Point group symmetries Why is it Important? reveal the

E. O

rlova

E

Orloo

vovvvavvaaa

Lect

ure 9

Imag

e pr

oces

sing

fo

r cry

o m

icro

scop

y Se

p 5

- Sep

t 15,

201

7 Bi

rkbe

ck C

olle

ge

Lond

on

Pra

ctic

al C

ours

e

Poin

t gro

up s

ymm

etrie

s

Sym

met

ry in

pla

ne

Sym

met

ry in

2D

spa

ce

Sym

met

ry in

3D

spa

ce

Po

int g

roup

sym

met

ries

Sym

met

ry in

spa

ce –

sym

met

ry in

pro

ject

ions

WH

AT IS

SYM

MET

RY

The

term

Sym

met

ry h

as G

reek

orig

ins.

For

the

Gre

eks

it m

eant

the

harm

ony

of p

arts

, pr

opor

tions

an

d rh

ythm

. An

othe

r mor

e m

athe

mat

ical

way

, tha

t is

rela

ted

to

our i

mag

e an

alys

is a

nd c

ryst

allo

grap

hy is

: An

obj

ect c

an b

e di

vide

d by

a p

oint

, or l

ine,

or

radi

atio

n lin

es o

r pla

nes

into

to o

r mor

e pa

rts

EXA

CTL

Y SI

MIL

AR

in s

ize

and

shap

e, a

nd in

po

sitio

n re

lativ

e to

the

divi

ding

ele

men

t. R

epet

ition

of

exa

ctly

sim

ilar p

arts

faci

ng e

ach

othe

r or a

cen

tre.

Why

is it

Impo

rtan

t?

We

can

clas

sify

mol

ecul

es (b

ioco

mpl

exes

) acc

ordi

ng to

the

leve

l of t

heir

sym

met

ry e

lem

ents

, so

they

can

be

grou

ped

toge

ther

hav

ing

the

sam

e se

t of s

ymm

etry

ele

men

ts.

This

cla

ssifi

catio

n is

ver

y im

porta

nt, b

ecau

se it

allo

ws

to

mak

e so

me

gene

ral c

oncl

usio

ns a

bout

mol

ecul

ar p

rope

rties

w

ithou

t ext

ra c

alcu

latio

n.

On

the

atom

ic le

vel i

t hel

ps to

reve

al th

e m

olec

ular

pro

perti

es

with

out a

ny c

alcu

latio

ns. W

e w

ill be

abl

e to

dec

ide

if a

mol

ecul

e ha

s a

dipo

le m

omen

t or

not

, an

d to

kno

w h

ow

thes

e p

rope

rty a

re re

flect

ed o

n th

eir s

urfa

ces.

Wha

t sor

t of

inte

ract

ion

hold

bio

mol

ecul

es to

geth

er in

hug

e bi

ocom

lexe

s su

ch a

s vi

ruse

s, s

ecre

tion

syst

ems,

etc

.

Sym

met

ry

arou

nd u

s

We

say

that

suc

h an

obj

ect i

s sy

mm

etric

with

re

spec

t to

a gi

ven

oper

atio

n if

this

ope

ratio

n,

whe

n ap

plie

d to

the

obje

ct, d

oes

not a

ppea

r to

cha

nge

it.

Sym

met

ry m

ay d

epen

d on

the

prop

ertie

s un

der c

onsi

dera

tion:

fo

r an

imag

e w

e m

ay c

onsi

der j

ust t

he

shap

es, o

r als

o th

e co

lors

; fo

r an

obje

ct (3

D),

we

may

add

ition

ally

co

nsid

er d

ensi

ty, c

hem

ical

com

posi

tion,

co

ntac

ts b

etw

een

dom

ains

, etc

.

Bio

logi

cal o

bjec

ts

The

pers

on -

here

afte

r ref

erre

d to

as

‘it’ h

as a

mirr

or p

lane

, pr

ovid

ed it

sta

nds

stra

ight

(and

th

at w

e ig

nore

its

inte

rnal

or

gans

). Ea

ch h

alf o

f the

figu

re is

an

asy

mm

etric

uni

t.

Mov

ing

an a

rm o

r leg

des

troys

th

e sy

mm

etry

and

the

who

le

figur

e ca

n th

en b

e tre

ated

as

an

asym

met

ric u

nit.

Th

is li

ttle

‘obj

ect’,

bot

h w

ith a

nd

with

out i

ts m

irror

pla

ne w

ill be

us

ed to

illu

stra

te fu

rther

sy

mm

etry

ele

men

ts, a

nd to

bu

ild m

ore

com

plic

ated

gro

ups.

1D

2D

3D

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

One

dim

ensi

onal

func

tion

-> a

cur

ve

1 5

2 3

4 6

10

7 8

9 11

12

14

13

15

1 5

2 3

4 6

10

7 8

9 11

12

14

13

15

Int

Int

X X X

Ther

e ar

e tw

o ty

pes

oper

atio

ns w

ith fu

nctio

ns in

on

e-di

men

sion

al s

pace

.

1. T

rans

latio

n

1D

2D

3D

)a

x(g)x(g

a g(

x)

g(x+

a)

a g(

x+2a

)

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

2.

Ref

lect

ion

(mirr

or)

Thes

e o

pera

tions

can

be

com

bine

d

)xb(g

)x(g

b

a

)xa

b(g)

ax(g

)a

x(g

)x(g

11

)xb(g

)x(g)x(

g1

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Ther

e ar

e th

ree

type

s op

erat

ions

with

im

ages

in p

lane

: 1.

Tran

slat

ion

in tw

o di

rect

ions

2.

Ref

lect

ion

3.

Rot

atio

n

in p

lane

1D

2

D

3D

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Tran

slat

ion

in O

NE

dire

ctio

n

)a

r(g)

r(g

a

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Tran

slat

ion

in tw

o di

rect

ions

a b

)bl

akr(g

)r(g

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Ref

lect

ion

= m

irror

)

ra(g

)r(g

Ther

e is

a p

lain

bet

wee

n ob

ject

s. S

o th

e di

stan

ce fr

om a

ny p

oint

of

the

obje

ct a

nd th

e pl

ain

is th

e sa

me

as th

e di

stan

ce b

etw

een

the

sam

e po

int o

f the

mirr

ored

obj

ect a

nd th

e pl

ain

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

T T

M

irror

-imag

e sy

mm

etry

Th

is is

the

mos

t fam

iliar a

nd

conv

entio

nally

taug

ht ty

pe o

f sy

mm

etry

. It a

pplie

s fo

r ins

tanc

e

for t

he le

tter T

: whe

n th

is le

tter i

s re

flect

ed a

long

a v

ertic

al a

xis,

it

appe

ars

the

sam

e. T

has

a

verti

cal s

ymm

etry

axi

s.

A re

flect

ion

"flip

s" a

n ob

ject

ove

r a

line

(in 2

D) o

r pla

ne (i

n 3D

), in

verti

ng

it to

its

mirr

or im

age,

as

if in

a m

irror

. If

the

resu

lt is

the

sam

e th

en w

e ha

ve m

irror

-imag

e sy

mm

etry

(als

o kn

own

in th

e te

rmin

olog

y of

mod

ern

phys

ics

as P

-sym

met

ry).

T

T

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Ref

lect

ion

(mirr

or o

pera

tion)

Tr

ansl

atio

n

Shift

and

refle

ctio

n

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

73o

Rot

atio

n in

pla

ne

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

Inve

rsio

n

g(r)

=g(-r

)

Ther

e is

a p

oint

bet

wee

n ob

ject

s. S

o th

e di

stan

ces

from

any

po

int o

f the

obj

ect a

nd th

e po

int a

re th

e sa

me

as th

e di

stan

ces

betw

een

the

sim

ilar p

oint

s of

the

inve

rsed

obj

ect a

nd th

e po

int

Ope

ratio

ns w

ith M

ultim

eric

Fun

ctio

ns

http

s://r

eadi

ngfe

ynm

an.o

rg/ta

g/p-

sym

met

ry/

Rot

atio

nal s

ymm

etry

Cn

A ro

tatio

n ro

tate

s an

obj

ect a

bout

a p

oint

(in

3D: a

bout

an

axis

). R

otat

iona

l sym

met

ry o

f ord

er n

, als

o ca

lled

n-fo

ld

rota

tiona

l sym

met

ry, w

ith re

spec

t to

a pa

rticu

lar p

oint

or a

xis

mea

ns th

at ro

tatio

n by

an

angl

e of

360

/n d

oes

not c

hang

e th

e ob

ject

.

(180

, 120

, 90

, 72

, 60

, 51.

43,4

5, 4

0 …

..

2

,

3,

4

, 5

, 6

,

7,

8

, 9

et

c )

For e

ach

poin

t or a

xis

of s

ymm

etry

the

sym

met

ry g

roup

is

isom

etric

with

the

cycl

ic g

roup

Cn o

f ord

er n

. The

fu

ndam

enta

l dom

ain

is a

sec

tor o

f 360

/n.

C

2,

C

3, C

4, C

5, C

6,

C

7,

C8,

C9

…

One

poi

nt re

mai

ns u

nmov

ed, w

hich

is th

e ro

tatio

n po

int

Rot

atio

nal s

ymm

etry

2-fo

ld

3-fo

ld

5-fo

ld

6-fo

ld

C2

C3

C5

C6

Sym

met

ry c

ombi

natio

ns

Mor

e co

mpl

ex s

ymm

etrie

s ar

e co

mbi

natio

ns o

f re

flect

iona

l, ro

tatio

nal,

trans

latio

nal,

and

glid

e re

flect

ion

sym

met

ry.

Mirr

or-im

age

sym

met

ry in

com

bina

tion

with

n-fo

ld

rota

tiona

l sym

met

ry, w

ith th

e po

int o

f sym

met

ry o

n th

e lin

e of

sym

met

ry, i

mpl

ies

mirr

or-im

age

sym

met

ry w

ith re

spec

t to

line

s of

refle

ctio

n ro

tate

d by

mul

tiple

s of

180

/n, i

.e. n

re

flect

ion

lines

whi

ch a

re ra

dial

ly s

pace

d ev

enly

(for

odd

n

this

alre

ady

follo

ws

from

app

lyin

g th

e ro

tatio

nal s

ymm

etry

to

a s

ingl

e re

flect

ion

axis

, but

it a

lso

hold

s fo

r eve

n n)

.

Com

bina

tion

of s

ymm

etrie

s in

pla

ne

P3

Com

bina

tion

of s

ymm

etrie

s in

pla

ne

P3

(from

R. H

ende

rson

)

The

five

basi

c la

ttice

type

s Th

ere

are

17 s

pace

gro

ups

in th

e pl

ane,

but

thei

r uni

t ce

lls fa

ll in

to o

ne o

f fiv

e ba

sic

shap

es a

s fo

llow

s:

repe

atin

g pa

ttern

s in

the

plan

e ca

n on

ly h

ave

1, 2

, 3, 4

or

6-fo

ld s

ymm

etry

. In

parti

cula

r, re

peat

ing

patte

rns

in th

e pl

ane

cann

ot h

ave

five-

fold

sym

met

ry.

W

hat i

s ab

out f

ive-

fold

sym

met

ry?

The

elem

ent o

f the

uni

t cel

l may

hav

e 5-

fold

sym

met

ry

but t

he o

vera

ll sy

mm

etry

of t

he p

atte

rn d

oes

not.

In

rece

nt y

ears

som

e m

ater

ials

hav

e tu

rned

up

with

fiv

e-fo

ld s

ymm

etry

. The

se m

ater

ials

, ter

med

qu

asic

ryst

als,

do

not h

ave

repe

atin

g pa

ttern

s.

http

://w

ww.

spsu

.edu

/mat

h/til

e/sy

mm

/iden

t17.

htm

Com

bina

tion

of s

ymm

etrie

s in

pla

ne

Prot

eins

don

’t do

this

– p

ack

by tr

ansl

atio

ns

Sym

met

ry in

3D

spa

ce

Thre

e ty

pes

oper

atio

ns w

ith o

bjec

ts in

spa

ce

Tran

slat

ion

in th

ree

dire

ctio

ns

Ref

lect

ion

(mirr

or)

in s

pace

R

otat

ion

(In

vers

ion

)

Rot

atio

ns c

an b

e co

mbi

ned

with

tran

slat

ions

an

d re

flect

ions

, how

ever

they

can

not b

e co

mbi

ned

in a

n ar

bitra

ry w

ay.

Sym

met

ry o

pera

tors

impo

se c

onst

rain

s.

Com

bina

tion

of o

pera

tors

can

gen

erat

e in

finite

latti

ces

as w

e ca

n se

e th

at in

cry

stal

s.

Basi

c sy

mm

etry

ope

ratio

ns in

spa

ce Tran

slat

ion

in

thre

e di

rect

ions

Unit

cell

a

B A

C

Cry

stal

l

Sym

met

ry in

3D

spa

ce

One

poi

nt re

mai

ns u

ncha

nged

. Th

ere

are

no tr

ansl

atio

nal o

pera

tors

C

ombi

natio

n of

rota

tion,

mirr

or a

nd in

vers

ion

give

s 32

com

bina

tions

Bu

t for

the

prot

eins

we

will

have

onl

y 11

co

mbi

natio

ns: n

o in

vers

ion

or m

irror

A sp

ace

rela

tions

hip

bet

wee

n el

emen

ts in

eac

h ol

igom

eric

mol

ecul

e ca

n be

des

crib

ed b

y a

set

sym

met

ry o

pera

tions

tha

t de

scrib

es t

he o

vera

ll m

olec

ular

sy

mm

etry

. Th

is

com

bina

tion

of

oper

atio

ns d

efin

e th

e PO

INT

GR

OU

P of

the

m

olec

ule.

Basi

c sy

mm

etry

ope

ratio

ns in

spa

ce

Mirr

or p

lane

Mirr

or p

lane

, sho

wn

as

dash

ed li

ne

Com

bina

tion

of

two-

fold

axi

s with

m

irror

pla

nes

Mirr

or p

lane

Ref

lect

ion

(mirr

or)

in

spa

ce

Rota

tions

axi

s

Basi

c sy

mm

etry

ope

ratio

ns in

spa

ce

Rot

atio

n

in s

pace

Mirr

or p

lane

Mirr

or p

lane

Basi

c sy

mm

etry

ope

ratio

ns in

spa

ce

The

grou

p do

es n

ot fo

rm a

mirr

or p

lane

Ref

lect

ion

(mirr

or)

in s

pace

R

otat

ion

in

spa

ce

C2

Cm

In th

ree

dim

ensi

ons

we

can

dist

ingu

ish

cylin

dric

al

sym

met

ry a

nd s

pher

ical

sym

met

ry (n

o ch

ange

whe

n ro

tatin

g ab

out o

ne a

xis,

or f

or a

ny ro

tatio

n).

Ther

e is

no

depe

nden

ce o

n th

e an

gle

usin

g cy

lindr

ical

coo

rdin

ates

and

no

depe

nden

ce o

n ei

ther

an

gle

usin

g sp

heric

al c

oord

inat

es.

Cyl

indr

ical

sym

met

ry is

ver

y of

ten

calle

d as

cyc

lical

sy

mm

etry

. Th

e si

mpl

est s

ymm

etry

is C1,

if th

e ob

ject

doe

s no

t ha

ve a

ny s

ymm

etry

at a

ll.

Sym

met

ry in

3D

spa

ce

Bas

ic s

ymm

etry

ope

ratio

ns in

spa

ce

Rota

tion

axi

s

Mirr

or

plan

e

Inve

rsio

n ce

ntre

– ro

tatio

n ce

ntre

C2m

http

://cs

i.che

mie

.tu d

arm

stad

t.de/

ak/im

mel

/tuto

rials

/sym

met

ry/in

dex4

.htm

l

Basi

c sy

mm

etry

ope

ratio

ns in

spa

ce

Cn

(for c

yclic

) ind

icat

es th

at th

e gr

oup

has

an n

-fold

ro

tatio

n ax

is..

Sn (f

or S

pieg

el, G

erm

an fo

r mirr

or) d

enot

es a

gro

up

that

con

tain

s on

ly a

n n-

fold

rota

tion-

refle

ctio

n ax

is.

D (

for d

ihed

ral,

or tw

o-si

ded)

indi

cate

s th

at th

e gr

oup

has

an n

-fold

rota

tion

axis

plu

s a

two-

fold

axi

s pe

rpen

dicu

lar t

o th

at a

xis.

In c

ryst

allo

grap

hy th

is a

pplie

s on

ly fo

r n =

1, 2

, 3, 4

, 6,

due

to th

e cr

ysta

llogr

aphi

c re

stric

tion

theo

rem

.

sym

met

ry C

2

Rot

atio

nal s

ymm

etry

is C

n, if

the

obje

ct h

as s

ever

al e

lem

ents

, tha

t arra

nged

in

a c

ircul

ar s

yste

m. T

he n

umbe

r of e

lem

ents

det

erm

ines

the

orde

r of

sym

met

ry.

Proj

ectio

ns

Sym

met

ry in

3D

spa

ce

sym

met

ry C

3

C 2 C 3

Proj

ectio

ns o

f the

obj

ect

rota

tiona

l sy

mm

etry

and

thei

r sym

met

ry.

Sym

met

ry in

3D

spa

ce

a-La

troto

xin

52

0kD

a C

a re

leas

e ch

anne

l 2.

4 M

da

C14

C13

C

4

Porta

l pro

tein

SPP

1

C5

Rot

atio

nal s

ymm

etry

Sy

mm

etry

in 3

D s

pace

sy

mm

etry

D2

Sym

met

ry in

3D

spa

ce

Dih

edra

l poi

nt g

roup

sym

met

ry D

n ar

e a

com

bina

tion

of c

yclic

al

sym

met

ries

with

a tw

o-fo

ld a

xis,

whi

ch is

per

pend

icul

ar to

the

axis

of

rota

tion.

22

2 D

2

32

D3

52

D5

Sym

met

ry in

3D

spa

ce

D3

Pal

iniru

s el

epha

s he

moc

yani

n (7

5kD

a x

6)

D5

Ke

yhol

e Li

mpe

t Hem

ocye

anin

Plat

o an

d Ar

istot

le

Plat

onic

and

Ar

chim

edea

n Po

lyhe

dra

Th

e Pl

aton

ic S

olid

s, d

isco

vere

d by

the

Pyth

agor

eans

but

de

scrib

ed b

y Pl

ato

(in th

e Ti

mae

us) a

nd u

sed

by h

im fo

r his

th

eory

of t

he 4

ele

men

ts, c

onsi

st o

f sur

face

s of

a s

ingl

e ki

nd

of re

gula

r pol

ygon

, with

iden

tical

ver

tices

. Th

e Ar

chim

edea

n So

lids,

con

sist

of s

urfa

ces

of m

ore

than

a

sing

le k

ind

of re

gula

r pol

ygon

, with

iden

tical

ver

tices

and

id

entic

al a

rrang

emen

ts o

f pol

ygon

s ar

ound

eac

h po

lygo

n.

Cub

ic p

oint

grou

p sy

mm

etrie

s T

- 2

3 T

etra

hedr

al s

ymm

etry

requ

eire

s a

min

imum

of 1

2 id

entic

al s

ubun

its

O -

432

Oct

ahed

ral p

oint

gro

uop

sym

met

ry,

ne

eds

24 s

ubun

its

I -

532

Ico

sahe

dral

sym

met

ry, 6

0 su

buni

ts

Tetra

hedr

on

3

2 3

3

2

3

Com

bina

tion

of s

ymm

etrie

s in

3D

Com

bina

tion

of s

ymm

etrie

s in

3D

EXPA

ND

ED

CO

MPA

CT

3

2

3

2

Hea

t sho

ck p

rote

in

Hsp

26

Cub

e 2 4 3

4 2

3

Oct

ahed

ron

Com

bina

tion

of s

ymm

etrie

s in

3D

Com

bina

tion

of s

ymm

etrie

s in

3D

Dod

ecah

edro

n

2 5 3

3

2 5

Icos

ahed

ron

Com

bina

tion

of s

ymm

etrie

s in

3D

http

://w

ww.

staf

f.ncl

.ac.

uk/j.

p.go

ss/s

ymm

etry

/

Icos

ahed

rael

sym

met

ry

Dod

ecah

edro

n Ic

osah

edro

n

3

5

2

3

RC

NM

V- R

ed

clov

er n

ecro

tic

mos

aic

viru

s

Her

pes

viru

s

Te

trahe

dron

Oc

tahe

dron

Cu

be

Dode

cahe

dron

Ico

sahe

dron

Grap

hics

Fa

ces

4 tria

ngles

8 t

riang

les

6 squ

ares

12

pen

tago

ns

20 tr

iangl

es

Verti

ces

4 6

8 20

12

Ed

ges

6 12

12

30

30

Po

int G

roup

Td

Oh

Oh

Ih Ih

For e

ach

of th

e po

int g

roup

s T d

, Oh,

and

I h th

ere

exis

ts s

ub-g

roup

s T,

O, a

nd I

whi

ch c

onta

in a

ll C

n sym

met

ry e

lem

ents

, but

non

e of

the

Sn o

pera

tions

(inc

ludi

ng

inve

rsio

n an

d re

flect

ion)

. Add

ing

a σ h

mirr

or p

lane

or a

n in

vers

ion

cent

er to

the

T gr

oup

yiel

ds T

h.

The

high

-sym

met

ry p

oint

gro

ups

in w

hich

mor

e th

an o

ne C

n axi

s w

ith n

≥ 3

is

pres

ent a

re b

est v

isua

lized

by

the

five

regu

lar p

olyh

edra

(Pla

toni

c so

lids)

as

show

n be

low.

In

thes

e ob

ject

s, a

ll fa

ces,

ver

tices

, and

edg

es a

re s

ymm

etry

re

late

d an

d th

us e

quiv

alen

t. T

he o

ctah

edro

n an

d th

e cu

be a

re c

lose

rela

ted

to

each

oth

er a

s th

ey c

onta

in th

e sa

me

sym

met

ry e

lem

ents

, but

in d

iffer

ent

orie

ntat

ions

. Th

e sa

me

appl

ies

to th

e do

deca

hedr

on a

nd ic

osah

edro

n.

1.Bu

rns,

G.;

Gla

zer,

A. M

. (19

90).

Spa

ce G

roup

s fo

r S

cien

tists

and

Eng

inee

rs (2

nd e

d.).

Bost

on: A

cade

mic

Pr

ess,

Inc.

ISBN

0-1

2-14

5761

-3.

2.C

legg

, W (1

998)

. Cry

stal

Stru

ctur

e D

eter

min

atio

n (O

xfor

d C

hem

istry

Prim

er).

Oxf

ord:

Oxf

ord

Uni

vers

ity

Pres

s. IS

BN 0

-19-

8559

01-1

. 3.

O'K

eeffe

, M.;

Hyd

e, B

. G. (

1996

). C

ryst

al S

truct

ures

; I.

Pat

tern

s an

d S

ymm

etry

. Was

hing

ton,

DC

: Min

eral

ogic

al

Soci

ety

of A

mer

ica,

Mon

ogra

ph S

erie

s. IS

BN 0

-939

950-

40-5

. 4.

Mille

r, W

illard

Jr.

(197

2). S

ymm

etry

Gro

ups

and

Thei

r A

pplic

atio

ns. N

ew Y

ork:

Aca

dem

ic P

ress

. OC

LC 5

8908

1.

Ret

rieve

d 20

09-0

9-28

. 5.

http

://en

.wik

iped

ia.o

rg/w

iki/S

pace

_gro

up

6.ht

tp://

en.w

ikip

edia

.org

/wik

i/Poi

nt_g

roup

Furth

er re

adin

g Sc

hoen

flies

not

atio

n In

Sch

oenf

lies

nota

tion,

poi

nt g

roup

s ar

e de

note

d by

a

lette

r sym

bol w

ith a

sub

scrip

t. Th

e sy

mbo

ls m

ean

the

follo

win

g:

The

lette

r I (f

or ic

osah

edro

n) in

dica

tes

that

the

grou

p ha

s th

e sy

mm

etry

of a

n ic

osah

edro

n.

The

lette

r O (f

or o

ctah

edro

n) in

dica

tes

that

the

grou

p ha

s th

e sy

mm

etry

of a

n oc

tahe

dron

(or c

ube)

. Th

e le

tter T

(for

tetra

hedr

on) i

ndic

ates

that

the

grou

p ha

s th

e sy

mm

etry

of a

tetra

hedr

on.

1.

Loo

k at

the

mol

ecul

e an

d se

e if

it se

ems

to b

e ve

ry

sym

met

ric o

r ver

y un

sym

met

ric. I

f so,

it p

roba

bly

belo

ngs

to

one

of th

e sp

ecia

l gro

ups

(low

sym

met

ry: C

1, C

s, C

i or l

inea

r, C

h, D

h) o

r hig

h sy

mm

etry

(Td,

Oh,

I h).

2. F

or a

ll ot

her m

olec

ules

find

the

rota

tion

axis

with

the

high

est n

, the

hig

hest

ord

er C

n ax

is o

f the

mol

ecul

e.

3. D

oes

the

mol

ecul

e ha

ve a

ny C

2 axe

s pe

rpen

dicu

lar t

o th

e C

n axi

s? If

it d

oes,

ther

e w

ill be

n o

f suc

h C

2 axe

s, a

nd th

e m

olec

ule

is in

one

of D

poi

nt g

roup

s. If

not

, it w

ill be

in o

ne o

f C

or S

poi

nt g

roup

s.

4. D

oes

it ha

ve a

ny m

irror

pla

ne (s

h) p

erpe

ndic

ular

to th

e C

n ax

is. I

f so,

it is

Cnh

or D

nh.

5. D

oes

it ha

ve a

ny m

irror

pla

ne (s

d,sv)?

If s

o, it

is C

nv o

r Dnd