<s>
In	O
group	O
theory	O
and	O
geometry	O
,	O
a	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
discrete	O
group	O
which	O
is	O
generated	O
by	O
a	O
set	O
of	O
reflections	B-Algorithm
of	O
a	O
finite-dimensional	O
Euclidean	O
space	O
.	O
</s>
<s>
The	O
symmetry	O
group	O
of	O
a	O
regular	O
polytope	O
or	O
of	O
a	O
tiling	B-Algorithm
of	O
the	O
Euclidean	O
space	O
by	O
congruent	O
copies	O
of	O
a	O
regular	O
polytope	O
is	O
necessarily	O
a	O
reflection	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Reflection	B-Algorithm
groups	I-Algorithm
also	O
include	O
Weyl	O
groups	O
and	O
crystallographic	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
While	O
the	O
orthogonal	O
group	O
is	O
generated	O
by	O
reflections	B-Algorithm
(	O
by	O
the	O
Cartan	O
–	O
Dieudonné	O
theorem	O
)	O
,	O
it	O
is	O
a	O
continuous	O
group	O
(	O
indeed	O
,	O
Lie	O
group	O
)	O
,	O
not	O
a	O
discrete	O
group	O
,	O
and	O
is	O
generally	O
considered	O
separately	O
.	O
</s>
<s>
A	O
finite	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
subgroup	O
of	O
the	O
general	O
linear	O
group	O
of	O
E	O
which	O
is	O
generated	O
by	O
a	O
set	O
of	O
orthogonal	O
reflections	B-Algorithm
across	O
hyperplanes	O
passing	O
through	O
the	O
origin	O
.	O
</s>
<s>
An	O
affine	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
discrete	O
subgroup	O
of	O
the	O
affine	O
group	O
of	O
E	O
that	O
is	O
generated	O
by	O
a	O
set	O
of	O
affine	O
reflections	B-Algorithm
of	O
E	O
(	O
without	O
the	O
requirement	O
that	O
the	O
reflection	B-Algorithm
hyperplanes	O
pass	O
through	O
the	O
origin	O
)	O
.	O
</s>
<s>
The	O
corresponding	O
notions	O
can	O
be	O
defined	O
over	O
other	O
fields	O
,	O
leading	O
to	O
complex	O
reflection	B-Algorithm
groups	I-Algorithm
and	O
analogues	O
of	O
reflection	B-Algorithm
groups	I-Algorithm
over	O
a	O
finite	O
field	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
finite	O
reflection	B-Algorithm
groups	I-Algorithm
are	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
,	O
which	O
are	O
generated	O
by	O
reflection	B-Algorithm
in	O
two	O
lines	O
that	O
form	O
an	O
angle	O
of	O
and	O
correspond	O
to	O
the	O
Coxeter	O
diagram	O
Conversely	O
,	O
the	O
cyclic	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
two	I-Algorithm
dimensions	I-Algorithm
are	O
not	O
generated	O
by	O
reflections	B-Algorithm
,	O
and	O
indeed	O
contain	O
no	O
reflections	B-Algorithm
–	O
they	O
are	O
however	O
subgroups	O
of	O
index	O
2	O
of	O
a	O
dihedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Infinite	O
reflection	B-Algorithm
groups	I-Algorithm
include	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
and	O
and	O
the	O
wallpaper	O
groups	O
,	O
,	O
,	O
and	O
.	O
</s>
<s>
If	O
the	O
angle	O
between	O
two	O
lines	O
is	O
an	O
irrational	O
multiple	O
of	O
pi	O
,	O
the	O
group	O
generated	O
by	O
reflections	B-Algorithm
in	O
these	O
lines	O
is	O
infinite	O
and	O
non-discrete	O
,	O
hence	O
,	O
it	O
is	O
not	O
a	O
reflection	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Finite	O
reflection	B-Algorithm
groups	I-Algorithm
are	O
the	O
point	B-Algorithm
groups	I-Algorithm
Cnv	O
,	O
Dnh	O
,	O
and	O
the	O
symmetry	O
groups	O
of	O
the	O
five	O
Platonic	O
solids	O
.	O
</s>
<s>
The	O
classification	O
of	O
finite	O
reflection	B-Algorithm
groups	I-Algorithm
of	O
R3	O
is	O
an	O
instance	O
of	O
the	O
ADE	O
classification	O
.	O
</s>
<s>
A	O
reflection	B-Algorithm
group	I-Algorithm
W	O
admits	O
a	O
presentation	O
of	O
a	O
special	O
kind	O
discovered	O
and	O
studied	O
by	O
H	O
.	O
S	O
.	O
M	O
.	O
Coxeter	O
.	O
</s>
<s>
The	O
reflections	B-Algorithm
in	O
the	O
faces	O
of	O
a	O
fixed	O
fundamental	O
"	O
chamber	O
"	O
are	O
generators	O
ri	O
of	O
W	O
of	O
order	O
2	O
.	O
</s>
<s>
expressing	O
the	O
fact	O
that	O
the	O
product	O
of	O
the	O
reflections	B-Algorithm
ri	O
and	O
rj	O
in	O
two	O
hyperplanes	O
Hi	O
and	O
Hj	O
meeting	O
at	O
an	O
angle	O
is	O
a	O
rotation	O
by	O
the	O
angle	O
fixing	O
the	O
subspace	O
Hi∩Hj	O
of	O
codimension	O
2	O
.	O
</s>
<s>
Thus	O
,	O
viewed	O
as	O
an	O
abstract	O
group	O
,	O
every	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
When	O
working	O
over	O
finite	O
fields	O
,	O
one	O
defines	O
a	O
"	O
reflection	B-Algorithm
"	O
as	O
a	O
map	O
that	O
fixes	O
a	O
hyperplane	O
(	O
otherwise	O
for	O
example	O
there	O
would	O
be	O
no	O
reflections	B-Algorithm
in	O
characteristic	O
2	O
,	O
as	O
so	O
reflections	B-Algorithm
are	O
the	O
identity	O
)	O
.	O
</s>
<s>
Geometrically	O
,	O
this	O
amounts	O
to	O
including	O
shears	B-Algorithm
in	O
a	O
hyperplane	O
.	O
</s>
<s>
Reflection	B-Algorithm
groups	I-Algorithm
over	O
finite	O
fields	O
of	O
characteristic	O
not	O
2	O
were	O
classified	O
by	O
.	O
</s>
<s>
Discrete	O
isometry	O
groups	O
of	O
more	O
general	O
Riemannian	B-Architecture
manifolds	I-Architecture
generated	O
by	O
reflections	B-Algorithm
have	O
also	O
been	O
considered	O
.	O
</s>
<s>
affine	O
reflection	B-Algorithm
groups	I-Algorithm
,	O
and	O
the	O
hyperbolic	O
space	O
Hn	O
,	O
where	O
the	O
corresponding	O
groups	O
are	O
called	O
hyperbolic	O
reflection	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
triangle	O
groups	O
include	O
reflection	B-Algorithm
groups	I-Algorithm
of	O
all	O
three	O
kinds	O
.	O
</s>
