Lecture 2: Crystallization &

Symmetrycrystal growth theory

symmetry

symmetry operator

symmetry operators

space groups

protein crystals

cellulasesubtilisin

The color you see is “birefringence”, the wavelength-dependent rotation of polarized light.

~1mm

Crystallization robotHigh-throughput crystallography labs use pipeting robots to explore thousnds of “conditions”. Each condition is a formulation of the crystal drop and the reservoir solution.

Conditions can have different:

•protein concentration

•pH

•precipitant, precipitant concentration

•detergents

•organic co-solvents

•metal ions

•ligands

•concentration gradient

Protein crystal growth

blue line = saturation of protein

red line = supersaturation limit

Crystal growth occurs between these two limits. Above the supersaturation limit, proteins form only disordered precipitate.

prot

ein

conc

entr

atio

n

precipitant concentration

Arrows indicate different diffusion experients.

A,B,D,F,G. Vapor diffusion.

E. Bulk

C. Microdialysis

L=liquidS=solidm=metastable state (supersaturated)

vapor diffusion setup

Volatiles (i.e. water) evaporate from one surface and condence on the other.

Drop has higher water concentration than reservoir, so drop slowly evaporates.

a Linbro plate

Other ways to supersaturate slowly

Sitting drops

Microdialysis

Gel filtration

precipitantsA precipitant (r) causes proteins (p) to stick to each other by competing for solvent.

pp

r r r

pp

r r r

r = EtOH, (NH4)2SO4, methylpentanediol, polyethylene glycol, etc

http://www.ccp14.ac.uk/ccp/web-mirrors/llnlrupp/crystal_lab/hampton_screen.htm

50 of the most successful crystallization conditions:

Crystallization theoryNucleation takes higher concentration than crystal growth.

R RR

R

R

R

RR

R

RR

R

Rslow slow fast

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR

R

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR

fastnot so slow

After nucleation, the large size of a face makes the weak bond more likely.

Periodic Bond Chain theory

Bonds A,B are stronger than P,Q. Dimensions of crystal at equilibrium are proportional.

More on Periodic Bond Chain theory: http://www.che.utoledo.edu/nadarajah/webpages/PBC.htm

Periodic Bond Chain theory

Growth is unfavorable directions increases as the crystal grows.

Weak bonds in Z favor growth in XY, forming “plate” xtal.

Growing cross-section in XY favors growth in Z.

Ratio of cross sections is inverse to ratio of bond strength.

diffusion depletionCrystal growth depletes the surrounding solution of protein, while concentrating impurities. Local depletion...

...prevents nucleations close to a growing crystal

...slows and eventually stops crystal growth

...concentrates impurities on the surface of the crystal

...causes convection currents.

Cobalt impurities in SiO2

(amethyst) are concentrated in the part of the crystal that

formed last (the tip).

Better crystals in micro gravity?

More at: http://science.nasa.gov/ssl/msad/pcg/

Higher concentration of protein = higher density

Differences in densty cause convection currents, which might cause crystal defects. Microgravity eliminates convection currents.

mounting crystalsThin-walled glass capillaries (<1mm in diameter) are filled with “mother liquor”(the fluid in which the crystal was grown) and a crystal is carefully dropped in. The mother liquor is removed using filter paper cut to fine strips. The crystal sticks to the glass, immobilized.

The xtal remains in vapor diffusion contact with the mother liquor. If not it will dryout and crack.

Protein crystals are extremely fragile!!! They may break upon sudden contact with a solid object. Tiny pipets are used to pull crystals from drops.

Crystal mounting

Xtal is mounted in a thin-walled glass capillary tube

Xtal is mounted on a thin film of water in a wire loop. The loop is fixed to a metal or glass rod.

If freezing (preferred)

eucentric goniometer head(made by Nonius)

If not freezing

Must freeze immediately or film will dry out.!

Mounted xtal is attached to a goniometer head for precise adjustment.

wax

Crystal must be kept at proper humidity and temperature!! Very fragile!

Low-melting hard wax is used to ‘glue’ the rod or capillary here.

Small wrenches fit here, here, here and here.

Why freeze?

Essentially eliminates X-ray damage to crystal. Crystals do not decay during data collection.

Why not?

Cryo equipment is expensive.

Ice crystals may form if freezing is not done properly, ruining data.

wire-loop crystal catcher

Crystals must be flash frozen

Water must be frozen to < –70°C very fast to prevent the formation of hexagonal ice. Water glass forms.

How? Crystals, mounted on loops, are flash frozen by dipping in liquid propane or freon at –70°, or by instant exposure to N2 gas at –70°C.

hexagonal ice

...to prevent glass->ice transition

Centering the crystal in the beam

“machine center” is the intersection of the beam and the two goniostat rotation axes. Must be set by manufacturer!

xrays

To place crystal at machine center, rotate and and watch the crystal. If it moves from side to side, it is off center.

If it is off-center, we adjust the screws on the goniometer head.

whoops it’s off center. Fix it!

Aligning crystal lattice with the beam.

Rotate the crystal until the zero-layer disappears and the 1-layer is centered on the beam.

Thi

s is

whe

re th

e a*

axi

s is

poi

nted

h=1h=0

h=-1 h=1

misaligned aligned

beam is here

Concentric circles around beammeans axis is aligned with beam.

Precession photograph

Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry.

Spacing of spotsis used to get unit celldimensions.

symmetry

An object or function is symmetrical if a spatial transformation of it looks identical to the original.

X

X

This is the original

This is rotated by 180°

Symmetry operatorsA spatial transformation can be expressed as an operator that changes the coordinates of every point in the object the same way. Symmetry operators do not distort the object. In other words, the distance between any two points is the same before and after being moved by the symmetry operation.

−1 0 0

0 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−x

−y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Here is the operator for a 180° rotation around Z.

equivalent positions

3x3 Matrix multiplicationby the way

a b c

d e f

g h i

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

ax+by+cz

dx+ey+fz

gx+hy+iz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

REMINDER

Types of symmetry operations

•Point of inversion

•mirror plane

•glide plane

•rotation (2,3,4 or 6-fold)

•screw axis

•lattice symmetry

Fractional coordinatesThe crystallographic coordinate system is defined by the unit cell. The location of a point is defined by fraction of traveled (from 0 to 1) along each unit cell axis.

Fractional coordinates are always measured parallel to each axis. The axes are not necessarilly 90° apart!

(0.33,0.25,0.55)

a

c

b

point of inversion

−1 0 0

0 −1 0

0 0 −1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−x

−y

−z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

centric symmetry

mirror plane

1 0 0

0 1 0

0 0 −1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

x

y

−z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

centric symmetry

glide plane

1 0 0

0 1 0

0 0 −1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

+

1/ 2

0

0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

x +1/2

y

−z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

centric symmetry

rotation

cosθ −sinθ 0

sinθ cosθ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

x'

y'

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

non-centric symmetry

screw-rotation

cosθ −sinθ 0

sinθ cosθ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

+

0

0

1/3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

x'

y'

z+1/3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

1/3 of a unit cell

non-centric symmetry

120°

Why proteins cannot have centric symmetry

Mirror images and points of inversion cannot be re-created by pure rotations.

Centric operations would change the chirality of chiral centers such as the alpha-carbon of amino acids or the ribosal carbons of RNA or DNA.

N

C

HR

C

O

Rotational symmetry

−1 0 0

0 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−x

−y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

A 2-fold (180°) rotation around the Z-axis

rotation

Rotation matrices... the mathematical description of a rotation.

In polar coordinates, a rotation is the addition of angles.

xr

(x,y)

(x’,y’)

y

axis of rotation

atom starts here...

...goes here..rotates by

REMINDER: sum of angles rules

cos (= cos cos sin sin

sin (= sin cos sin cos

Adding angles in Cartesian space

x

y

r

(x,y)

(x’,y’)

x' = |r| cos (

= |r|(cos cos sin sin

= (|r| cos cos |r| sin sin

= xcos y sin

y' = |r| sin (

= |r|(sin cos sin cos

= (|r| sin cos |r| cos sin

= ycos x sin

x = |r|cos

y = |r|sin

x '

y '

⎛

⎝

⎜⎞

⎠

=

cos − sin

sin cos

⎛

⎝

⎜⎞

⎠

x

y

⎛

⎝

⎜⎞

⎠

in matrix notation...

converting internal motion to Cartesian motion

rotation matrix

2D rotation using matrix notation

x '

y '

⎛

⎝

⎜⎞

⎠

=

cos − sin

sin cos

⎛

⎝

⎜⎞

⎠

x

y

⎛

⎝

⎜⎞

⎠

x' = xcos y sin

= (|r| cos cos |r| sin sin

= |r| cos (

y' = ycos x sin

= (|r| sin cos |r| cos sin

= |r| sin (

“row times column”

Transposing the matrix reverses the rotation

x '

y '

⎛

⎝

⎜⎞

⎠

=

cos sin

− sin cos

⎛

⎝

⎜⎞

⎠

x

y

⎛

⎝

⎜⎞

⎠

To rotate the opposite direction, flip the matrix about the diagonal.

inverse rotation matrix = transposed rotation matrix.

cos sin

− sin cos

⎛

⎝

⎜⎞

⎠

cos − sin

sin cos

⎛

⎝

⎜⎞

⎠

=

1 0

0 1

⎛

⎝

⎜⎞

⎠

A B

C D

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

T

=A C

B D

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ the “transpose”

...because cos cos + sin sin = 1

A 3D rotation matrix

Is the product of 2D rotation matrices.

cos −sin 0

sin cos 0

0 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

cos γ 0 −sin γ

0 1 0

sin γ 0 cos γ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

cos cos γ −sin cos

sin cosγ cos −sin sin γ

sin γ 0 cosγ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Rotation around z-axis

Rotation around y-axis

3D rotation

Example:Rotate v=(1.,2.,3.) around Z by 60°, then rotate around Y by -60°

cos60° −sin60° 0

sin60° cos60° 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

1

2

3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

1(0.5)−2(0.866)+3(0)

1(0.866) +2(0.5)+3(0)

0+0+3(1)

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−1.232

1.866

3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cos60° 0 −sin60°

0 1 0

sin60° 0 cos60°

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

−1.232

1.866

3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−1.232(0.5)+1.866(0)−3(0.866)

−1.232(0)+1.866(1)+3(0)

−1.232(0.866)+1.866(0)+3(0.5)

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−3.214

1.866

0.433

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Examples:

x

y0 −1 0

1 0 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

z

z

x

y

0 0 1

0 1 0

−1 0 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

y

z

x

1 0 0

0 0 −1

0 1 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

90° rotation around

X

Y

Z

For a R-handed rotation, the minus sine is the one on the “Right.”Helpful hint:

Euler angles, γcosγ −sinγ 0

sinγ cosγ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

1 0 0

0 cosβ −sinβ

0 sinβ cosβ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cosα −sinα 0

sinα cosα 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Order of rotations: 123

axis of rotation:

z’’ x’ z

cosφ −sinφ 0

sinφ cosφ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cosϕ 0 −sinϕ

0 1 0

sinϕ 0 cosϕ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cosκ −sinκ 0

sinκ cosκ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cosϕ 0 sinϕ

0 1 0

−sinϕ 0 cosϕ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

cosφ sinφ 0

−sinφ cosφ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

123

z’’ -z-y’y’’’z’’’’

45

Polar angles,

3D angle conventions:

Net rotation =

Properties of rotation matrices

More at http://mathworld.wolfram.com/RotationMatrix.html

•Square, 2x2 or 3x3

•The product of any two rotation matrices is a rotation matrix

•The inverse equals the transpose, R-1 = RT

•orthogonality

•The dot-product of any row or column with itself is one.

•The dot-product of any row or column with a different row or column is zero.

•|x| equals |Rx|, for any rotation R.

2-fold rotation

RR

−1 0 0

0 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−x

−y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

180° rotation. Called a 2-fold because doing it twice brings you back to where you started.

2-fold symbol

Equivalent positions in fractional coordinates:

x,y,z -x,-y,z

P2

3-fold rotation

R

cos120° −sin120° 0

−sin120° cos120° 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

xcos120°−ysin120°

xsin120°+ycos120°

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

3-fold symbol

R

R

In fractional coordinates:0 −1 0

1 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−y

x−y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

P3

Equivalent positions :

x,y,z -y,x-y,z -x+y,-x,z

4-fold rotation

R

cos90° −sin90° 0

−sin90° cos90° 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

xcos90°−ysin90°

xsin90°+ycos90°

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

4-fold symbol

R

In fractional coordinates (same as orthogonal coords):0 −1 0

1 0 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−y

x

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

R

R

Equivalent positions:

x,y,z -x, -y,z-y, x,z y,-x,z

P4

6-fold rotation

R

cos60° −sin60° 0

−sin60° cos60° 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

xcos60° −ysin60°

xsin60°+ycos60°

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

6-fold symbol

R

In fractional coordinates:−1 1 0

−1 0 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

−x +y

−x

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

RRRR

Equivalent positions:

x,y,z -y,x-y,z -x+y,-x,z-x,-y,z y,-x+y,z x-y,x,z

P6

In class exercise: rotating a point

Choose a point r=(0.1,0.2,0.3) [orthogonal coordinates]

Rotate the point by 30° in x.

Then rotate it by -90° in y.

What are the new coordinates?

(a)

(b)Choose a point r=(0.1,0.2,0.3) [fractional coordinates]

Multiply by the symmetry operator:

What are the new fractional coordinates?

0 −1 0

1 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

No 5-fold symmetry in crystals??

A crystal lattice must be space-filling and periodic.

This “Penrose tile pattern” is spacefilling but not periodic.

Look for translational symmetry in this image. Is there any?

Quasicrystals: 5-fold point group symmetry, but no space group symmetry

The poliovirus crystal structurehas 5-fold, 3-fold, and 2-fold point-group symmetry.

Screw symmetry

Equivalent positions are related by rotation AND translation

21 31 61

32

62

41 6463

42 4365

Example: 6-fold in the projection. Screw moves up and to the right 4/6 units.

Screw axes

A right-handed 3-fold screw

cos120° −sin120° 0

−sin120° cos120° 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

+

0

0

1/3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

xcos120°−ysin120°

xsin120°+ycos120°

z+1/3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

A screw rotation is a rotation of 2/n plus a translation along the axis of rotation by 1/n (right-handed screw) or -1/n (left-handed screw).

P31

Symbol for 3-fold screw

translational symmetryThe crystal lattice is an example of translational symmetry. Equivalent positions are

(x,y,z) and (x+1,y+1,z+1), in fractional coordinates.

Space groups that have no other translational symmetry operations are called “primitive”. Space group letter “P”

Space groups have letters indicating the type of translational symmetry:

C (centered)

F (face-centered)

I (body-centered)

Centered latticesCentered: “C”

Translational symmetry operator (1/2,1/2,0)

This is “face-centered” but only on one face.

Centered latticesFace-centered: “F”

Translational symmetry operators:(1/2,1/2,0),(0,1/2,1/2),(1/2,0,1/2)

Centered latticesBody-centered: “I”

Translational symmetry operator:(1/2,1/2,1/2)

Space groups

A symmetry “group” is a set of symmetry operators that is closed, meaning any two operations, applied in succession, create a third operation that is part of the group. A “space group” is a symmetry group that includes lattice symmetry operators.

All space groups implicitly include lattice operators: (±1,±1,±1)

Equivalent positions: (x, y, z), (-x,-y,z+1/2)

The International Tables for Crystallography

I4122

Space groupsCell type P,C,I or F

I=body centered

Secondary axes of symmetry are proper 2-folds

Principle axis of symmetry is 4-fold screw

Group theory

Equivs for P21: (x, y, z), (-x,-y,z+1/2)

A space group is a closed set of operators.

If you apply any two operators in succession. the result is another one of the operators in the group.

−1 0 0

0 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

+

0

012

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

=

−x

−y

z+12

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

−1 0 0

0 −1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

−x

−y

z+12

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

+

0

012

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

=

x

y

z+1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Cubic space group P213 x,y,z

-x+1/2,-y,z+1/2

-x,y+1/2,z+1/2

x+1/2,-y+1/2,-z

z,x,y

z+1/2,-x+1/2,-y

-z+1/2,-x,y+1/2

-z,x+1/2,-y+1/2

y,z,x

-y,z+1/2,-x+1/2

y+1/2,-z+1/2,-x

-y+1/2,-z,x+1/2

marie.epfl.ch/escher/

In class exercise: Use the Escher web sketch applet to find the equivalent positions for cm, p4mm, and p6.

In class exercise: Find equivalent positions

Draw a dot at fractional coordinates (0.1, 0.2, 0.3)What are the fractional coordinates of the equivalent positions?

Write the 2D symmetry operators (matrix and vector)

Point group symbols, etc.

Finding symmetry in an image

Plane groups

space group p1