chapter 3 the world of symmetry, crystal classes and plane groups

8/12/2019 CHAPTER 3 the World of Symmetry, Crystal Classes and Plane Groups

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The Fascination of Crystals and Symmetry

Unit 3.1


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Entering the World of Symmetry E n t e r i n g t h e W o r l d o f S y m m e t r y

external symmetry of macroscopic objects


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Systematization of Crystal Structures

1. Step: Crystal systemsmetric + symmetry of the UC 7

2. Step: Bravais latticesprimitive + centered 14

3. Step: Crystal classescrystallographic PG 32

4. Step: Space groups

complete symmetry 230


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Symmetry

from Greek (symmetria)meaning agreement in dimensions, due proportion, arrangement

Regularity / Harmony


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Symmetry operation and Symmetry element

a = b = c

a=

b=

g= 90

90

a = b = c

a=

b=

g= 90

indistinguishable

axis of rotation 360 / 90 = 4 4-fold axis of rotation

Symmetry element (SE)

is the geometrical object (point, line, plane) on which the SO is carried out


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a = b = c

a=

b=

g= 90

90

a = b = c

a=

b=

g= 90

indistinguishable




it comprises at least all invariant spatial points (fixed points) of the operation


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a = b = c

a=

b=

g= 90

180

a = b = c

a=

b=

g= 90

indistinguishable




it comprises at least all invariant spatial points (fixed points) of the operation

usually on one SE several different SO can be carried out


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Symmetry concerning macroscopic object external symmetry

Symmetry elements of macroscopic objects

1. Identity

2. Mirror plane

3. Axis of rotation

4. Center of inversion

5. Rotoinversion axisidentity

1-fold axis of rotation(rotation by 360 )

symbol E

even the most asymmetric

objects have at least one SE


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Mirror symmetry

mirror plane

also called line symmetry or reflection symmetry or bilateral symmetry

an object which does not change upon undergoing a reflection has mirror symmetry,

it is mirror symmetric

In 2D there is a line of symmetry or mirror line, in 3D a plane of symmetry or mirror plane

symbol m

mm


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Excursus: Mirror symmetry and Beauty

original right side mirrored left side mirrored

Kelly GeorgeMiss Arkansas 2007


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Excursus: Mirror symmetry and Beauty

1995

Prof. Dr. rer. nat. Karl GrammerUniversity of Vienna

http://evolution.anthro.univie.ac.at/institutes/urbanethology/staff/grammer.html

AcademiaResearchGateGoogle Scholars

2007
http://evolution.anthro.univie.ac.at/institutes/urbanethology/staff/grammer.htmlhttp://evolution.anthro.univie.ac.at/institutes/urbanethology/staff/grammer.htmlhttp://evolution.anthro.univie.ac.at/institutes/urbanethology/staff/grammer.html


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Picture Credits

If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain

alephcomo1 - Fotolia.com CC-BY-SA 3.0 de | US Air Force | Friedrich Graf with kind permission of Karl Grammer


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Unit 3.2


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Rotational symmetry

25 -fold axis of rotation

16 -fold axis of rotation

!

4-fold axis of rotation


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Rotational symmetry

single objects can have rotational symmetry of any order

-fold axis of rotation

rotational symmetry may or may not be combined with mirror symmetry

SO = Rotation by 360 / n SE = n-fold axis of rotation symbol n (1, 2, 3)

5-fold axis of rotation3-fold axis of rotation

120 120

120


3 3 5

72 120


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Combination of Mirror and Rotational symmetry



1 unique mirror plane 2 unique mirror planes6-fold axis of rotation

5m 6mmmm

single objects can have rotational symmetry of any order


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Combination of Mirror and Rotational symmetry

single objects can have any rotational symmetry whatsoever


2mm

mm

any pair of two orthogonal mirror planes generate atwo-fold axis of rotation at their intersection line


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Axis of Rotation Summary

Axis of Rotation

rotation around a axis (= fixed points ofthe rotation) with an angle of rotation

after n rotations by the startingposition is reached

n = order of the axis

in crystallography

n = 2, 3, 4, 6

2-fold

3-fold

4-fold

6-fold

180

120

90

60

the number of crystal classes is limited to 32because of the restrictions of rotationalsymmetry in crystals


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Picture Credits


Wikipedia user Stephanb | CC-BY-SA 3.0 Sergey Nivens | Fotolia.com

zentilia - Fotolia.com by-studio - Fotolia.com

with kind permission by Der Linkshnder

www.derlinkshaender.com
http://www.derlinkshaender.com/http://www.derlinkshaender.com/


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Unit 3.3


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Point symmetry Center of Inversion

also called origin symmetry or center of symmetry

180

symbol i or 1 (one -bar) x, y, z -x, -y, -z

there is always a matching part, which has the same distance from a central point

but in the opposite direction

in the plane it is identical with rotational symmetry of order 2

(2-fold axis of rotation)


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Rotoinversion axis

mirror planes

2-fold axes of rotation

3-fold axes of rotation

no center of inversion

Tetrahedron

rotoinversion axis


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Rotoinversion axis

90

i

a rotoinversion is a combined SO, where two transformations have to be carried out

(1) rotation around 360 /n

(2) immediately followed by an inversion at a center of symmetry, which lies on therotoinversion axis

Tetrahedron

4-fold rotoinversion axis 4 1= 4 +symbol


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Rotoinversion axes of order 1, 2, and 3

1

2

1 = i

2

1

2 = m

odd rotoinversions possess automatically a center of inversion

3 = 3 + 1

1

3

5

2

4

6


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Rotoinversion axes of order 4 and 6

4

1 3

2

4

even rotoinversions contain automatically an axis of rotation of the half order(4-bar contains a 2-fold, and 6-bar contains a 3-fold axis of rotation)

1

3

5

2

4

6

6 = 3 m1= 4 + 1= 6 +

if the order n is even, but not divisible by 4, then there is automatically amirror plane perpendicular to the rotoinversion axis


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Rotoinversions vs. Rotary reflections


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Picture Credits


eldadcarin - Fotolia.com


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Unit 3.4


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S h fli H M i


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Schoenflies vs. Hermann-Mauguin

World of molecules World of crystals

Symmetry notation system according toArtur Moritz Schoenflies

Symmetry notation system according toCarl Hermann und Charles-Victor Mauguin

Schoenflies symbolism Hermann-Mauguin symbolism

R i i R fl i


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Schoenflies Hermann-Mauguin

Symmetry elements of macroscopic objects

Identity

Mirror plane

Axis of rotation

Center of inversion

Rotoinversion axis

Identity

Mirror plane

Axis of rotation

Center of inversion

Rotation-reflection axis

(Improper axis of rotation)

E 1

m

C n n

i 1

Sn n

R t i i R t fl ti


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90 i

Rotoinversion

4

h

h90

Rotary reflection

S 4

rotation-reflection axis

rotoinversion axis

improper axis of rotation

R t i i R t fl ti


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3 S 6

R t i i R t fl ti


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3 S 6

Assignment

Try to determine the order of

(a) the rotoinversion and

(b) the rotary reflection of

these two arrangements of locomotives!


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Unit 3.5

Systematization of crystal structures


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Systematization of crystal structures

1. Step: Crystal Systemsmetric + symmetry of the UC 7

2. Step: Bravais latticesprimitive + centered 14

3. Step: Crystal classescrystallographic PG 32

Crystal Classes


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Crystal Classes

everyday objects can have any symmetry, and

symmetry elements can be combined, in principle,

arbitrarily

the symmetry of crystals i.e. the symmetry of the external

shape of crystals is limited

they can be classified into 32 classes only

the symmetry has to be compatible with the repeating

pattern of the crystal lattice

infinite number of symmetry classes 32 symmetry classes (point groups)

see later

Classify Crystals into Classes


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1. What kind of symmetry elements (not operations) does the outer shape crystal sample possess?

4. You will see that the crystal classes can be categorized according to

the crystal systems (or crystal families, see below).

Recipe

2. Write down only those SE, which are unique (i.e. a SE that exists by

itself and is not created by other SE)!

3. Crystals that have the same SE belong to the same crystal class!



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1st Exampleno mno n

i

ba

c

1crystal class

pinacoidal

microcline (potassium feldspar),turquoise, and wollastonite

plagioclase



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2nd Example



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2nd Exampleone mone 2 perpendicular to m

http://webmineral.com/data/Gypsum.shtml

2

m

perpendicular to

speak: 2 over m

m

2

m

2

2/mcrystal class
http://webmineral.com/data/Gypsum.shtmlhttp://webmineral.com/data/Gypsum.shtml


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2nd Exampleone mone 2 perpendicular to m


azurite, chlorite, clinopyroxene,epidote, malachite, kaolinite,

orthoclase, and talc

CaSO4 2 H2O

prismatic

2

m

perpendicular to

speak: 2 over m

Gypsum

2/mcrystal class



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one mone 2 perpendicular to m


prismatic

2

m

perpendicular to

speak: 2 over m

2nd Example

azurite, chlorite, clinopyroxene,epidote, malachite, kaolinite,

orthoclase, and talc

2/mcrystal class

Fe3(PO4)2 8 H2OVivianite

Picture Credits


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Picture Credits


Rob Lavinsky, iRocks.com CC-BY-SA-3.0 Rob Lavinsky, iRocks.com CC-BY-SA-3.0 alephcomo1 - Fotolia.com by-studio - Fotolia.com
http://www.irocks.com/http://www.irocks.com/http://www.irocks.com/http://www.irocks.com/


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Unit 3.6



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y y

3 rd Example

a

b

c



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y y

3 rd Example

a

b

c

Introducing viewing directions, here a b c



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y y

m

3 rd Example

a

b

c



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y y

Struvite

MgNH4PO4 6 H2O

m

m2 m m 2crystal class

3 rd Example

(ortho)rhombic-pyramidal

a

b

c

2 m msymmetry elements

by convention

http://webmineral.com/data/Struvite.shtml

a b c

http://webmineral.com/data/Struvite.shtmlhttp://webmineral.com/data/Struvite.shtml


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y y

Zn4Si2O7(OH)2 H2O

m

m2 mm2crystal class

3 rd Example

Hemimorphite

a

b

c

(ortho)rhombic-pyramidal



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y y

4 th Example

ab

c

120

6 viewing directions(hexagonal crystal system)

c a

6

[210]



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4 th Example

ab

c

120

6

m

m

viewing directions(hexagonal crystal system)

c a [210]

6m

m



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4 th Example

ab

c

120

6

m

viewing directions(hexagonal crystal system)

c a [210]

6m

m m



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4 th Example

ab

c

120 (Mg, Graphite, Nickeline)

CuSCovellite

6/mmmcrystal class

dihexagonal-dipyramidal

Crystal Classes A collectible


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2014 FIFA World Cup Brazil

Collectible

2014 International Year of Crystallography

Collectible

Crystal Classes Poster



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Mapping Crystal Classes to Crystal Systems/Crystal Families


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crystal system

triclinic

orthorhombic

trigonal

monoclinic

tetragonal

hexagonal

cubic

crystal classes (point groups)

1, 1

2, m, 2/m

222, mm2, 2/m 2/m 2/m

4, 4, 4/m, 422, 4mm, 42m, 4/m 2/m 2/m

3, 3, 32, 3m, 32/m

6, 6, 6/m, 622, 6mm, 6m2, 6/m 2/m 2/m

23, 2/m3, 432, 43m, 4/m 3 2/m

charact. SE

1 or 1

one 2 and/or m

three 2 and/or m

one 4 or 4

one 6 or 6

one 3 or 3

four 3 or 3

hexagonal crystal family

Picture Credits


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McArthurGlen Designer Outlets | Flickr | CC-BY-SA-2.0

Didier Descouenss | CC-BY-SA-3.0

Walter Klle | CC-BY-SA-3.0

Rob Lavinsky, iRocks.com CC-BY-SA-3.0

with kind permission bythe PANINI group
http://www.irocks.com/http://www.irocks.com/


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Unit 3.7


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Translational symmetry elements and Wallpaper groups

1. Translations2. Glide planes / Glide axes

3. Screw axes

Symmetric Patterns in 2D Plane groups


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Translational Symmetry


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There are three symmetry elements, which have a translational component

1. Translation (in units of whole unit cells along the lattice vectors)

translation

repeating unit (unit cell)

mirror plane

m

translation repeating unit (unit cell)

Glide planes/lines


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glide line

(a) reflection at a plane / line(b) translation (usually by 1/ 2 of the unit cell)

g

glide reflection

Glide planes/lines


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g

glide reflection

translationreflection

Glide planes/lines


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g

glide reflection

translationreflection

There are three symmetry elements, which have a translational component

1. Translations (in units of whole unit cells along the lattice vectors)

2. Glide planes / glide axes

3. Screw axes

Kaiser's spotted newt

Repeating Patterns in the Plane


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TranslationRotation

ReflectionGlide

With the help of these symmetryoperations every conceivablerepeating pattern in the plane canbe generated and characterized.

wallpaperstextile patterns

tilingspavements

gift wrap papersEscher drawings

M.C. Escher


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Unicorn (No. 78)1950 Colored pencil, watercolor.

Lizard / Fish / Bat (No. 85)1952 Ink, pencil, watercolor.

All M.C. Escher works 2014 The M.C. Escher Company - the Netherlands. All rights reserved. Used by permission. www.mcescher.com

Repeating Patterns in the Plane
http://www.mcescher.com/http://www.mcescher.com/


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TranslationRotation

ReflectionGlide

With the help of these symmetryoperations every conceivablerepeating pattern in the plane canbe generated and characterized.

wallpaperstextile patterns

tilingspavements

gift wrap papersEscher drawings

Bravais latticesprimitive + centered 5

Plane groupscomplete plane symmetry 17

Bravais Lattices in 2D and Plane Symmetry Groups


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The 5 Bravais lattices of the plane

obliquep

rectangularp

squarep

centered rectangularc

hexagonalp

90

90 90

120

a = b

a = b

Notation of Plane groups


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Notation of Wallpaper groups

plus two symbols indicating mirrors ( m ), glides ( g)

perpendicular to a) the x -axis and b) the y -axis

p2mg

symmetry elements

Bravais type


mirror plane

glide plane

in full notation always 4 symbols

begins with p or c according to the Bravais lattice type

followed by the digit n indicating the rotationalsymmetry order

if there are no such operators a ( 1)is denoted

Notation of Plane groups


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Full and Short Notation of Wallpaper groups

Full p111 p211 p311 p3m1p1m1 p1g1 c1m1 p2mm p2mg p2gg c2mm p411

p4g

p4gm p611 p6mm

Short p1 pm pg cm p2 pmm pmg pgg cmm p3 p3m1

p31m

p31m p4

p4mm

p4m p6 p6m

p2mgthe short notation drops digits n or a m thatcan be deduced, so long as that leaves noconfusion with another plane group

Optional assignment : Overlay this pattern

with the unit cell and the respectivegraphical symbols of the symmetry elementsat their correct positions within this pattern!

Picture Credits


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with kind permission byDr. Richard Bartlett , all rights reserved.

All M.C. Escher works 2014 The M.C. Escher Company - the Netherlands.All rights reserved. Used by permission. www.mcescher.com
http://www.mcescher.com/http://www.mcescher.com/


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Unit 3.8

Summary and Outlook


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Space Symmetry (3D)Plane Symmetry (2D)

TranslationInversion (point mirroring)

Rotation-

ReflectionGlide (reflect, then translate, 2D)

5 Bravais lattices

TranslationInversion (point mirroring)

RotationRoto-Inversion (rotate, then invert)

ReflectionGlide (reflect, then translate, 3D)

Screw (rotate, then translate)

230 space groups

32 crystal classes

14 Bravais lattices

17 plane groups

Point Symmetry (3D)

-Inversion (point mirroring)

RotationRoto-Inversion (rotate, then invert)

Reflection-

-

32 crystal classes

Tim White Beauty, Form & Function: An Exploration of Symmetry


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A course on Coursera Feb 17 th to Apr 14 th

Algorithm to determine the plane symmetry group


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Brian Sanderson's Pattern Recognition Algorithm:

http://www.math.toronto.edu/~drorbn/Gallery/Symmetry/Tilings/Sanderson/index.html

Is the maximum rotation order 1,2,3,4 or 6?

Is there a mirror (m)?Is there an indecomposable glide reflection (g)?Is there a rotation axis on a mirror?

Is there a rotation axis not on a mirror?

Web and other Tools regarding plane groups
http://www.math.toronto.edu/~drorbn/Gallery/Symmetry/Tilings/Sanderson/index.htmlhttp://www.math.toronto.edu/~drorbn/Gallery/Symmetry/Tilings/Sanderson/index.html


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http://www.scienceu.com/geometry/handson/kali/ http://escher.epfl.ch/escher/ http://www.mathsisfun.com/geometry/symmetry-artist.html

Web and other Tools regarding plane groups
http://www.scienceu.com/geometry/handson/kali/http://escher.epfl.ch/escher/http://www.mathsisfun.com/geometry/symmetry-artist.htmlhttp://www.mathsisfun.com/geometry/symmetry-artist.htmlhttp://www.mathsisfun.com/geometry/symmetry-artist.htmlhttp://www.mathsisfun.com/geometry/symmetry-artist.htmlhttp://escher.epfl.ch/escher/http://www.scienceu.com/geometry/handson/kali/


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http://imaginary.org/program/morenaments/applet http://weavesilk.com/

The Great Pattern Collection
http://imaginary.org/program/morenaments/applethttp://weavesilk.com/http://weavesilk.com/http://imaginary.org/program/morenaments/applet


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http://www.thegreatpatterncollection.com/

Photo Contest Crystallography in Everyday Life
http://www.thegreatpatterncollection.com/http://www.thegreatpatterncollection.com/


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http://www.iycr2014.org/participate/photo-competition

crystalmooc

3 trigonal pyramidal

3 rhombohedral

_
http://www.iycr2014.org/participate/photo-competitionhttp://www.iycr2014.org/participate/photo-competition


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Crystal Classes and their Hermann -Mauguin symbols

organized by their crystal systems

triclinic pedial

1 1 triclinic pinacoidal

2 monoclinic sphenoidal

m monoclinic doma c

2/m monoclinic prisma c

222

orthorhombic sphenoidal

mm2 orthorhombic pyramidal

mmm orthorhombic dipyramidal

4 tetragonal pyramidal

4 tetragonal disphenoidal

4/m tetragonal dipyramidal

422 tetragonal trapezoidal

4mm ditetragonal pyramidal

4m2 tetragonal scalenoidal

4/mmm ditetragonal dipyramidal

triclinic

monoclinic

orthorhombic

tetragonal

321 trigonal trapezoidal

3m1 ditrigonal pyramidal

3m1 ditrigonal scalahedral

6 hexagonal pyramidal

6 trigonal dipyramidal

6/m hexagonal dipyramidal

622 hexagonal trapezoidal 6mm

dihexagonal pyramidal 6m2

ditrigonal dipyramidal 6/mmm

dihexagonal dipyramidal

23 tetrahedral 43m hextetrahedral

m3 diploidal 432 gyroidal hexoctahedral

hexagonal

trigonal

cubic

m3m

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chapter 3 the world of symmetry, crystal classes and plane groups

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