<s>
A	O
line	B-Algorithm
group	I-Algorithm
is	O
a	O
mathematical	O
way	O
of	O
describing	O
symmetries	O
associated	O
with	O
moving	O
along	O
a	O
line	O
.	O
</s>
<s>
However	O
,	O
line	B-Algorithm
groups	I-Algorithm
may	O
have	O
more	O
than	O
one	O
dimension	O
,	O
and	O
they	O
may	O
involve	O
those	O
dimensions	O
in	O
its	O
isometries	O
or	O
symmetry	O
transformations	O
.	O
</s>
<s>
One	O
constructs	O
a	O
line	B-Algorithm
group	I-Algorithm
by	O
taking	O
a	O
point	O
group	O
in	O
the	O
full	O
dimensions	O
of	O
the	O
space	O
,	O
and	O
then	O
adding	O
translations	O
or	O
offsets	O
along	O
the	O
line	O
to	O
each	O
of	O
the	O
point	O
group	O
's	O
elements	O
,	O
in	O
the	O
fashion	O
of	O
constructing	O
a	O
space	O
group	O
.	O
</s>
<s>
There	O
are	O
2	O
one-dimensional	O
line	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
They	O
are	O
the	O
infinite	O
limits	O
of	O
the	O
discrete	O
two-dimensional	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
Cn	O
and	O
Dn	O
:	O
</s>
<s>
There	O
are	O
7	O
frieze	B-Algorithm
groups	I-Algorithm
,	O
which	O
involve	O
reflections	O
along	O
the	O
line	O
,	O
reflections	O
perpendicular	O
to	O
the	O
line	O
,	O
and	O
180°	O
rotations	O
in	O
the	O
two	O
dimensions	O
.	O
</s>
<s>
There	O
are	O
13	O
infinite	O
families	O
of	O
three-dimensional	O
line	B-Algorithm
groups	I-Algorithm
,	O
derived	O
from	O
the	O
7	O
infinite	O
families	O
of	O
axial	O
three-dimensional	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
As	O
with	O
space	O
groups	O
in	O
general	O
,	O
line	B-Algorithm
groups	I-Algorithm
with	O
the	O
same	O
point	O
group	O
can	O
have	O
different	O
patterns	O
of	O
offsets	O
.	O
</s>
<s>
There	O
appears	O
to	O
be	O
no	O
comparable	O
notation	O
for	O
the	O
line	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
These	O
groups	O
can	O
also	O
be	O
interpreted	O
as	O
patterns	O
of	O
wallpaper	O
groups	O
wrapped	O
around	O
a	O
cylinder	O
n	O
times	O
and	O
infinitely	O
repeating	O
along	O
the	O
cylinder	O
's	O
axis	O
,	O
much	O
like	O
the	O
three-dimensional	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
and	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Point	O
group	O
Line	B-Algorithm
group	I-Algorithm
H-M	O
Schönf	O
.	O
</s>
<s>
If	O
the	O
point	O
group	O
is	O
constrained	O
to	O
be	O
a	O
crystallographic	O
point	O
group	O
,	O
a	O
symmetry	O
of	O
some	O
three-dimensional	O
lattice	O
,	O
then	O
the	O
resulting	O
line	B-Algorithm
group	I-Algorithm
is	O
called	O
a	O
rod	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
There	O
are	O
75	O
rod	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Going	O
to	O
the	O
continuum	B-Algorithm
limit	I-Algorithm
,	O
with	O
n	O
to	O
∞	O
,	O
the	O
possible	O
point	O
groups	O
become	O
C∞	O
,	O
C∞h	O
,	O
C∞v	O
,	O
D∞	O
,	O
and	O
D∞h	O
,	O
and	O
the	O
line	B-Algorithm
groups	I-Algorithm
have	O
the	O
appropriate	O
possible	O
offsets	O
,	O
with	O
the	O
exception	O
of	O
zigzag	O
.	O
</s>
