<s>
In	O
mathematics	O
,	O
an	O
exceptional	B-Algorithm
isomorphism	I-Algorithm
,	O
also	O
called	O
an	O
accidental	B-Algorithm
isomorphism	I-Algorithm
,	O
is	O
an	O
isomorphism	O
between	O
members	O
ai	O
and	O
bj	O
of	O
two	O
families	O
,	O
usually	O
infinite	O
,	O
of	O
mathematical	O
objects	O
,	O
which	O
is	O
incidental	O
,	O
in	O
that	O
it	O
is	O
not	O
an	O
instance	O
of	O
a	O
general	O
pattern	O
of	O
such	O
isomorphisms	O
.	O
</s>
<s>
The	O
exceptional	B-Algorithm
isomorphisms	I-Algorithm
between	O
the	O
series	O
of	O
finite	O
simple	O
groups	O
mostly	O
involve	O
projective	O
special	O
linear	O
groups	O
and	O
alternating	B-Algorithm
groups	I-Algorithm
,	O
and	O
are	O
:	O
</s>
<s>
(	O
The	O
above	O
isomorphisms	O
for	O
and	O
are	O
linked	O
via	O
the	O
exceptional	B-Algorithm
isomorphism	I-Algorithm
.	O
)	O
</s>
<s>
There	O
are	O
also	O
some	O
coincidences	O
with	O
symmetries	O
of	O
regular	O
polyhedra	O
:	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A5	O
agrees	O
with	O
the	O
icosahedral	O
group	O
(	O
itself	O
an	O
exceptional	O
object	O
)	O
,	O
and	O
the	O
double	B-Algorithm
cover	I-Algorithm
of	O
the	O
alternating	B-Algorithm
group	I-Algorithm
A5	O
is	O
the	O
binary	O
icosahedral	O
group	O
.	O
</s>
<s>
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
on	O
0	O
,	O
1	O
,	O
or	O
2	O
letters	O
;	O
</s>
<s>
There	O
are	O
some	O
exceptional	B-Algorithm
isomorphisms	I-Algorithm
of	O
Dynkin	O
diagrams	O
,	O
yielding	O
isomorphisms	O
of	O
the	O
corresponding	O
Coxeter	B-Algorithm
groups	I-Algorithm
and	O
of	O
polytopes	O
realizing	O
the	O
symmetries	O
,	O
as	O
well	O
as	O
isomorphisms	O
of	O
Lie	O
algebras	O
whose	O
root	O
systems	O
are	O
described	O
by	O
the	O
same	O
diagrams	O
.	O
</s>
