<s>
In	O
mathematical	O
analysis	O
,	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
is	O
a	O
curve	O
whose	O
range	B-Algorithm
reaches	O
every	O
point	O
in	O
a	O
higher	O
dimensional	O
region	O
,	O
typically	O
the	O
unit	O
square	O
(	O
or	O
more	O
generally	O
an	O
n-dimensional	O
unit	O
hypercube	B-Operating_System
)	O
.	O
</s>
<s>
Because	O
Giuseppe	O
Peano	O
(	O
1858	O
–	O
1932	O
)	O
was	O
the	O
first	O
to	O
discover	O
one	O
,	O
space-filling	B-Algorithm
curves	I-Algorithm
in	O
the	O
2-dimensional	O
plane	O
are	O
sometimes	O
called	O
Peano	B-Algorithm
curves	I-Algorithm
,	O
but	O
that	O
phrase	O
also	O
refers	O
to	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
,	O
the	O
specific	O
example	O
of	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
found	O
by	O
Peano	O
.	O
</s>
<s>
In	O
the	O
most	O
general	O
form	O
,	O
the	O
range	B-Algorithm
of	O
such	O
a	O
function	O
may	O
lie	O
in	O
an	O
arbitrary	O
topological	O
space	O
,	O
but	O
in	O
the	O
most	O
commonly	O
studied	O
cases	O
,	O
the	O
range	B-Algorithm
will	O
lie	O
in	O
a	O
Euclidean	O
space	O
such	O
as	O
the	O
2-dimensional	O
plane	O
(	O
a	O
planar	O
curve	O
)	O
or	O
the	O
3-dimensional	O
space	O
(	O
space	O
curve	O
)	O
.	O
</s>
<s>
In	O
1890	O
,	O
Peano	O
discovered	O
a	O
continuous	O
curve	O
,	O
now	O
called	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
,	O
that	O
passes	O
through	O
every	O
point	O
of	O
the	O
unit	O
square	O
.	O
</s>
<s>
Peano	O
was	O
motivated	O
by	O
Georg	O
Cantor	O
's	O
earlier	O
counterintuitive	O
result	O
that	O
the	O
infinite	O
number	O
of	O
points	O
in	O
a	O
unit	O
interval	O
is	O
the	O
same	O
cardinality	B-Application
as	O
the	O
infinite	O
number	O
of	O
points	O
in	O
any	O
finite-dimensional	O
manifold	B-Architecture
,	O
such	O
as	O
the	O
unit	O
square	O
.	O
</s>
<s>
Peano	O
's	O
solution	O
does	O
not	O
set	O
up	O
a	O
continuous	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
between	O
the	O
unit	O
interval	O
and	O
the	O
unit	O
square	O
,	O
and	O
indeed	O
such	O
a	O
correspondence	O
does	O
not	O
exist	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
It	O
was	O
common	O
to	O
associate	O
the	O
vague	O
notions	O
of	O
thinness	O
and	O
1-dimensionality	O
to	O
curves	O
;	O
all	O
normally	O
encountered	O
curves	O
were	O
piecewise	B-Algorithm
differentiable	I-Algorithm
(	O
that	O
is	O
,	O
have	O
piecewise	B-Algorithm
continuous	I-Algorithm
derivatives	O
)	O
,	O
and	O
such	O
curves	O
cannot	O
fill	O
up	O
the	O
entire	O
unit	O
square	O
.	O
</s>
<s>
Therefore	O
,	O
Peano	O
's	O
space-filling	B-Algorithm
curve	I-Algorithm
was	O
found	O
to	O
be	O
highly	O
counterintuitive	O
.	O
</s>
<s>
From	O
Peano	O
's	O
example	O
,	O
it	O
was	O
easy	O
to	O
deduce	O
continuous	O
curves	O
whose	O
ranges	O
contained	O
the	O
n-dimensional	O
hypercube	B-Operating_System
(	O
for	O
any	O
positive	O
integer	O
n	O
)	O
.	O
</s>
<s>
Most	O
well-known	O
space-filling	B-Algorithm
curves	I-Algorithm
are	O
constructed	O
iteratively	O
as	O
the	O
limit	O
of	O
a	O
sequence	O
of	O
piecewise	B-Algorithm
linear	O
continuous	O
curves	O
,	O
each	O
one	O
more	O
closely	O
approximating	O
the	O
space-filling	O
limit	O
.	O
</s>
<s>
Peano	O
's	O
ground-breaking	O
article	O
contained	O
no	O
illustrations	O
of	O
his	O
construction	O
,	O
which	O
is	O
defined	O
in	O
terms	O
of	O
ternary	B-Algorithm
expansions	I-Algorithm
and	O
a	O
mirroring	O
operator	O
.	O
</s>
<s>
Peano	O
's	O
article	O
also	O
ends	O
by	O
observing	O
that	O
the	O
technique	O
can	O
be	O
obviously	O
extended	O
to	O
other	O
odd	O
bases	O
besides	O
base	B-Algorithm
3	I-Algorithm
.	O
</s>
<s>
A	O
year	O
later	O
,	O
David	O
Hilbert	B-Algorithm
published	O
in	O
the	O
same	O
journal	O
a	O
variation	O
of	O
Peano	O
's	O
construction	O
.	O
</s>
<s>
Hilbert	B-Algorithm
's	O
article	O
was	O
the	O
first	O
to	O
include	O
a	O
picture	O
helping	O
to	O
visualize	O
the	O
construction	O
technique	O
,	O
essentially	O
the	O
same	O
as	O
illustrated	O
here	O
.	O
</s>
<s>
The	O
analytic	O
form	O
of	O
the	O
Hilbert	B-Algorithm
curve	I-Algorithm
,	O
however	O
,	O
is	O
more	O
complicated	O
than	O
Peano	O
's	O
.	O
</s>
<s>
(	O
The	O
restriction	O
of	O
the	O
Cantor	B-Algorithm
function	I-Algorithm
to	O
the	O
Cantor	O
set	O
is	O
an	O
example	O
of	O
such	O
a	O
function	O
.	O
)	O
</s>
<s>
Since	O
the	O
Cantor	O
set	O
is	O
homeomorphic	O
to	O
the	O
product	O
,	O
there	O
is	O
a	O
continuous	O
bijection	B-Algorithm
from	O
the	O
Cantor	O
set	O
onto	O
.	O
</s>
<s>
A	O
non-self-intersecting	O
continuous	O
curve	O
cannot	O
fill	O
the	O
unit	O
square	O
because	O
that	O
will	O
make	O
the	O
curve	O
a	O
homeomorphism	O
from	O
the	O
unit	O
interval	O
onto	O
the	O
unit	O
square	O
(	O
any	O
continuous	O
bijection	B-Algorithm
from	O
a	O
compact	O
space	O
onto	O
a	O
Hausdorff	O
space	O
is	O
a	O
homeomorphism	O
)	O
.	O
</s>
<s>
For	O
the	O
classic	O
Peano	O
and	O
Hilbert	B-Algorithm
space-filling	B-Algorithm
curves	I-Algorithm
,	O
where	O
two	O
subcurves	O
intersect	O
(	O
in	O
the	O
technical	O
sense	O
)	O
,	O
there	O
is	O
self-contact	O
without	O
self-crossing	O
.	O
</s>
<s>
A	O
space-filling	B-Algorithm
curve	I-Algorithm
can	O
be	O
(	O
everywhere	O
)	O
self-crossing	O
if	O
its	O
approximation	O
curves	O
are	O
self-crossing	O
.	O
</s>
<s>
A	O
space-filling	B-Algorithm
curve	I-Algorithm
's	O
approximations	O
can	O
be	O
self-avoiding	O
,	O
as	O
the	O
figures	O
above	O
illustrate	O
.	O
</s>
<s>
Space-filling	B-Algorithm
curves	I-Algorithm
are	O
special	O
cases	O
of	O
fractal	B-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
No	O
differentiable	O
space-filling	B-Algorithm
curve	I-Algorithm
can	O
exist	O
.	O
</s>
<s>
Michał	O
Morayne	O
proved	O
that	O
the	O
continuum	O
hypothesis	O
is	O
equivalent	O
to	O
the	O
existence	O
of	O
a	O
Peano	B-Algorithm
curve	I-Algorithm
such	O
that	O
at	O
each	O
point	O
of	O
real	O
line	O
at	O
least	O
one	O
of	O
its	O
components	O
is	O
differentiable	O
.	O
</s>
<s>
Spaces	O
that	O
are	O
the	O
continuous	O
image	O
of	O
a	O
unit	O
interval	O
are	O
sometimes	O
called	O
Peano	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Wiener	O
pointed	O
out	O
in	O
The	O
Fourier	O
Integral	O
and	O
Certain	O
of	O
its	O
Applications	O
that	O
space-filling	B-Algorithm
curves	I-Algorithm
could	O
be	O
used	O
to	O
reduce	O
Lebesgue	O
integration	O
in	O
higher	O
dimensions	O
to	O
Lebesgue	O
integration	O
in	O
one	O
dimension	O
.	O
</s>
