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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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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
it comprises at least all invariant spatial points (fixed points) of the operation
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Symmetry operation and Symmetry element
a = b = c
a=
b=
g= 90
180
a = b = c
a=
b=
g= 90
indistinguishable
axis of rotation 360 / 180 = 2 2-fold axis of rotation
Symmetry element (SE)
is the geometrical object (point, line, plane) on which the SO is carried out
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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The Fascination of Crystals and Symmetry
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-fold axis of rotation
3 3 5
72 120
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Combination of Mirror and Rotational symmetry
5-fold axis of rotation
rotational symmetry may or may not be combined with mirror 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
rotational symmetry may or may not be combined with mirror symmetry
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
If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain
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
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The Fascination of Crystals and Symmetry
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
If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain
eldadcarin - Fotolia.com
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The Fascination of Crystals and Symmetry
Unit 3.4
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Rotoinversions vs. Rotary reflections
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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Rotoinversions vs. Rotary reflections
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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Rotoinversions vs. Rotary reflections
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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Rotoinversions vs. Rotary reflections
3 S 6
R t i i R t fl ti
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Rotoinversions vs. Rotary reflections
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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The Fascination of Crystals and Symmetry
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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Classify Crystals into Classes
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!
Classify Crystals into Classes
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Classify Crystals into Classes
1st Exampleno mno n
i
ba
c
1crystal class
pinacoidal
microcline (potassium feldspar),turquoise, and wollastonite
plagioclase
Classify Crystals into Classes
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Classify Crystals into Classes
2nd Example
Classify Crystals into Classes
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Classify Crystals into Classes
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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Classify Crystals into Classes
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Classify Crystals into Classes
2nd Exampleone mone 2 perpendicular to m
http://webmineral.com/data/Gypsum.shtml
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
Classify Crystals into Classes
http://webmineral.com/data/Gypsum.shtmlhttp://webmineral.com/data/Gypsum.shtml -
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Classify Crystals into Classes
one mone 2 perpendicular to m
http://webmineral.com/data/Gypsum.shtml
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
http://webmineral.com/data/Gypsum.shtmlhttp://webmineral.com/data/Gypsum.shtml -
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Picture Credits
If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain
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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The Fascination of Crystals and Symmetry
Unit 3.6
Classify Crystals into Classes
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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
Classify Crystals into Classes
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y y
m
3 rd Example
a
b
c
Classify Crystals into Classes
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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
Classify Crystals into Classes
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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
Classify Crystals into Classes
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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
Crystal Classes Poster
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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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If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain
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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The Fascination of Crystals and Symmetry
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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(a) reflection at a plane / line(b) translation (usually by 1/ 2 of the unit cell)
g
glide reflection
translationreflection
Glide planes/lines
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(a) reflection at a plane / line(b) translation (usually by 1/ 2 of the unit cell)
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
2-fold axis of rotation
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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If not otherwise stated pictures, images, sketches, clip arts are self-taken/self-drawn or public domain
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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The Fascination of Crystals and Symmetry
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
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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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