Crystal Form

8/13/2019 Crystal Form

1/16

Types of Crystals

Shapes and Structures

By Anne Marie Helmenstine, Ph.D., About.com Guide

See More About:

crystallography

crystal lattices

physical properties

chemical properties

crystals

Copper sulfate has a triclinic crystal structure.

Stephanb, wikipedia.org

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Crystal Grouped by Lattices (Shape)

8here are se;en crystal lattice systems. only one lattice point per unit cell? or nonprimiti;e >more than one lattice

point per unit cell?. Combining the crystal systems !ith the lattice types yields the *4Bra;ais attices >named after Auguste Bra;ais, !ho !or#ed out lattice structures in

*-)?. 8he structure of real crystals is pretty complicated: 1aCl?is an e%ample of this type of crystal.
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A specified motif !hich is translated linearly and repeated many times !ill

produce a lattice. A lattice is an array of points !hich define a repeated spatial

entity called a unit cell. 8he unit cell of a lattice is the smallest unit !hich can

be repeated in three dimensions in order to construct the lattice. 8he corners of

the unit cell ser;e as points !hich are repeated to form the lattice arrayE these

points are termed lattice points.

8he number of possible lattices is limited. 'n the plane only fi;e different

lattices may be produced by translation. 8he 6rench crystallographer Auguste

Bra;ais >*-***-3@? established that in threedimensional space only fourteen

different lattices may be constructed. 8hese fourteen different lattice structures

are thus termed the ra%ais lattices.

8he reflection, rotation, in;ersion, and rotoin;ersion symmetry operations

may be combined in a ;ariety of different !ays. 8here are thirtyt!o possible

uniue combinations of symmetry operations. Minerals possessing the different

combinations are therefore categori0ed as members of thirtyt!o crystal

classesE each crystal class corresponds to a uniue set of symmetry operations.

$ach of the crystal classes is named according to the ;ariant of a crystal form

!hich it displays. $ach crystal class is grouped as one of the si% different

crystal systemsaccording to !hich characteristic symmetry operation it

possesses.

A crystal formis a set of planar faces !hich are geometrically eui;alent

and !hose spatial positions are related to one another by a specified set of

symmetry operations. 'f one face of a crystal form is defined, the specified set

of point symmetry operations !ill determine all of the other faces of the crystalform.

A simple crystal may consist of only a single crystal form. A more

complicated crystal may be a combination of se;eral different forms. 8he

crystal forms of the fi;e nonisometric crystal systems are the monohedron or

pedion, parallelohedron or pinacoid, dihedron, or dome and sphenoid,

disphenoid, prism, pyramid, dipyramid, trape0ohedron, scalenohedron,

rhombohedron and tetrahedron. 6ifteen different forms are possible !ithin the

isometric system.

$ach crystal class is a member of one of si% crystal systems. 8hese systems

include the isometric, he%agonal, tetragonal, orthorhombic, monoclinic, andtriclinic systems. 8he he%agonal crystal system is further bro#en do!n into the

he%agonal and rhombohedral di;isions. $;ery crystal of a certain crystal

system !ill share a characteristic symmetry element !ith the other members of

its system. 8he crystal system of a mineral species may sometimes be

determined ;isually by e%amining a particularly !ellformed crystal of the

species.


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*' Symmetry and Lattices

8op

Symmetry

Crystals possess a regular, repetiti;e internal structure. 8he concept of

symmetrydescribes the repetition of structural features. Crystals therefore

possess symmetry, and much of the discipline of crystallography is concerned

!ith describing and cataloging different types of symmetry.

8!o general types of symmetry e%ist. 8hese consist of translational

symmetryand point symmetry. 8ranslational symmetry describes the periodic

repetition of a structural feature across a length or through an area or ;olume.

Point symmetry, on the other hand, describes the periodic repetition of astructural feature around a point. 5eflection, rotation, and in;ersion are all

point symmetries.

Lattices

8he concept of a latticeis directly related to the idea of translational

symmetry. A lattice is a net!or# or array composed of single motif !hich has

been translated and repeated at fi%ed inter;als throughout space. 6or e%ample, a

suare !hich is translated and repeated many times across the plane !ill

produce a planar suare lattice.8he unit cellof a lattice is the smallest unit !hich can be repeated in three

dimensions in order to construct the lattice. 'n a crystal, the unit cell consists of

a specific group of atoms !hich are bonded to one another in a set geometrical

arrangement. 8his unit and its constituent atoms are then repeated o;er and

o;er in order to construct the crystal lattice. 8he surroundings in any gi;en

direction of one corner of a unit cell must be identical to the surroundings in the

same direction of all the other corners. 8he corners of the unit cell therefore

ser;e as points !hich are repeated to form a lattice arrayE these points are

termed lattice points. 8he ;ectors !hich connect a straight line of eui;alent

lattice points and delineate the edges of the unit cell are #no!n as thecrystalloraphic a#es.

8he number of possible lattices is limited. 'n the plane only fi;e different

lattices may be produced by translation. ne of these lattices possesses a suare

unit cell !hile another possesses a rectangular unit cell. 8he third possible

planar lattice possesses a centered rectangular unit cell, !hich contains a lattice

point in the center as !ell as lattice points on the corners. 8he unit cell of the
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fourth possible planar lattice is a parallelogram, and that of the final planar

lattice is a he%agonal unit cell !hich may alternately be considered a rhombus.

Bravais Lattices

8op

8he 6rench crystallographer Auguste Bra;ais >*-***-3@? established that

in threedimensional space only fourteen different lattices may be constructed.

8he fourteen ra%ais latticesmay be di;ided among si% crystal systems. 8hese

are the isometric or cubic, tetragonal, orthorhombic, monoclinic, triclinic, and

he%agonal systems. >8he si% crystal systemsare discussed belo!.? 8he Bra;ais

lattices are furthermore of three different types. A primiti%e latticehas only a

lattice point at each corner of the threedimensional unit cell. A body+centered

latticecontains 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 face+

centered latticepossesses not only lattice points at the corners of the unit cell

but also at either the centers of "ust one pair of faces or else at the centers of all

three pairs of faces. 8he fourteen Bra;ais lattices are therefore the primiti;e

cubic, bodycentered cubic, facecentered cubic, primiti;e tetragonal, body

centered tetragonal, primiti;e orthorhombic, bodycentered orthorhombic,

single facecentered orthorhombic, multiple facecentered orthorhombic,

primiti;e monoclinic, single facecentered monoclinic, primiti;e triclinic,

single facecentered he%agonal, and rhombohedral lattices. >8he rhombohedral

lattice is a subset of the he%agonal crystal system.?

Point Symmetry Operations

$oint symmetrydescribes the repetition of a motif or structural feature

around a single reference point, commonly the center of a unit cell or a crystal.

8he different pointsymmetry operations are reflection, rotation, in;ersion, and

the combined operation rotoin;ersion.

A reflectionoccurs !hen 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. 8he plane across !hich the reflection occurs is then

termed a mirror plane.

otational symmetryarises !hen a structural element is rotated a fi%ed

number of degrees about a central point and then repeated. A suare, for

e%ample, possesses 4fold rotational symmetry because it may be rotated four

times by ()F about its central point before it is returned to its original position.

$ach time it is rotated by ()F the resultant suare !ill be identical in

appearance to the original suare.

'f a crystal possesses in%ersion symmetry, then any line !hich is dra!n
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through the origin at the center of the crystal !ill connect t!o identical features

on opposite sides of the crystal.

otoin%ersionis a compound symmetry operation !hich is produced by

performing a rotation follo!ed by an in;ersion. *fold, fold, @fold, 4fold,

and 3fold rotoin;ersion operations e%ist. Most of these rotoin;ersions may

alternately be described by a specified set of rotation, reflection and in;ersion

operations. A *fold rotoin;ersion is eui;alent to rotation by @3)F follo!ed by

in;ersion. 8his procedure is ultimately eui;alent to a single in;ersion. A

fold rotoin;ersion a%is is eui;alent to reflection through a mirror plane

perpendicular to the rotoin;ersion a%is. A crystal !hich possesses a @fold

rotoin;ersion a%is is eui;alent to one !hich possesses both @fold rotational

symmetry and in;ersion symmetry. A 3fold rotoin;ersion is eui;alent to @

fold rotation and reflection across a mirror plane !hich lies at right angles to

the rotation a%is. 8he only rotoin;ersion operation !hich cannot be replaced by

a combination of rotations, reflections and in;ersions is 4fold rotoin;ersion.


may be combined in a ;ariety of different !ays. 8here are thirtyt!o different

possible combinations of these symmetry elements. Minerals possessing the

different combinations are therefore categori0ed as members of @ possible

crystal classes. According to this schema, each crystal class corresponds to a

uniue set of symmetry operations. $ach crystal class is then placed into one of

the si% different crystal systems so that se;eral different classes are members of

each system.

,' Crystal Systems

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$;ery crystal class is a member of one of the si% crystal systems. 8hese

systems include the isometric, he%agonal, tetragonal, orthorhombic,

monoclinic, and triclinic crystal systems. 8he he%agonal crystal system is

further bro#en do!n into he%agonal and rhombohedral di;isions.

$;ery crystal class !hich belongs to a certain crystal system !ill share a

characteristic symmetry element !ith the other members of its system. 6ore%ample, all crystals of the isometric system possess four @fold a%es of

symmetry !hich proceed diagonally from corner to corner through the center

of the cubic unit cell. 'n contrast, all crystals of the he%agonal di;ision of the

he%agonal system possess a single si%fold a%is of rotation.

'n addition to the characteristic symmetry element, a crystal class may

possess other symmetry elements !hich are not necessarily present in all


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members of the same system. 8he crystal class !hich possesses the highest

possible symmetry or the highest number of symmetry elements !ithin each

system is termed the holomorphic classof the system. 6or e%ample, crystals of

the holomorphic class of the isometric system possess in;ersion symmetry,

three 4fold a%es of rotational symmetry, the characteristic set of four @fold

a%es of rotational symmetry !hich is indicati;e of the isometric crystal system,

si% fold a%es of rotational symmetry, and nine different mirror planes. 'n

contrast, a crystal !hich is not a member of the holomorphic class yet still

belongs to the isometric system may possess only three fold a%es of rotational

symmetry and the characteristic four @fold a%es of rotational symmetry.

8he crystal system of a mineral species may sometimes be determined in

the field by ;isually e%amining a particularly !ellformed crystal of the

species.

Isometric

8he isometriccrystal system is also #no!n as the cubicsystem. 8he

crystallographic a%es used in this system are of eual length and are

mutually perpendicular, occurring at right angles to one another.

All crystals of the isometric system possess four @fold a%es of

symmetry, each of !hich 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 4fold a%es of rotational

symmetry. 8hese a%es, if present, proceed from the center of each face

through the origin to the center of the opposite face and correspond to

the crystallographic a%es. 6urthermore crystals of the isometric systemmay possess si% fold a%es of symmetry !hich e%tend 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.

$%amples of minerals !hich crystalli0e in the isometric system are

halite, magnetite, and garnet. Minerals of this system tend to produce

crystals of euidimensional or euant habit. >Please refer to ection for

more information on crystal habit.?

"e#aonal

Minerals of the he#aonalcrystal system are referred to threecrystallographic a%es !hich intersect at *)F and a fourth !hich is

perpendicular to the other three. 8his fourth a%is is usually depicted

;ertically.

8he he%agonal crystal system is di;ided into the he#aonaland

rhombohedralor trionaldi;isions. All crystals of the he%agonal

di;ision possess a single 3fold a%is of rotation. 'n addition to the single
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3fold a%is of rotation, crystals of the he%agonal di;ision may possess up

to si% fold a%es of rotation. 8hey may demonstrate a center of

in;ersion symmetry and up to se;en mirror planes. Crystals of the

trigonal di;ision all possess a single @fold a%is of rotation rather than

the 3fold a%is of the he%agonal di;ision. Crystals of this di;ision may

possess up to three fold a%es of rotation and may demonstrate a center

of in;ersion and up to three mirror planes.

Minerals species !hich crystalli0e in the he%agonal di;ision are

apatite, beryl, and high uart0. Minerals of this di;ision tend to produce

he%agonal prisms and pyramids. $%ample species !hich crystalli0e in

the rhombohedral di;ision are calcite, dolomite, lo! uart0, and

tourmaline. uch minerals tend to produce rhombohedra and triangular

prisms.

Tetraonal

Minerals of the tetraonalcrystal system are referred to three

mutually perpendicular a%es. 8he t!o hori0ontal a%es are of eual

length, !hile the ;ertical a%is is of different length and may be either

shorter or longer than the other t!o. Minerals of this system all possess a

single 4fold symmetry a%is. 8hey may possess up to four fold a%es of

rotation, a center of in;ersion, and up to fi;e mirror planes.

Mineral species !hich crystalli0e in the tetragonal crystal system are

0ircon and cassiterite. 8hese minerals tend to produce short crystals of

prismatic habit.

!rthorhombic

Minerals of the orthorhombiccrystal system are referred to threemutually perpendicular a%es, each of !hich is of a different length than

the others.

Crystals of this system uniformly possess three fold rotation a%es

and/or three mirror planes. 8he holomorphic class demonstrates three

fold symmetry a%es and three mirror planes as !ell as a center of

in;ersion. ther classes may demonstrate three fold a%es of rotation or

one fold rotation a%is and t!o mirror planes.

pecies !hich belong to the orthorhombic system are oli;ine and

barite. Crystals of this system tend to be of prismatic, tabular, or acicular

habit.Monoclinic

Crystals of the monoclinicsystem are referred to three uneual a%es.

8!o of these a%es are inclined to!ard each other at an obliue angleE

these are usually depicted ;ertically. 8he third a%is is perpendicular to

the other t!o. 8he t!o ;ertical a%es therefore do not intersect one

another at right angles, although both are perpendicular to the hori0ontal


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a%is.

Monoclinic crystals demonstrate a single fold rotation a%is and/or a

single mirror plane. 8he holomorphic class possesses the single fold

rotation a%is, a mirror plane, and a center of symmetry. ther classes

display "ust the fold rotation a%is or "ust the mirror plane.

Mineral species !hich adhere to the monoclinic crystal system

include pyro%ene, amphibole, orthoclase, a0urite, and malachite, among

many others. 8he minerals of the monoclinic system tend to produce

long prisms.

Triclinic

Crystals of the triclinicsystem are referred to three uneual a%es, all

of !hich intersect at obliue angles. 1one of the a%es are perpendicular

to any other a%is.

Crystals of the triclinic system may be said to possess only a *fold

symmetry a%is, !hich is eui;alent to possessing no symmetry at all.

Crystals of this system possess no mirror planes. 8he holomorphic class

demonstrates a center of in;ersion symmetry.

Mineral species of the triclinic class include plagioclase and a%initeE

these species tend to be of tabular habit.

-' Crystal .orms

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A crystal formis a set of faces !hich are geometrically eui;alent and

!hose spatial positions are related to one another according to the symmetry of

the crystal. 'f one face of a crystal form is defined, the point symmetry

operations !hich specify the class to !hich the crystal belongs also determine

the other faces of the crystal form.

6ifteen different forms are possible !ithin the isometric or cubic system.

8hese include the he%octahedron, gyroid, he%tetrahedron, diploid, and tetartoid,

among others. 8he crystal forms of the remaining fi;e crystal systems are the

monohedron or pedion, parallelohedron or pinacoid, dihedron, or dome andsphenoid, disphenoid, prism, pyramid, dipyramid, trape0ohedron,

scalenohedron, rhombohedron, and tetrahedron.

8he crystal forms !hich occur in each crystal class and system must

possess a symmetry complementary to that of the associated crystal class and

system. 6or e%ample, a monohedron, !hich possesses only one face, !ill ne;er

occur in a crystal !ith in;ersion symmetry because the in;ersion operation


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reuires that an eui;alent 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 se;eral different forms. All forms

!hich occur in a crystal of a particular system must be compatible !ith that

crystal system.

Monohedron

8he monohedralcrystal form is also called a pedion. 't consists of a

single face !hich is geometrically uniue for the crystal and is not

repeated by any set of symmetry operations. Members of the triclinic

crystal system produce monohedral crystal forms.

$arallelohedron

8he parallelohedralcrystal form is also called a pinacoid. 't consists of

t!o and only t!o geometrically eui;alent faces !hich occupy opposite

sides of a crystal. 8he t!o faces are parallel and are related to one

another only by a reflection or an in;ersion. Members of the triclinic

crystal system produce parallelohedral crystal forms.

/ihedron

8he dihedronconsists of t!o and only t!o nonparallel geometrically

eui;alent faces. 8he t!o faces may be related by a reflection or by a

rotation. 8he dihedron is termed a domeif the t!o faces are related only

by reflection across a mirror plane. 'f the t!o faces are related instead by

a fold rotation a%is then the dihedron is termed a sphenoid. Members

of the monoclinic crystal system produce dihedral crystal forms.

/isphenoidMembers of the orthorhombic and tetragonal crystal systems produce

rhombic and tetragonal disphenoids, !hich possess t!o sets of

nonparallel geometrically eui;alent faces, each of !hich is related by a

fold rotation. 8he faces of the upper sphenoid alternate !ith the faces

of the lo!er sphenoid in such forms.

$rism

A prismis composed of a set of @, 4, 3, -, or * geometrically

eui;alent faces !hich are all parallel to the same a%is. $ach of these

faces intersects !ith the t!o faces ad"acent to it to produce a set of

parallel edges. 8he mutually parallel edges of all intersections of theprism sides then form a tube. Prisms are gi;en names based on the shape

of their cross section. ariants of the prism form include the rhombic

prism, tetragonal prism, trigonal prism, and he%agonal prism. A prism in

!hich the large faces are di;ided into t!o mirrorimage faces !hich

intersect !ith one another at an obliue angle is called a ditetragonal


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prism, a ditrigonal prism, or a dihe%agonal prism. Prisms are associated

!ith the members of the monoclinic crystal system.

$yramid

A pyramidis composed of a set of @, 4, 3, -, or * faces !hich are not

parallel but instead intersect at a point. 8he orthorhombic, tetragonal and

he%agonal crystal systems all produce pyramids. 8hese pyramids are

named according to the shape of their crosssection in the same !ay that

prisms are. 8hus are produced the rhombic pyramid, tetragonal pyramid,

trigonal pyramid, and he%agonal pyramid. $ach large face of the

ditetragonal pyramid, ditrigonal pyramid, and dihe%agonal pyramids is

di;ided into t!o mirrorimage faces !hich occupy an obliue angle !ith

respect to one another.

/ipyramid

8he dipyramidalcrystal form is composed of t!o pyramids placed

basetobase and related by reflection across a mirror plane !hich runs

parallel to and ad"acent to the pyramid bases. 8he upper and lo!er

pyramids may each ha;e @, 4, 3, -, or * facesE the dipyramidal form

therefore possesses a total of 3, -, *, *3, or 4 faces. 8he orthorhombic,

tetragonal and he%agonal crystal systems all produce dipyramids. 8hese

dipyramids are named for the shape of their crosssection "ust as prisms

and pyramids are, resulting in the rhombic dipyramid, trigonal

dipyramid, tetragonal dipyramid, and he%agonal dipyramid. 8he large

faces of the ditetragonal, ditrigonal and dihe%agonal dipyramids are

di;ided into t!o mirrorimage faces !hich intersect one another at an

obliue angle.Trape0ohedron

A trape0ohedronis a crystal form possessing 3, -, or * trape0oidal

faces. 8he tetragonal crystal system and both the trigonal and he%agonal

di;isions of the he%agonal crystal system produce trape0ohedral crystal

forms. 8rigonal trape0ohedra possess three trape0oidal faces on the top

and three on the bottom for a total of si% facesE tetragonal trape0ohedra

ha;e four faces on top and four on the bottom for a total of eight facesE

and he%agonal trape0ohedra ha;e si% faces on top and si% on the bottom,

resulting in t!el;e faces total.

ScalenohedronA scalenohedronconsists of - or * faces, each of !hich is a scalene

triangle. 8he faces appear to be grouped into symmetric pairs. 8he

tetragonal and he%agonal crystal systems produce the scalenohedral

crystal form, of !hich e%amples may be further described as trigonal,

tetragonal and he%agonal scalenohedra.

hombohedron


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8he rhombohedralcrystal form possesses si% rhombusshaped faces. A

rhombohedron resembles in appearance a cube !hich is poised upright

upon one corner and has been either flattened or elongated along an a%is

!hich runs diagonally from corner to corner through the center. 8he

rhombohedral crystal form is produced only by members of the trigonal

and rhombohedral di;isions of the he%agonal crystal system.

Tetrahedron

A tetrahedronis composed of four triangular faces. 'n crystals of the

isometric system each face is an identical euilateral triangle. 'n crystals

of the tetragonal system each face is an identical isoceles triangleE this

;ariant of the tetrahedron is called a tetragonal tetrahedron. 'n crystals of

the orthorhombic system the faces consist of t!o pairs of different

isoceles trianglesE the crystal is then termed a rhombic tetrahedron.

1' Crystal Classes


may be combined in thirtyt!o different !ays. 8hirtyt!o different crystal

classesare therefore defined so that each crystal class corresponds to a uniue

set of symmetry operations. $ach of the crystal classes is named according to

the ;ariant of a crystal form!hich it displays. 6or e%ample, the isometric

he%octahedral class belongs to the isometric crystal system and demonstrates

the he%octahedral crystal form. 8he rhombic pyramidal, tetragonal pyramidal,trigonal pyramidal and he%agonal pyramidal classes each display a ;ariant of

the crystal form !hich is called a pyramid.

$ach crystal class is a member of one of the si% different crystal systems

according to !hich characteristic symmetry operation it possesses. 6or

e%ample, all crystals of the isometric system possess four @fold a%es of

symmetry, !hile minerals of the tetragonal system possess a single 4fold

symmetry a%is and crystals of the triclinic class sho! no symmetry at all. 8he

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 he%agonal pyramidal classes are members of the

rhombohedral >trigonal? and he%agonal di;isions of the he%agonal crystal

system respecti;ely.


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Table of the 32 Crystal Classes

8he follo!ing table lists in bold type the si% crystal systems. 'ncluded are

the isometric, he%agonal, tetragonal, orthorhombic, monoclinic, and triclinic

systems. 8he tetragonal crystal system is further separated into the he%agonal

and trigonal or rhombohedral di;isions. nder each crystal system the tablelists by name the crystal classes !hich occur !ithin that system. 6or e%ample,

the crystal classes !hich occur !ithin the trigonal crystal system are the

trigonal monohedral and trigonal parallelohedral crystal classes. Ad"acent to the

listing of each crystal class is the symmetry of the class.

=hen listing the symmetry of each crystal class an a%is of rotational

symmetry is represented by the capital letter A. =hether this a%is is a fold, @

fold, or 4fold a%is is indicated by a subscript follo!ing the letter A. 8he

number of such a%es present is indicated by a numeral preceding the capital A.

*A, A@, and @A4thus represent one fold a%is of rotation, t!o @fold a%es, and

three 4fold a%es respecti;ely. A center of in;ersion is noted by the lo!ercase

letter 9i9 !hile a mirror plane is denoted by 9m9. 8he numeral preceding the m

indicates ho! many mirror planes are present. A%es of rotary in;ersion are

usually replaced by the eui;alent rotations and reflections. 6or e%ample, a

fold rotoin;ersion a%is is eui;alent to reflection through a mirror plane

perpendicular to the rotoin;ersion a%is. A crystal !hich possesses a @fold

rotoin;ersion a%is is eui;alent to one !hich possesses both @fold rotational

symmetry and in;ersion symmetry. A 3fold rotoin;ersion is eui;alent to @

fold rotation and reflection across a mirror plane at right angles to the rotation

a%is. 8he only rotoin;ersion operation !hich cannot be thus replaced is 4foldrotoin;ersion, !hich is indicated by 54.

8he class !hich possesses the highest possible symmetry !ithin each

crystal system is termed the holomorphic class of that system. 8he holomorphic

class of each crystal system is indicated in the table by bold type. 6or e%ample,

the triclinic parallelohedron is the holomorphic class of the triclinic crystal

system !hile the isometric he%octahedron is the holomorphic class of the

isomorphic or cubic crystal system. 8he characteristic symmetry element of

each crystal system is listed in bold type. 't is thus apparent that the

characteristic symmetry element of the isometric crystal system is the

possession of four @fold a%es of rotational symmetry, !hile the characteristicsymmetry element of the rhombohedral di;ision of the he%agonal crystal

system is the possession of a single @fold a%is of rotational symmetry.

Crystal System Crystal Class 2 Symmetry of


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Crystal .orm Class

Isometric System

he#octahedron

gyroid

he%tetrahedron

diploidtetartoid

i, @A4, -A,, 3A,

(m

@A4, -A,, 3A@A, -A,, 3m

i, @A, -A,, @m@A, -A,

"e#aonal SystemHe%agonal

Di;ision

dihe#aonal

dipyramid

he%agonal

trape0ohedron

dihe%agonal pyramid

ditrigonal dipyramid

he%agonal dipyramid

he%agonal pyramidtrigonal dipyramid

i, &A3, 3A, m

&A3, 3A

&A3, 3m

&3, @A, @m

i, &A3, *m

&A3

&3

5hombohedral

Di;ision

he#aonal

scalenohedron

trigonal trape0ohedron

ditrigonal pyramid

rhombohedron

trigonal pyramid

i, &A,, @A, @m

&A,, @A

&A,, @m

i, &A,

&A,

Tetraonal

System

ditetraonal

dipyramidtetragonal

trap0ohedron

ditetragonal pyramid

tetragonal

scalenohedron

tetragonal dipyramid

tetragonal pyramid

tetragonal disphenoid

i, &A-, 4A, m

&A-, 4A

&A-, 4m

&-, A, m

i, &A-, *m

&A-

&-

!rthorhombicSystem

rhombic dipyramid

rhombic disphenoid

rhombic pyramid

i, @A, @m@A

*A, m

Monoclinic

System

prism

sphenoid

dome

i, *A, *m

*A*m

Triclinic System parallellohedron i


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Crystal Form