<s>
+	O
Example	O
polyhedra	O
with	O
rotoreflection	B-Algorithm
symmetryGroup	O
S4	O
S6	O
S8	O
S10	O
S12Subgroups	O
C2	O
C3	O
,	O
S2	O
=	O
Ci	O
C4	O
,	O
C2	O
C5	O
,	O
S2	O
=	O
Ci	O
C6	O
,	O
S4	O
,	O
C3	O
,	O
C2Example80pxbeveled	O
digonal	O
antiprism	O
80pxtriangular	O
antiprism	O
80pxsquare	O
antiprism	O
100pxpentagonal	O
antiprism	O
100pxhexagonal	O
antiprism	O
Antiprisms	O
with	O
directed	O
edges	O
have	O
rotoreflection	B-Algorithm
symmetry.p-antiprisms	O
for	O
odd	O
p	O
contain	O
inversion	B-Algorithm
symmetry	I-Algorithm
,	O
Ci	O
.	O
</s>
<s>
In	O
geometry	O
,	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
,	O
also	O
called	O
rotation-reflection	O
,	O
rotoreflection	B-Algorithm
,	O
rotary	B-Algorithm
reflection	I-Algorithm
,	O
or	O
rotoinversion	B-Algorithm
is	O
an	O
isometry	O
in	O
Euclidean	O
space	O
that	O
is	O
a	O
combination	O
of	O
a	O
rotation	B-General_Concept
about	O
an	O
axis	O
and	O
a	O
reflection	B-Algorithm
in	O
a	O
plane	O
perpendicular	O
to	O
that	O
axis	O
.	O
</s>
<s>
Reflection	B-Algorithm
and	O
inversion	B-Algorithm
are	O
each	O
special	O
case	O
of	O
improper	B-Algorithm
rotation	I-Algorithm
.	O
</s>
<s>
Any	O
improper	B-Algorithm
rotation	I-Algorithm
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
and	O
,	O
in	O
cases	O
that	O
keep	O
the	O
coordinate	O
origin	O
fixed	O
,	O
a	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
It	O
is	O
used	O
as	O
a	O
symmetry	O
operation	O
in	O
the	O
context	O
of	O
geometric	O
symmetry	O
,	O
molecular	O
symmetry	O
and	O
crystallography	O
,	O
where	O
an	O
object	O
that	O
is	O
unchanged	O
by	O
a	O
combination	O
of	O
rotation	B-General_Concept
and	O
reflection	B-Algorithm
is	O
said	O
to	O
have	O
improper	B-Algorithm
rotation	I-Algorithm
symmetry	O
.	O
</s>
<s>
In	O
3	O
dimensions	O
,	O
improper	B-Algorithm
rotation	I-Algorithm
is	O
equivalently	O
defined	O
as	O
a	O
combination	O
of	O
rotation	B-General_Concept
about	O
an	O
axis	O
and	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
on	O
the	O
axis	O
.	O
</s>
<s>
For	O
this	O
reason	O
it	O
is	O
also	O
called	O
a	O
rotoinversion	B-Algorithm
or	O
rotary	O
inversion	B-Algorithm
.	O
</s>
<s>
The	O
two	O
definitions	O
are	O
equivalent	O
because	O
rotation	B-General_Concept
by	O
an	O
angle	O
θ	O
followed	O
by	O
reflection	B-Algorithm
is	O
the	O
same	O
transformation	O
as	O
rotation	B-General_Concept
by	O
θ+180°	O
followed	O
by	O
inversion	B-Algorithm
(	O
taking	O
the	O
point	O
of	O
inversion	B-Algorithm
to	O
be	O
in	O
the	O
plane	O
of	O
reflection	B-Algorithm
)	O
.	O
</s>
<s>
A	O
three-dimensional	O
symmetry	O
that	O
has	O
only	O
one	O
fixed	O
point	O
is	O
necessarily	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
.	O
</s>
<s>
An	O
improper	B-Algorithm
rotation	I-Algorithm
of	O
an	O
object	O
thus	O
produces	O
a	O
rotation	B-General_Concept
of	O
its	O
mirror	O
image	O
.	O
</s>
<s>
The	O
axis	O
is	O
called	O
the	O
rotation-reflection	B-Algorithm
axis	I-Algorithm
.	O
</s>
<s>
This	O
is	O
called	O
an	O
n-fold	O
improper	B-Algorithm
rotation	I-Algorithm
if	O
the	O
angle	O
of	O
rotation	B-General_Concept
,	O
before	O
or	O
after	O
reflexion	O
,	O
is	O
360°	O
/n	O
(	O
where	O
n	O
must	O
be	O
even	O
)	O
.	O
</s>
<s>
There	O
are	O
several	O
different	O
systems	O
for	O
naming	O
individual	O
improper	B-Algorithm
rotations	I-Algorithm
:	O
</s>
<s>
In	O
the	O
Schoenflies	O
notation	O
the	O
symbol	O
Sn	O
(	O
German	O
,	O
,	O
for	O
mirror	O
)	O
,	O
where	O
n	O
must	O
be	O
even	O
,	O
denotes	O
the	O
symmetry	O
group	O
generated	O
by	O
an	O
n-fold	O
improper	B-Algorithm
rotation	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
symmetry	O
operation	O
S6	O
is	O
the	O
combination	O
of	O
a	O
rotation	B-General_Concept
of	O
(	O
360°	O
/6	O
)	O
=	O
60°	O
and	O
a	O
mirror	B-Algorithm
plane	I-Algorithm
reflection	B-Algorithm
.	O
</s>
<s>
(	O
This	O
should	O
not	O
to	O
be	O
confused	O
with	O
the	O
same	O
notation	O
for	O
symmetric	B-Algorithm
groups	I-Algorithm
)	O
.	O
</s>
<s>
In	O
Hermann	O
–	O
Mauguin	O
notation	O
the	O
symbol	O
is	O
used	O
for	O
an	O
n-fold	O
rotoinversion	B-Algorithm
;	O
i.e.	O
,	O
rotation	B-General_Concept
by	O
an	O
angle	O
of	O
rotation	B-General_Concept
of	O
360°	O
/n	O
with	O
inversion	B-Algorithm
.	O
</s>
<s>
(	O
Note	O
that	O
would	O
be	O
simply	O
a	O
reflection	B-Algorithm
,	O
and	O
is	O
normally	O
denoted	O
"	O
m	O
"	O
,	O
for	O
"	O
mirror	O
"	O
.	O
)	O
</s>
<s>
When	O
n	O
is	O
odd	O
this	O
corresponds	O
to	O
a	O
2n-fold	O
improper	B-Algorithm
rotation	I-Algorithm
(	O
or	O
rotary	O
reflexion	O
)	O
.	O
</s>
<s>
The	O
direct	O
subgroup	O
of	O
S2n	O
is	O
Cn	O
,	O
order	O
n	O
,	O
index	O
2	O
,	O
being	O
the	O
rotoreflection	B-Algorithm
generator	O
applied	O
twice	O
.	O
</s>
<s>
For	O
odd	O
n	O
,	O
S2n	O
contains	O
an	O
inversion	B-Algorithm
,	O
denoted	O
Ci	O
or	O
S2	O
.	O
</s>
<s>
In	O
a	O
wider	O
sense	O
,	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
may	O
be	O
defined	O
as	O
any	O
indirect	O
isometry	O
;	O
i.e.	O
,	O
an	O
element	O
of	O
E(3 )	O
\E+( 3	O
)	O
:	O
thus	O
it	O
can	O
also	O
be	O
a	O
pure	O
reflection	B-Algorithm
in	O
a	O
plane	O
,	O
or	O
have	O
a	O
glide	B-Algorithm
plane	I-Algorithm
.	O
</s>
<s>
An	O
indirect	O
isometry	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
with	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
that	O
has	O
a	O
determinant	O
of	O
−1	O
.	O
</s>
<s>
A	O
proper	B-Algorithm
rotation	I-Algorithm
is	O
an	O
ordinary	O
rotation	B-General_Concept
.	O
</s>
<s>
In	O
the	O
wider	O
sense	O
,	O
a	O
proper	B-Algorithm
rotation	I-Algorithm
is	O
defined	O
as	O
a	O
direct	O
isometry	O
;	O
i.e.	O
,	O
an	O
element	O
of	O
E+( 3	O
)	O
:	O
it	O
can	O
also	O
be	O
the	O
identity	O
,	O
a	O
rotation	B-General_Concept
with	O
a	O
translation	O
along	O
the	O
axis	O
,	O
or	O
a	O
pure	O
translation	O
.	O
</s>
<s>
A	O
direct	O
isometry	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
with	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
that	O
has	O
a	O
determinant	O
of	O
1	O
.	O
</s>
<s>
In	O
either	O
the	O
narrower	O
or	O
the	O
wider	O
senses	O
,	O
the	O
composition	O
of	O
two	O
improper	B-Algorithm
rotations	I-Algorithm
is	O
a	O
proper	B-Algorithm
rotation	I-Algorithm
,	O
and	O
the	O
composition	O
of	O
an	O
improper	O
and	O
a	O
proper	B-Algorithm
rotation	I-Algorithm
is	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
.	O
</s>
<s>
When	O
studying	O
the	O
symmetry	O
of	O
a	O
physical	O
system	O
under	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
(	O
e.g.	O
,	O
if	O
a	O
system	O
has	O
a	O
mirror	O
symmetry	O
plane	O
)	O
,	O
it	O
is	O
important	O
to	O
distinguish	O
between	O
vectors	O
and	O
pseudovectors	O
(	O
as	O
well	O
as	O
scalars	O
and	O
pseudoscalars	O
,	O
and	O
in	O
general	O
between	O
tensors	B-Device
and	O
pseudotensors	O
)	O
,	O
since	O
the	O
latter	O
transform	O
differently	O
under	O
proper	O
and	O
improper	B-Algorithm
rotations	I-Algorithm
(	O
in	O
3	O
dimensions	O
,	O
pseudovectors	O
are	O
invariant	O
under	O
inversion	B-Algorithm
)	O
.	O
</s>
