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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
