<s>
In	O
mathematics	O
,	O
the	O
tricorn	B-Algorithm
,	O
sometimes	O
called	O
the	O
Mandelbar	B-Algorithm
set	O
,	O
is	O
a	O
fractal	B-Application
defined	O
in	O
a	O
similar	O
way	O
to	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
but	O
using	O
the	O
mapping	O
instead	O
of	O
used	O
for	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
John	O
Milnor	O
found	O
tricorn-like	O
sets	O
as	O
a	O
prototypical	O
configuration	O
in	O
the	O
parameter	O
space	O
of	O
real	O
cubic	O
polynomials	O
,	O
and	O
in	O
various	O
other	O
families	O
of	O
rational	O
maps	O
.	O
</s>
<s>
The	O
characteristic	O
three-cornered	O
shape	O
created	O
by	O
this	O
fractal	B-Application
repeats	O
with	O
variations	O
at	O
different	O
scales	O
,	O
showing	O
the	O
same	O
sort	O
of	O
self-similarity	O
as	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
In	O
addition	O
to	O
smaller	O
tricorns	B-Algorithm
,	O
smaller	O
versions	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
are	O
also	O
contained	O
within	O
the	O
tricorn	B-Algorithm
fractal	B-Application
.	O
</s>
<s>
In	O
analogy	O
with	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
the	O
tricorn	B-Algorithm
is	O
defined	O
as	O
the	O
set	O
of	O
all	O
parameters	O
for	O
which	O
the	O
forward	O
orbit	O
of	O
the	O
critical	O
point	O
is	O
bounded	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
saying	O
that	O
the	O
tricorn	B-Algorithm
is	O
the	O
connectedness	O
locus	O
of	O
the	O
family	O
of	O
quadratic	O
antiholomorphic	O
polynomials	O
;	O
i.e.	O
</s>
<s>
the	O
set	O
of	O
all	O
parameters	O
for	O
which	O
the	O
Julia	B-Language
set	I-Language
is	O
connected	O
.	O
</s>
<s>
The	O
higher	O
degree	O
analogues	O
of	O
the	O
tricorn	B-Algorithm
are	O
known	O
as	O
the	O
multicorns	O
.	O
</s>
<s>
The	O
tricorn	B-Algorithm
is	O
compact	O
,	O
and	O
connected	O
.	O
</s>
<s>
In	O
fact	O
,	O
Nakane	O
modified	O
Douady	O
and	O
Hubbard	O
's	O
proof	O
of	O
the	O
connectedness	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
to	O
construct	O
a	O
dynamically	O
defined	O
real-analytic	B-Language
diffeomorphism	O
from	O
the	O
exterior	O
of	O
the	O
tricorn	B-Algorithm
onto	O
the	O
exterior	O
of	O
the	O
closed	O
unit	O
disc	O
in	O
the	O
complex	O
plane	O
.	O
</s>
<s>
One	O
can	O
define	O
external	O
parameter	O
rays	O
of	O
the	O
tricorn	B-Algorithm
as	O
the	O
inverse	O
images	O
of	O
radial	O
lines	O
under	O
this	O
diffeomorphism	O
.	O
</s>
<s>
Every	O
hyperbolic	O
component	O
of	O
the	O
tricorn	B-Algorithm
is	O
simply	O
connected	O
.	O
</s>
<s>
The	O
boundary	O
of	O
every	O
hyperbolic	O
component	O
of	O
odd	O
period	O
of	O
the	O
tricorn	B-Algorithm
contains	O
real-analytic	B-Language
arcs	O
consisting	O
of	O
quasi-conformally	O
equivalent	O
but	O
conformally	O
distinct	O
parabolic	O
parameters	O
.	O
</s>
<s>
Such	O
an	O
arc	O
is	O
called	O
a	O
parabolic	O
arc	O
of	O
the	O
tricorn	B-Algorithm
.	O
</s>
<s>
This	O
is	O
in	O
stark	O
contrast	O
with	O
the	O
corresponding	O
situation	O
for	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
where	O
parabolic	O
parameters	O
of	O
a	O
given	O
period	O
are	O
known	O
to	O
be	O
isolated	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
boundary	O
of	O
every	O
hyperbolic	O
component	O
of	O
odd	O
period	O
of	O
the	O
tricorn	B-Algorithm
is	O
a	O
simple	O
closed	O
curve	O
consisting	O
of	O
exactly	O
three	O
parabolic	O
cusp	O
points	O
as	O
well	O
as	O
three	O
parabolic	O
arcs	O
,	O
each	O
connecting	O
two	O
parabolic	O
cusps	O
.	O
</s>
<s>
The	O
tricorn	B-Algorithm
is	O
not	O
path	O
connected	O
.	O
</s>
<s>
Hubbard	O
and	O
Schleicher	O
showed	O
that	O
there	O
are	O
hyperbolic	O
components	O
of	O
odd	O
period	O
of	O
the	O
tricorn	B-Algorithm
that	O
cannot	O
be	O
connected	O
to	O
the	O
hyperbolic	O
component	O
of	O
period	O
one	O
by	O
paths	O
.	O
</s>
<s>
A	O
stronger	O
statement	O
to	O
the	O
effect	O
that	O
no	O
two	O
(	O
non-real	O
)	O
odd	O
period	O
hyperbolic	O
components	O
of	O
the	O
tricorn	B-Algorithm
can	O
be	O
connected	O
by	O
a	O
path	O
was	O
proved	O
by	O
Inou	O
and	O
Mukherjee	O
.	O
</s>
<s>
It	O
is	O
well	O
known	O
that	O
every	O
rational	O
parameter	O
ray	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
lands	O
at	O
a	O
single	O
parameter	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
rational	O
parameter	O
rays	O
at	O
odd-periodic	O
(	O
except	O
period	O
one	O
)	O
angles	O
of	O
the	O
tricorn	B-Algorithm
accumulate	O
on	O
arcs	O
of	O
positive	O
length	O
consisting	O
of	O
parabolic	O
parameters	O
.	O
</s>
<s>
Moreover	O
,	O
unlike	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
the	O
dynamically	O
natural	O
straightening	O
map	O
from	O
a	O
baby	O
tricorn	B-Algorithm
to	O
the	O
original	O
tricorn	B-Algorithm
is	O
discontinuous	O
at	O
infinitely	O
many	O
parameters	O
.	O
</s>
