Symmetry

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)

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!

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?

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.

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?

Which one is the asymmetric unit?

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 ?

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

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.

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?

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’.

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

, 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?

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?

Moving Objects By 4-fold Rotation & Reflection 5 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

Moving Objects By 3-fold Rotation & Reflection x 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

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

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

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

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

Creating a Plane Group - Translation + Reflection x 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.

Crystallographic Representation of a Plane Group x 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

Creating a Plane Group - Reflection + Glide x 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

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 ,

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

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

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

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

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

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

Formal Representation of Plane Projections x 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 , ,

Summarising Crystallographic Information x 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

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

Centering , , , , , , , , , Rectangular pm No. 3 Rectangular cm No. 5 primitive Rectangular cm No. 5 centered , , , , , ½ ½ , , , ,

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

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

17 Plane Groups = 2D Symmetry Groups = Wallpaper Groups No. Point Group Lattice Full Symbol Short Symbol 1 1 Oblique p1 p1 2 2 Oblique p211 p2 3 m Rectangular p1m1 pm 4 Rectangular p1g1 pg 5 Rectangular c1m1 cm 6 2mm Rectangular p2mm pmm 7 Rectangular p2mg pmg 8 Rectangular p2gg pgg 9 Rectangular c2mm cmm 10 4 Square p4 p4 11 4mm Square p4mm p4m 12 Square p4gm p4g 13 3 Hexagonal p3 p3 14 3m Hexagonal p3m1 p3m1 15 Hexagonal p31m p31m 16 6 Hexagonal p6 p6 17 6mm Hexagonal p6mm p6m