<s>
In	O
mathematics	O
,	O
a	O
frieze	O
or	O
frieze	B-Algorithm
pattern	I-Algorithm
is	O
a	O
two-dimensional	O
design	O
that	O
repeats	O
in	O
one	O
direction	O
.	O
</s>
<s>
Frieze	B-Algorithm
patterns	I-Algorithm
can	O
be	O
classified	O
into	O
seven	O
types	O
according	O
to	O
their	O
symmetries	O
.	O
</s>
<s>
The	O
set	O
of	O
symmetries	O
of	O
a	O
frieze	B-Algorithm
pattern	I-Algorithm
is	O
called	O
a	O
frieze	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Frieze	B-Algorithm
groups	I-Algorithm
are	O
two-dimensional	O
line	B-Algorithm
groups	I-Algorithm
,	O
having	O
repetition	O
in	O
only	O
one	O
direction	O
.	O
</s>
<s>
Formally	O
,	O
a	O
frieze	B-Algorithm
group	I-Algorithm
is	O
a	O
class	O
of	O
infinite	O
discrete	O
symmetry	O
groups	O
of	O
patterns	O
on	O
a	O
strip	O
(	O
infinitely	O
wide	O
rectangle	O
)	O
,	O
hence	O
a	O
class	O
of	O
groups	O
of	O
isometries	O
of	O
the	O
plane	O
,	O
or	O
of	O
a	O
strip	O
.	O
</s>
<s>
A	O
symmetry	O
group	O
of	O
a	O
frieze	B-Algorithm
group	I-Algorithm
necessarily	O
contains	O
translations	O
and	O
may	O
contain	O
glide	B-Algorithm
reflections	I-Algorithm
,	O
reflections	O
along	O
the	O
long	O
axis	O
of	O
the	O
strip	O
,	O
reflections	O
along	O
the	O
narrow	O
axis	O
of	O
the	O
strip	O
,	O
and	O
180°	O
rotations	O
.	O
</s>
<s>
There	O
are	O
seven	O
frieze	B-Algorithm
groups	I-Algorithm
,	O
listed	O
in	O
the	O
summary	O
table	O
.	O
</s>
<s>
Many	O
authors	O
present	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
in	O
a	O
different	O
order	O
.	O
</s>
<s>
The	O
actual	O
symmetry	O
groups	O
within	O
a	O
frieze	B-Algorithm
group	I-Algorithm
are	O
characterized	O
by	O
the	O
smallest	O
translation	B-Algorithm
distance	O
,	O
and	O
,	O
for	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
with	O
vertical	O
line	O
reflection	B-Algorithm
or	O
180°	O
rotation	O
(	O
groups	O
2	O
,	O
5	O
,	O
6	O
,	O
and	O
7	O
)	O
,	O
by	O
a	O
shift	O
parameter	O
locating	O
the	O
reflection	B-Algorithm
axis	O
or	O
point	O
of	O
rotation	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
symmetry	O
groups	O
in	O
the	O
plane	O
,	O
additional	O
parameters	O
are	O
the	O
direction	O
of	O
the	O
translation	B-Algorithm
vector	O
,	O
and	O
,	O
for	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
with	O
horizontal	O
line	O
reflection	B-Algorithm
,	O
glide	B-Algorithm
reflection	I-Algorithm
,	O
or	O
180°	O
rotation	O
(	O
groups	O
3	O
–	O
7	O
)	O
,	O
the	O
position	O
of	O
the	O
reflection	B-Algorithm
axis	O
or	O
rotation	O
point	O
in	O
the	O
direction	O
perpendicular	O
to	O
the	O
translation	B-Algorithm
vector	O
.	O
</s>
<s>
For	O
two	O
of	O
the	O
seven	O
frieze	B-Algorithm
groups	I-Algorithm
(	O
groups	O
1	O
and	O
4	O
)	O
the	O
symmetry	O
groups	O
are	O
singly	O
generated	O
,	O
for	O
four	O
(	O
groups	O
2	O
,	O
3	O
,	O
5	O
,	O
and	O
6	O
)	O
they	O
have	O
a	O
pair	O
of	O
generators	O
,	O
and	O
for	O
group	O
7	O
the	O
symmetry	O
groups	O
require	O
three	O
generators	O
.	O
</s>
<s>
A	O
symmetry	O
group	O
in	O
frieze	B-Algorithm
group	I-Algorithm
1	O
,	O
2	O
,	O
3	O
,	O
or	O
5	O
is	O
a	O
subgroup	O
of	O
a	O
symmetry	O
group	O
in	O
the	O
last	O
frieze	B-Algorithm
group	I-Algorithm
with	O
the	O
same	O
translational	O
distance	O
.	O
</s>
<s>
A	O
symmetry	O
group	O
in	O
frieze	B-Algorithm
group	I-Algorithm
4	O
or	O
6	O
is	O
a	O
subgroup	O
of	O
a	O
symmetry	O
group	O
in	O
the	O
last	O
frieze	B-Algorithm
group	I-Algorithm
with	O
half	O
the	O
translational	O
distance	O
.	O
</s>
<s>
This	O
last	O
frieze	B-Algorithm
group	I-Algorithm
contains	O
the	O
symmetry	O
groups	O
of	O
the	O
simplest	O
periodic	O
patterns	O
in	O
the	O
strip	O
(	O
or	O
the	O
plane	O
)	O
,	O
a	O
row	O
of	O
dots	O
.	O
</s>
<s>
Any	O
transformation	O
of	O
the	O
plane	O
leaving	O
this	O
pattern	O
invariant	O
can	O
be	O
decomposed	O
into	O
a	O
translation	B-Algorithm
,	O
,	O
optionally	O
followed	O
by	O
a	O
reflection	B-Algorithm
in	O
either	O
the	O
horizontal	O
axis	O
,	O
,	O
or	O
the	O
vertical	O
axis	O
,	O
,	O
provided	O
that	O
this	O
axis	O
is	O
chosen	O
through	O
or	O
midway	O
between	O
two	O
dots	O
,	O
or	O
a	O
rotation	O
by	O
180°	O
,	O
(	O
ditto	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
a	O
way	O
,	O
this	O
frieze	B-Algorithm
group	I-Algorithm
contains	O
the	O
"	O
largest	O
"	O
symmetry	O
groups	O
,	O
which	O
consist	O
of	O
all	O
such	O
transformations	O
.	O
</s>
<s>
the	O
group	O
of	O
horizontal	B-Algorithm
translations	I-Algorithm
by	O
rational	O
distances	O
)	O
.	O
</s>
<s>
the	O
group	O
consisting	O
of	O
the	O
identity	O
and	O
reflection	B-Algorithm
in	O
the	O
horizontal	O
axis	O
(	O
isomorphic	O
to	O
C2	O
,	O
the	O
cyclic	O
group	O
of	O
order	O
2	O
)	O
.	O
</s>
<s>
There	O
are	O
seven	O
distinct	O
subgroups	O
(	O
up	O
to	O
scaling	O
and	O
shifting	O
of	O
patterns	O
)	O
in	O
the	O
discrete	O
frieze	B-Algorithm
group	I-Algorithm
generated	O
by	O
a	O
translation	B-Algorithm
,	O
reflection	B-Algorithm
(	O
along	O
the	O
same	O
axis	O
)	O
and	O
a	O
180°	O
rotation	O
.	O
</s>
<s>
Each	O
of	O
these	O
subgroups	O
is	O
the	O
symmetry	O
group	O
of	O
a	O
frieze	B-Algorithm
pattern	I-Algorithm
,	O
and	O
sample	O
patterns	O
are	O
shown	O
in	O
Fig	O
.	O
</s>
<s>
They	O
are	O
identified	O
in	O
the	O
table	O
below	O
using	O
Hermann	O
–	O
Mauguin	O
notation	O
(	O
or	O
IUC	O
notation	O
)	O
,	O
Coxeter	O
notation	O
,	O
Schönflies	O
notation	O
,	O
orbifold	O
notation	O
,	O
nicknames	O
created	O
by	O
mathematician	O
John	O
H	O
.	O
Conway	O
,	O
and	O
finally	O
a	O
description	O
in	O
terms	O
of	O
translation	B-Algorithm
,	O
reflections	O
and	O
rotations	O
.	O
</s>
<s>
Of	O
the	O
seven	O
frieze	B-Algorithm
groups	I-Algorithm
,	O
there	O
are	O
only	O
four	O
up	O
to	O
isomorphism	O
.	O
</s>
