces which are geometrically equivalent together with whose spatial positions are related to 1 some other yesteryear a specified laid of symmetry operations. If 1 facial expression upward of a crystal shape is defined, the specified laid of betoken symmetry operations volition create upward one’s heed all of the other faces of the crystal form.

A elementary crystal may consist of exclusively a unmarried crystal form. Influenza A virus subtype H5N1 to a greater extent than complicated crystal may live on a combination of several dissimilar forms. The crystal forms of the v non-isometric crystal systems are the monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome together with sphenoid, disphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron together with tetrahedron. Fifteen dissimilar forms are possible inside the isometric system.

Each crystal course of written report is a fellow member of 1 of six **crystal systems**. These systems include the isometric, hexagonal, tetragonal, orthorhombic, monoclinic, together with triclinic systems. The hexagonal crystal scheme is farther broken downward into the hexagonal together with rhombohedral divisions. Every crystal of a sure enough crystal scheme volition percentage a characteristic symmetry chemical element alongside the other members of its system. The crystal scheme of a mineral species may sometimes live on determined visually yesteryear examining a peculiarly well-formed crystal of the species.

#### 2. Symmetry together with Lattices

**symmetry**describes the repetition of structural features. Crystals thence possess symmetry, together with much of the plain of written report of crystallography is concerned alongside describing together with cataloging dissimilar types of symmetry.

Two full general types of symmetry exist. These consist of

**translational symmetry**and

**point symmetry**. Translational symmetry describes the periodic repetition of a structural characteristic across a length or through an expanse or volume. Point symmetry, on the other hand, describes the periodic repetition of a structural characteristic unopen to a point. Reflection, rotation, together with inversion are all betoken symmetries.

### __Lattices__

**lattice**is straight related to the see of translational symmetry. Influenza A virus subtype H5N1 lattice is a network or array composed of unmarried motif which has been translated together with repeated at fixed intervals throughout space. For example, a foursquare which is translated together with repeated many times across the bird volition hit a planar foursquare lattice.

The

**unit cell**of a lattice is the smallest unit of measurement which tin live on repeated inward iii dimensions inward guild to create the lattice. In a crystal, the unit of measurement jail cellular telephone consists of a specific grouping of atoms which are bonded to 1 some other inward a laid geometrical arrangement. This unit of measurement together with its ingredient atoms are together with so repeated over together with over inward guild to create the crystal lattice. The environs inward whatsoever given direction of 1 corner of a unit of measurement jail cellular telephone must live on identical to the environs inward the same direction of all the other corners. The corners of the unit of measurement jail cellular telephone thence serve every bit points which are repeated to shape a lattice array; these points are termed

**lattice points**. The vectors which connect a straight line of piece of job of equivalent lattice points together with delineate the edges of the unit of measurement jail cellular telephone are known every bit the

**crystallographic axes**.

The publish of possible lattices is limited. In the bird exclusively v dissimilar lattices may live on produced yesteryear translation. One of these lattices possesses a foursquare unit of measurement jail cellular telephone field some other possesses a rectangular unit of measurement cell. The tertiary possible planar lattice possesses a centered rectangular unit of measurement cell, which contains a lattice betoken inward the oculus every bit good every bit lattice points on the corners. The unit of measurement jail cellular telephone of the 4th possible planar lattice is a parallelogram, together with that of the concluding planar lattice is a hexagonal unit of measurement jail cellular telephone which may alternately live on considered a rhombus.

## 3. __Crystal Systems__

Every crystal course of written report is a fellow member of 1 of the half-dozen crystal systems. These systems include the isometric, hexagonal, tetragonal, orthorhombic, monoclinic, together with triclinic crystal systems. The hexagonal crystal scheme is farther broken downward into hexagonal together with rhombohedral divisions.

Every crystal course of written report which belongs to a sure enough crystal scheme volition percentage a characteristic symmetry chemical element alongside the other members of its system. For example, all crystals of the isometric scheme possess 4 3-fold axes of symmetry which maintain diagonally from corner to corner through the oculus of the cubic unit of measurement cell. In contrast, all crystals of the hexagonal sectionalisation of the hexagonal scheme possess a unmarried six-fold axis of rotation.

In add-on to the characteristic symmetry element, a crystal course of written report may possess other symmetry elements which are non necessarily acquaint inward all members of the same system. The crystal course of written report which possesses the highest possible symmetry or the highest publish of symmetry elements inside each scheme is termed the holomorphic course of written report of the system. For example, crystals of the holomorphic course of written report of the isometric scheme possess inversion symmetry, iii 4-fold axes of rotational symmetry, the characteristic laid of 4 3-fold axes of rotational symmetry which is indicative of the isometric crystal system, half-dozen 2-fold axes of rotational symmetry, together with nine dissimilar mirror planes. In contrast, a crystal which is non a fellow member of the holomorphic course of written report yet yet belongs to the isometric scheme may possess exclusively iii 2-fold axes of rotational symmetry together with the characteristic 4 3-fold axes of rotational symmetry.

**The crystal scheme of a mineral species may sometimes live on determined inward the plain yesteryear visually examining a peculiarly well-formed crystal of the species.**

**Isometric**

The isometric crystal scheme is likewise known every bit the cubic system. The crystallographic axes used inward this scheme are of equal length together with are mutually perpendicular, occurring at correct angles to 1 another.All crystals of the isometric scheme possess 4 3-fold axes of symmetry, each of which proceeds diagonally from corner to corner through the oculus of the cubic unit of measurement cell. Crystals of the isometric scheme may likewise demonstrate upward to iii separate 4-fold axes of rotational symmetry. These axes, if present, maintain from the oculus of each facial expression upward through the rootage to the oculus of the reverse facial expression upward together with lucifer to the crystallographic axes. Furthermore crystals of the isometric scheme may possess half-dozen 2-fold axes of symmetry which extend from the oculus of each border of the crystal through the rootage to the oculus of the reverse edge. Minerals of this scheme may demonstrate upward to nine dissimilar mirror planes. Examples of minerals which crystallize inward the isometric scheme are halite, magnetite, together with garnet. Minerals of this scheme tend to hit crystals of equidimensional or equant habit.

**Hexagonal**

Minerals of the hexagonal crystal scheme are referred to iii crystallographic axes which intersect at 120° together with a 4th which is perpendicular to the other three. This 4th axis is unremarkably depicted vertically. The hexagonal crystal scheme is divided into the hexagonal together with rhombohedral or trigonal divisions. All crystals of the hexagonal sectionalisation possess a unmarried 6-fold axis of rotation. In add-on to the unmarried 6-fold axis of rotation, crystals of the hexagonal sectionalisation may possess upward to half-dozen 2-fold axes of rotation. They may demonstrate a oculus of inversion symmetry together with upward to 7 mirror planes. Crystals of the trigonal sectionalisation all possess a unmarried 3-fold axis of rotation rather than the 6-fold axis of the hexagonal division. Crystals of this sectionalisation may possess upward to iii 2-fold axes of rotation together with may demonstrate a oculus of inversion together with upward to iii mirror planes.Minerals species which crystallize inward the hexagonal sectionalisation are apatite, beryl, together with high quartz. Minerals of this sectionalisation tend to hit hexagonal prisms together with pyramids. Example species which crystallize inward the rhombohedral sectionalisation are calcite, dolomite, depression quartz, together with tourmaline. Such minerals tend to hit rhombohedra together with triangular prisms.

**Tetragonal**

Minerals of the tetragonal crystal scheme are referred to iii mutually perpendicular axes. The 2 horizontal axes are of equal length, field the vertical axis is of dissimilar length together with may live on either shorter or longer than the other two. Minerals of this scheme all possess a unmarried 4-fold symmetry axis. They may possess upward to 4 2-fold axes of rotation, a oculus of inversion, together with upward to v mirror planes.Mineral species which crystallize inward the tetragonal crystal scheme are zircon together with cassiterite. These minerals tend to hit curt crystals of prismatic habit.

**Orthorhombic**

Minerals of the orthorhombic crystal scheme are referred to iii mutually perpendicular axes, each of which is of a dissimilar length than the others.Crystals of this scheme uniformly possess iii 2-fold rotation axes and/or iii mirror planes. The holomorphic course of written report demonstrates iii 2-fold symmetry axes together with iii mirror planes every bit good every bit a oculus of inversion. Other classes may demonstrate iii 2-fold axes of rotation or 1 2-fold rotation axis together with 2 mirror planes. Species which belong to the orthorhombic scheme are olivine together with barite. Crystals of this scheme tend to live on of prismatic, tabular, or acicular habit.

**Monoclinic**

Crystals of the monoclinic scheme are referred to iii unequal axes. Two of these axes are inclined toward each other at an oblique angle; these are unremarkably depicted vertically. The tertiary axis is perpendicular to the other two. The 2 vertical axes thence hit non intersect 1 some other at correct angles, although both are perpendicular to the horizontal axis.Monoclinic crystals demonstrate a unmarried 2-fold rotation axis and/or a unmarried mirror plane. The holomorphic course of written report possesses the unmarried 2-fold rotation axis, a mirror plane, together with a oculus of symmetry. Other classes display simply the 2-fold rotation axis or simply the mirror plane.Mineral species which adhere to the monoclinic crystal scheme include pyroxene, amphibole, orthoclase, azurite, together with malachite, amidst many others. The minerals of the monoclinic scheme tend to hit long prisms.

**Triclinic**

Crystals of the triclinic scheme are referred to iii unequal axes, all of which intersect at oblique angles. None of the axes are perpendicular to whatsoever other axis.Crystals of the triclinic scheme may live on said to possess exclusively a 1-fold symmetry axis, which is equivalent to possessing no symmetry at all. Crystals of this scheme possess no mirror planes. The holomorphic course of written report demonstrates a oculus of inversion symmetry. Mineral species of the triclinic course of written report include plagioclase together with axinite; these species tend to live on of tabular habit.

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