1. Symmetry

2. Rotational Symmetry in Flowers
rotation angle = 360o/2 = 180o
rotation angle = 360o/4 = 90o
4 mirror lines (2 are unique)
2 mirror lines (both unique)
2mm (both unique)
4mm
rotation angle = 360o/3 = 120o
rotation angle = 360o/6 = 60o
3 mirror lines (1 is unique)
6 mirror lines (2 are unique)
3m (not 3mmm)
6mm
rotation angle = 360o/5 = 72o
But....
5-fold rotation is never
used in crystallography.
Why is this so?
5 mirror lines (1 is unique)

4. Repeating Patterns in Two Dimensions
What is the size and shape of the asymmetric unit? y x a b
Each unit consists of two non- linear translational vectors and four edges (because we are concerned with crystals).
Here the asymmetric unit is also the repeating tile.
The tile has no fixed origin, i.e. the corner can be anywhere, but each corner must be identical!

5. Recognizing Two Dimensional (Planar) Tilings
Valid tiles for this flower garden include: 2 3 1 2 1 3
Are there any mirror lines or glide lines in the garden?
What is the point symmetry of the garden?

6. Creating Two Dimensional (Planar) Tilings
What are the key properties of a tiling?
The tiles are congruent-consistent - they join without gaps
The tile is a polygon. Any polygon could be used, but in crystallography four sided parallelograms where both pairs of opposite sides are parallel are used exclusively.
Tessellation is the process of filling 2-dimensional space with a collection of plane figures with no overlaps or gaps.

8. Creating Two Dimensional (Planar) Tilings
Find the repeating tile in the rose garden. y x a b
How many flowers are in the tile?
What is the point symmetry?

9. Which one is the asymmetric unit?

10. Finding Symmetry in Planar Tilings
Find the repeating tile in the garden. y x a b
What is the symmetry element?
How many flowers are in the tile?
What is the asymmetric unit?
How many flowers are in the asymmetric unit?
Point Symmetry is ?

12. Finding the Asymmetric Unit in Planar Tilings (1)
Find the repeating tile. y x a b
What are the symmetry elements?
X and Y
2-fold rotation
Flowers in the tile? 4
Point symmetry = 2mm

15. Finding the Asymmetric Unit in Planar Tilings (2)
Find the asymmetric unit.
Flowers in the asymmetric unit? 1
Create the tile.
Consecutive symmetry transformations which can be applied in different sequences to obtain the same result are commutative.

16. Locating Compound Symmetry in Planar Tilings
Find the repeating tile. y x a b
What are the symmetry elements?
Flowers in the tile?
Flowers in
asymmetric unit?

17. Low and High Symmetry Tiles
Point Symmetry is 1
Point Symmetry is m
Point Symmetry is 2mm
Lowest
Symmetry
Highest
Symmetry
More symmetry elements in an object creates ‘higher symmetry’.

18. Achiral and Chiral Halogenated Methanes
fluoro
fluorochloro
Fluorochlorobromo methane
has no mirror planes and is a chiral molecule
achiral
molecules
chiral
molecule

19. , Chiral Objects are Produced by Reflection Reflection is a
Symmetry Operation
A mirror line is a
Symmetry Element -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5
point symmetry
is m -5
point symmetry
is m -5 -4
-x,y -4 -3 -3 -2 -2 -1 y -1 y
1 object
generated 1 1 2 2 3
x,y 3
x,-y
x,y 4 4
2 objects
generated 5 x 5 x
What happens when an object lies on a mirror?

20. Moving Objects By 2-fold Rotation and Reflection
Rotation is a
Symmetry Operation
A rotation point is a
Symmetry Element 5 4 3 2 1 -1 -2 -3 -4 -5 x y -5 -4 -3 -2 -1 1 2 3 4 5 -5
point symmetry
is 2
point symmetry
is 2mm -4 -3
-x,-y
x,-y -2
-x,-y -1 y
x,y 1
x,y
x,-y 2
Rotation does not change chirality. 3 4 5 x
What happens when an object lies on a rotation point?

21. Moving Objects By 4-fold Rotation & Reflection5 4 3 2 1 -1 -2 -3 -4 -5 x y -5 -4 -3 -2 -1 1 2 3 4 5
point symmetry
is 4
point symmetry
is 4mm -5 -4
4 symmetry
related objects
-x,-y
-x,y -3
-x,-y
y,-x
-y,-x
-y,x -2 -1 y
8 symmetry
related objects
y,x 1
-y,x
y,-x 2 3
x,y
x,-y
x,y 4 5 x
anti-clockwise
rotation
Add perpendicular mirror lines.
How many new objects are generated? 4
How many mirror lines are there in total? 4

24. Moving Objects By 3-fold Rotation & Reflectionx 5 4 3 2 1 -1 -2 -3 -4 -5
point symmetry
is 3
identity operation
y-x,-x
-y,x-y
x,y
120o rotation operation
anticlockwise
240o rotation operation

25. Moving Objects By 3-fold Rotation & Reflection
point symmetry is 3m
3m1 -3 -3 -2 -1 1 2 3 -3 -3 -2 -1 1 2 3
-x,y-x
-y,-x
-y,x-y -2 -2
y-x,-x
-y,x-y -1 -1 y y
y-x,y
y-x,-x 1
x-y,-y
y,x 1 2 2
x,x-y
x,y
x,y 3 x 3 x
reflection in 31m
reflection in 3m1
Reflections are parallel to the axes x and y
Reflections are perpendicular to the axes x and y

26. 10 Point Symmetry Groups 3 3m 1 1m 6
6mm 2
2mm 4
4mm

28. Summary of Symmetry Symbols
Translation
To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.
Reflection
To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is "R ".
2-fold
diad
3-fold
triad
4-fold
tetrad
6-fold
hexad
Rotation
To rotate an object means to turn it around. Every rotation has a center and an angle.
Glide
A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry operation that involves more than one step.
Let’s look at more Escher Tessellations

29. Translation in 2 Dimensions (Bravais Lattices)
Square Lattice
a=b; g = 90o
Rectangular Lattice a¹b; g = 90o a b
120o a b
Oblique Lattice
a¹b; g ¹ 90o
Hexagonal Lattice
a=b; g = 120o

30. Creating a Plane Group - Translation + Reflectionx y b
Take a rectangular lattice and impose translational periodicity in the x and y directions a
-x,y y
The translational repeats are ‘a’ along x and ‘b’ along y.
x,y
Place an object at (x,y) and insert a mirror line on the y-axis (x = 0). x
How many positions (objects) are related through symmetry?
Fill the grid with symmetrically equivalent positions.
What additional symmetry operator is created?
What is the extent of the asymmetric unit?
In crystallography a unit cell (tile) contains all the atoms (objects) of a compound and its repetition creates an ‘infinite’ structure.

31. Crystallographic Representation of a Plane Groupx y 1 b
Create a symmetry diagram of a unit cell with superimposed operators. a
mirror
lines y
x,y
-x,y 2 x
Create a general position diagram in quadrants and standard symbols. 3
Create a plane group symbol.
1. Lattice Symbol (p or c)
2. Rotations perpendicular to the a-b plane
3. Operations perpendicular to x
4. Operations perpendicular to y
symmetry
related
objects
(equipoints)
p1m1 pm or

32. Creating a Plane Group - Reflection + Glidex y
Place an object at (x,y) and insert a mirror line on the y-axis and a glide line (translation y= ½) at x = ¼. b
-x+ ½,y+½ a
-x,y y
How many positions (objects) are related through symmetry?
x,y
What additional symmetry operators are created?
x+ ½,y+½ x
What is the extent of the asymmetric unit?
c1m1
mirror
lines
glide
lines or cm
symmetry
related objects

33. Formal Analysis of the Escher Cuttlefish Pattern (2),
Formal Analysis of the Escher Cuttlefish Pattern (2) , 2
Rotate to standard orientation 1
Analyze in accord with plane symmetry principles , x y b a 3
Reveal general position diagram ,

34. Formal Analysis of the Escher Cuttlefish Pattern (3)
Plane Symmetry pm , , , , , ,
Mirror Relationship pm
Glide Relationship cm

36. In short In crystallography ‘tiles’ are called ‘unit cells’.
There are 5 - and only 5 - two dimensional Bravais lattices
Introducing one symmetry operation (e.g. a mirror) may lead to the creation of additional symmetry operations.
The fractional co-ordinates of objects created through the action of symmetry operators can be arrived at by inspection
Even the most complex Escher drawing will conform to the principles of plane symmetry

37. General and Special Positions
Rectangular lattice with 2 mirror lines + 2-fold axes
4 per unit cell
General Position
Created with TESS

38. General and Special Positions
Rectangular lattice with 2 mirror lines + 2-fold axes
4 per unit cell
General Position
2 per unit cell
Special Positions
(horizontal mirror)
Created with TESS

39. General and Special Positions
Rectangular lattice with 2 mirror lines + 2-fold axes
4 per unit cell
General Position
2 per unit cell
Special Positions
(horizontal mirror)
(vertical mirror)
Created with TESS

40. General and Special Positions
Rectangular lattice with 2 mirror lines + 2-fold axes
4 per unit cell
General Position
2 per unit cell
Special Positions
(horizontal mirror)
(vertical mirror)
1 per unit cell
(diad axis)
Created with TESS

42. Formal Representation of Plane Projectionsx y b
p2mm a
1 - start with unit cell
2D translational repeat
2 - add symmetry elements
mirrors overlaid as bold lines , , , ,
3 - add general positions
x,y
x,y ,
chiral positions
(1) x,y (2) -x,-y (3) -x,y (4) x,-y , ,
x,y
x,y , ,

43. Summarising Crystallographic Informationx y ,
mirror
diad
Multiplicity
Wyckoff
letter
Site
symmetry
Coordinates 4 i 1
(1) x,y (2) -x,-y (3) -x,y (4) x,-y 2 h
.m.
½,y ½,-y 2 g
.m.
0,y 0,-y 2 f
..m
x,½ -x,½ 2 e
..m
x,0 -x,0 1 d
2mm
½,½ 1 c
2mm
½,0 1 b
2mm
0,½ 1 a
2mm
0,0

44. International Tables of Crystallography
plane
lattice type
plane
group
point group symmetry
international notation,
full form
convention for choice of origin
arrangement of
symmetry elements
in the unit cell
constraint on choice of general position x,y
location and type of x,y transformation
diffraction
data
general and special positions
using fractional co-ordinates

45. Centering , , , , , , , , , Rectangular pm No. 3 Rectangular cm No. 5
primitive
Rectangular cm No. 5
centered , , , , , , , , ,

46. Multiplicity and Object Co-ordinates-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -6 -5
plane symmetry
is p3 -4 -3 -2
y-x,-x
General Position -1
-y,x-y y
x,y
-y,x-y 1
y-x,-x 2
x,y
Special Positions 3
2/3, 1/3 4
1/3, 2/3 5 6
0,0 x

47. Wyckoff Letters (Again)p3
Multiplicity
Wyckoff
letter
Site
symmetry
Coordinates
Wyckoff letters assigned arbitrarily from bottom to top starting at ‘a’ 3 d
All rotation at right angles to the page 1
(1) x,y (2) -y,x-y (3) -x+y,-x 1 c
3..
2/3, 1/3 1 b
3..
1/3, 2/3 1 a
3..
0,0

48. 17 Plane Groups = 2D Symmetry Groups = Wallpaper Groups
No. Point Group Lattice Full Symbol Short Symbol
Oblique p p1
Oblique p p2
m Rectangular p1m pm
Rectangular p1g pg
Rectangular c1m cm
mm Rectangular p2mm pmm
Rectangular p2mg pmg
Rectangular p2gg pgg
Rectangular c2mm cmm
Square p p4
mm Square p4mm p4m
Square p4gm p4g
Hexagonal p p3
m Hexagonal p3m p3m1
Hexagonal p31m p31m
Hexagonal p p6
mm Hexagonal p6mm p6m