<s>
In	O
geometry	O
,	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
is	O
the	O
first	O
example	O
of	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
to	O
be	O
discovered	O
,	O
by	O
Giuseppe	O
Peano	O
in	O
1890	O
.	O
</s>
<s>
Peano	B-Algorithm
's	I-Algorithm
curve	I-Algorithm
is	O
a	O
surjective	B-Algorithm
,	O
continuous	O
function	O
from	O
the	O
unit	O
interval	O
onto	O
the	O
unit	O
square	O
,	O
however	O
it	O
is	O
not	O
injective	O
.	O
</s>
<s>
Peano	O
was	O
motivated	O
by	O
an	O
earlier	O
result	O
of	O
Georg	O
Cantor	O
that	O
these	O
two	O
sets	O
have	O
the	O
same	O
cardinality	B-Application
.	O
</s>
<s>
Because	O
of	O
this	O
example	O
,	O
some	O
authors	O
use	O
the	O
phrase	O
"	O
Peano	B-Algorithm
curve	I-Algorithm
"	O
to	O
refer	O
more	O
generally	O
to	O
any	O
space-filling	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
Peano	B-Algorithm
's	I-Algorithm
curve	I-Algorithm
may	O
be	O
constructed	O
by	O
a	O
sequence	O
of	O
steps	O
,	O
where	O
the	O
ith	O
step	O
constructs	O
a	O
set	O
Si	O
of	O
squares	O
,	O
and	O
a	O
sequence	O
Pi	O
of	O
the	O
centers	O
of	O
the	O
squares	O
,	O
from	O
the	O
set	O
and	O
sequence	O
constructed	O
in	O
the	O
previous	O
step	O
.	O
</s>
<s>
The	O
Peano	B-Algorithm
curve	I-Algorithm
itself	O
is	O
the	O
limit	B-Algorithm
of	O
the	O
curves	O
through	O
the	O
sequences	O
of	O
square	O
centers	O
,	O
as	O
i	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
In	O
the	O
definition	O
of	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
perform	O
some	O
or	O
all	O
of	O
the	O
steps	O
by	O
making	O
the	O
centers	O
of	O
each	O
row	O
of	O
three	O
squares	O
be	O
contiguous	O
,	O
rather	O
than	O
the	O
centers	O
of	O
each	O
column	O
of	O
squares	O
.	O
</s>
<s>
These	O
choices	O
lead	O
to	O
many	O
different	O
variants	O
of	O
the	O
Peano	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
curve	I-Algorithm
is	O
a	O
simpler	O
variant	O
of	O
the	O
same	O
idea	O
,	O
based	O
on	O
subdividing	O
squares	O
into	O
four	O
equal	O
smaller	O
squares	O
instead	O
of	O
into	O
nine	O
equal	O
smaller	O
squares	O
.	O
</s>
