<s>
In	O
mathematics	O
,	O
a	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
is	O
a	O
certain	O
type	O
of	O
fractal	B-Algorithm
curve	I-Algorithm
named	O
in	O
honor	O
of	O
Georges	O
de	O
Rham	O
.	O
</s>
<s>
The	O
Cantor	B-Algorithm
function	I-Algorithm
,	O
Cesàro	O
curve	O
,	O
Minkowski	O
's	O
question	O
mark	O
function	O
,	O
the	O
Lévy	B-Algorithm
C	I-Algorithm
curve	I-Algorithm
,	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
,	O
and	O
Koch	B-Algorithm
curve	I-Algorithm
are	O
all	O
special	O
cases	O
of	O
the	O
general	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
where	O
denotes	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
The	O
collection	O
of	O
points	O
,	O
parameterized	O
by	O
a	O
single	O
real	O
parameter	O
x	O
,	O
is	O
known	O
as	O
the	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
self-symmetries	O
of	O
all	O
of	O
the	O
de	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
are	O
given	O
by	O
the	O
monoid	O
that	O
describes	O
the	O
symmetries	O
of	O
the	O
infinite	O
binary	O
tree	O
or	O
Cantor	O
set	O
.	O
</s>
<s>
the	O
set	O
of	O
points	O
,	O
can	O
be	O
obtained	O
by	O
an	O
Iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
using	O
the	O
set	O
of	O
contraction	O
mappings	O
.	O
</s>
<s>
But	O
the	O
result	O
of	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
with	O
two	O
contraction	O
mappings	O
is	O
a	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
if	O
and	O
only	O
if	O
the	O
contraction	O
mappings	O
satisfy	O
the	O
continuity	O
condition	O
.	O
</s>
<s>
Detailed	O
,	O
worked	O
examples	O
of	O
the	O
self-similarities	O
can	O
be	O
found	O
in	O
the	O
articles	O
on	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
and	O
on	O
Minkowski	O
's	O
question-mark	O
function	O
.	O
</s>
<s>
Precisely	O
the	O
same	O
monoid	O
of	O
self-similarities	O
,	O
the	O
dyadic	O
monoid	O
,	O
apply	O
to	O
every	O
de	B-Algorithm
Rham	I-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
Cesàro	O
curves	O
(	O
or	O
Cesàro	O
–	O
Faber	O
curves	O
)	O
are	O
De	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
generated	O
by	O
affine	B-Algorithm
transformations	I-Algorithm
conserving	O
orientation	O
,	O
with	O
fixed	O
points	O
and	O
.	O
</s>
<s>
For	O
the	O
value	O
of	O
,	O
the	O
resulting	O
curve	O
is	O
the	O
Lévy	B-Algorithm
C	I-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
In	O
a	O
similar	O
way	O
,	O
we	O
can	O
define	O
the	O
KochPeano	O
family	O
of	O
curves	O
as	O
the	O
set	O
of	O
De	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
generated	O
by	O
affine	B-Algorithm
transformations	I-Algorithm
reversing	O
orientation	O
,	O
with	O
fixed	O
points	O
and	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
curve	I-Algorithm
is	O
obtained	O
by	O
setting	O
:	O
</s>
<s>
while	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
corresponds	O
to	O
:	O
</s>
<s>
The	O
Cesàro	O
–	O
Faber	O
and	O
Peano	O
–	O
Koch	B-Algorithm
curves	I-Algorithm
are	O
both	O
special	O
cases	O
of	O
the	O
general	O
case	O
of	O
a	O
pair	O
of	O
affine	O
linear	O
transformations	O
on	O
the	O
complex	O
plane	O
.	O
</s>
<s>
The	O
blancmange	B-Algorithm
curve	I-Algorithm
of	O
parameter	O
can	O
be	O
obtained	O
by	O
setting	O
,	O
and	O
.	O
</s>
<s>
Since	O
the	O
blancmange	B-Algorithm
curve	I-Algorithm
of	O
parameter	O
is	O
the	O
parabola	O
of	O
equation	O
,	O
this	O
illustrate	O
the	O
fact	O
that	O
in	O
some	O
occasion	O
,	O
de	B-Algorithm
Rham	I-Algorithm
curves	I-Algorithm
can	O
be	O
smooth	O
.	O
</s>
<s>
Such	O
a	O
generalization	O
allows	O
,	O
for	O
example	O
,	O
to	O
produce	O
the	O
Sierpiński	O
arrowhead	O
curve	O
(	O
whose	O
image	O
is	O
the	O
Sierpiński	O
triangle	O
)	O
,	O
by	O
using	O
the	O
contraction	O
mappings	O
of	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
that	O
produces	O
the	O
Sierpiński	O
triangle	O
.	O
</s>
<s>
Ornstein	O
and	O
others	O
describe	O
a	O
multifractal	B-Algorithm
system	I-Algorithm
,	O
where	O
instead	O
of	O
working	O
in	O
a	O
fixed	O
base	O
,	O
one	O
works	O
in	O
a	O
variable	O
base	O
.	O
</s>
