<s>
In	O
the	O
geometry	O
of	O
tessellations	B-Algorithm
,	O
a	O
rep-tile	B-Algorithm
or	O
reptile	O
is	O
a	O
shape	O
that	O
can	O
be	O
dissected	O
into	O
smaller	O
copies	O
of	O
the	O
same	O
shape	O
.	O
</s>
<s>
The	O
term	O
was	O
coined	O
as	O
a	O
pun	B-Application
on	O
animal	O
reptiles	O
by	O
recreational	O
mathematician	O
Solomon	O
W	O
.	O
Golomb	O
and	O
popularized	O
by	O
Martin	O
Gardner	O
in	O
his	O
"	O
Mathematical	O
Games	O
"	O
column	O
in	O
the	O
May	O
1963	O
issue	O
of	O
Scientific	O
American	O
.	O
</s>
<s>
In	O
2012	O
a	O
generalization	O
of	O
rep-tiles	B-Algorithm
called	O
self-tiling	O
tile	O
sets	O
was	O
introduced	O
by	O
Lee	O
Sallows	O
in	O
Mathematics	O
Magazine	O
.	O
</s>
<s>
A	O
rep-tile	B-Algorithm
is	O
labelled	O
rep-n	O
if	O
the	O
dissection	O
uses	O
n	O
copies	O
.	O
</s>
<s>
Such	O
a	O
shape	O
necessarily	O
forms	O
the	O
prototile	O
for	O
a	O
tiling	B-Algorithm
of	O
the	O
plane	O
,	O
in	O
many	O
cases	O
an	O
aperiodic	O
tiling	B-Algorithm
.	O
</s>
<s>
A	O
rep-tile	B-Algorithm
dissection	O
using	O
different	O
sizes	O
of	O
the	O
original	O
shape	O
is	O
called	O
an	O
irregular	O
rep-tile	B-Algorithm
or	O
irreptile	O
.	O
</s>
<s>
If	O
all	O
these	O
sub-tiles	O
are	O
of	O
different	O
sizes	O
then	O
the	O
tiling	B-Algorithm
is	O
additionally	O
described	O
as	O
perfect	O
.	O
</s>
<s>
The	O
order	O
of	O
a	O
shape	O
,	O
whether	O
using	O
rep-tiles	B-Algorithm
or	O
irrep-tiles	O
is	O
the	O
smallest	O
possible	O
number	O
of	O
tiles	O
which	O
will	O
suffice	O
.	O
</s>
<s>
The	O
Gosper	B-Algorithm
island	I-Algorithm
is	O
rep-7	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
is	O
irrep-7	O
:	O
six	O
small	O
snowflakes	O
of	O
the	O
same	O
size	O
,	O
together	O
with	O
another	O
snowflake	O
with	O
three	O
times	O
the	O
area	O
of	O
the	O
smaller	O
ones	O
,	O
can	O
combine	O
to	O
form	O
a	O
single	O
larger	O
snowflake	O
.	O
</s>
<s>
A	O
right	O
triangle	O
with	O
side	O
lengths	O
in	O
the	O
ratio	O
1:2	O
is	O
rep-5	O
,	O
and	O
its	O
rep-5	O
dissection	O
forms	O
the	O
basis	O
of	O
the	O
aperiodic	O
pinwheel	O
tiling	B-Algorithm
.	O
</s>
<s>
A	O
rectangle	O
(	O
or	O
parallelogram	O
)	O
is	O
rep-n	O
if	O
its	O
aspect	B-Device
ratio	I-Device
is	O
:1	O
.	O
</s>
<s>
Some	O
rep-tiles	B-Algorithm
,	O
like	O
the	O
square	O
and	O
equilateral	O
triangle	O
,	O
are	O
symmetrical	O
and	O
remain	O
identical	O
when	O
reflected	O
in	O
a	O
mirror	O
.	O
</s>
<s>
Dissection	O
of	O
the	O
sphinx	O
and	O
some	O
other	O
asymmetric	O
rep-tiles	B-Algorithm
requires	O
use	O
of	O
both	O
the	O
original	O
shape	O
and	O
its	O
mirror-image	O
.	O
</s>
<s>
Some	O
rep-tiles	B-Algorithm
are	O
based	O
on	O
polyforms	O
like	O
polyiamonds	O
and	O
polyominoes	O
,	O
or	O
shapes	O
created	O
by	O
laying	O
equilateral	O
triangles	O
and	O
squares	O
edge-to-edge	O
.	O
</s>
<s>
If	O
a	O
polyomino	O
is	O
rectifiable	O
,	O
that	O
is	O
,	O
able	O
to	O
tile	O
a	O
rectangle	O
,	O
then	O
it	O
will	O
also	O
be	O
a	O
rep-tile	B-Algorithm
,	O
because	O
the	O
rectangle	O
will	O
have	O
an	O
integer	O
side	O
length	O
ratio	O
and	O
will	O
thus	O
tile	O
a	O
square	O
.	O
</s>
<s>
Two	O
copies	O
of	O
some	O
octominoes	O
will	O
tile	O
a	O
square	O
;	O
therefore	O
these	O
octominoes	O
are	O
also	O
rep-16	O
rep-tiles	B-Algorithm
.	O
</s>
<s>
Four	O
copies	O
of	O
some	O
nonominoes	O
and	O
nonakings	O
will	O
tile	O
a	O
square	O
,	O
therefore	O
these	O
polyforms	O
are	O
also	O
rep-36	O
rep-tiles	B-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
if	O
a	O
polyiamond	O
tiles	O
an	O
equilateral	O
triangle	O
,	O
it	O
will	O
also	O
be	O
a	O
rep-tile	B-Algorithm
.	O
</s>
<s>
Two	O
particular	O
forms	O
of	O
right	O
triangle	O
have	O
attracted	O
the	O
attention	O
of	O
rep-tile	B-Algorithm
researchers	O
,	O
the	O
45°	O
-90	O
°	O
-45	O
°	O
triangle	O
and	O
the	O
30°	O
-60	O
°	O
-90	O
°	O
triangle	O
.	O
</s>
<s>
An	O
infinite	O
number	O
of	O
them	O
are	O
rep-tiles	B-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
the	O
simplest	O
of	O
all	O
rep-tiles	B-Algorithm
is	O
a	O
single	O
isosceles	O
right	O
triangle	O
.	O
</s>
<s>
Rep-2	O
rep-tiles	B-Algorithm
are	O
also	O
rep-2n	O
and	O
the	O
rep-4	O
,	O
8	O
,	O
16+	O
triangles	O
yield	O
further	O
rep-tiles	B-Algorithm
.	O
</s>
<s>
One	O
of	O
these	O
new	O
rep-tiles	B-Algorithm
is	O
reminiscent	O
of	O
the	O
fish	O
formed	O
from	O
three	O
equilateral	O
triangles	O
.	O
</s>
<s>
Many	O
of	O
the	O
common	O
rep-tiles	B-Algorithm
are	O
rep	O
-	O
for	O
all	O
positive	O
integer	O
values	O
of	O
.	O
</s>
<s>
In	O
particular	O
this	O
is	O
true	O
for	O
three	O
trapezoids	O
including	O
the	O
one	O
formed	O
from	O
three	O
equilateral	O
triangles	O
,	O
for	O
three	O
axis-parallel	O
hexagons	B-Application
(	O
the	O
L-tromino	O
,	O
L-tetromino	O
,	O
and	O
P-pentomino	O
)	O
,	O
and	O
the	O
sphinx	O
hexiamond	O
.	O
</s>
<s>
In	O
addition	O
,	O
many	O
rep-tiles	B-Algorithm
,	O
particularly	O
those	O
with	O
higher	O
rep-n	O
,	O
can	O
be	O
self-tiled	O
in	O
different	O
ways	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
rep-9	O
L-tetramino	O
has	O
at	O
least	O
fourteen	O
different	O
rep-tilings	B-Algorithm
.	O
</s>
<s>
The	O
most	O
familiar	O
rep-tiles	B-Algorithm
are	O
polygons	O
with	O
a	O
finite	O
number	O
of	O
sides	O
,	O
but	O
some	O
shapes	O
with	O
an	O
infinite	O
number	O
of	O
sides	O
can	O
also	O
be	O
rep-tiles	B-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
an	O
example	O
of	O
a	O
fractal	B-Application
rep-tile	B-Algorithm
.	O
</s>
<s>
Triangular	O
and	O
quadrilateral	O
(	O
four-sided	O
)	O
rep-tiles	B-Algorithm
are	O
common	O
,	O
but	O
pentagonal	O
rep-tiles	B-Algorithm
are	O
rare	O
.	O
</s>
<s>
These	O
pentagonal	O
rep-tiles	B-Algorithm
are	O
illustrated	O
on	O
the	O
pages	O
overseen	O
by	O
the	O
American	O
mathematician	O
Erich	O
Friedman	O
.	O
</s>
<s>
Rep-tiles	B-Algorithm
can	O
be	O
used	O
to	O
create	O
fractals	B-Application
,	O
or	O
shapes	O
that	O
are	O
self-similar	O
at	O
smaller	O
and	O
smaller	O
scales	O
.	O
</s>
<s>
A	O
rep-tile	B-Algorithm
fractal	B-Application
is	O
formed	O
by	O
subdividing	O
the	O
rep-tile	B-Algorithm
,	O
removing	O
one	O
or	O
more	O
copies	O
of	O
the	O
subdivided	O
shape	O
,	O
and	O
then	O
continuing	O
recursively	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
Sierpinski	O
carpet	O
is	O
formed	O
in	O
this	O
way	O
from	O
a	O
rep-tiling	B-Algorithm
of	O
a	O
square	O
into	O
27	O
smaller	O
squares	O
,	O
and	O
the	O
Sierpinski	O
triangle	O
is	O
formed	O
from	O
a	O
rep-tiling	B-Algorithm
of	O
an	O
equilateral	O
triangle	O
into	O
four	O
smaller	O
triangles	O
.	O
</s>
<s>
When	O
one	O
sub-copy	O
is	O
discarded	O
,	O
a	O
rep-4	O
L-triomino	O
can	O
be	O
used	O
to	O
create	O
four	O
fractals	B-Application
,	O
two	O
of	O
which	O
are	O
identical	O
except	O
for	O
orientation	O
.	O
</s>
<s>
Because	O
fractals	B-Application
are	O
self-similar	O
on	O
smaller	O
and	O
smaller	O
scales	O
,	O
many	O
may	O
be	O
decomposed	O
into	O
copies	O
of	O
themselves	O
like	O
a	O
rep-tile	B-Algorithm
.	O
</s>
<s>
However	O
,	O
if	O
the	O
fractal	B-Application
has	O
an	O
empty	O
interior	O
,	O
this	O
decomposition	O
may	O
not	O
lead	O
to	O
a	O
tiling	B-Algorithm
of	O
the	O
entire	O
plane	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Sierpinski	O
triangle	O
is	O
rep-3	O
,	O
tiled	O
with	O
three	O
copies	O
of	O
itself	O
,	O
and	O
the	O
Sierpinski	O
carpet	O
is	O
rep-8	O
,	O
tiled	O
with	O
eight	O
copies	O
of	O
itself	O
,	O
but	O
repetition	O
of	O
these	O
decompositions	O
does	O
not	O
form	O
a	O
tiling	B-Algorithm
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
dragon	B-Algorithm
curve	I-Algorithm
is	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
with	O
a	O
non-empty	O
interior	O
;	O
it	O
is	O
rep-4	O
,	O
and	O
does	O
form	O
a	O
tiling	B-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
the	O
Gosper	B-Algorithm
island	I-Algorithm
is	O
rep-7	O
,	O
formed	O
from	O
the	O
space-filling	O
Gosper	B-Algorithm
curve	I-Algorithm
,	O
and	O
again	O
forms	O
a	O
tiling	B-Algorithm
.	O
</s>
<s>
By	O
construction	O
,	O
any	O
fractal	B-Application
defined	O
by	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
of	O
n	O
contracting	O
maps	O
of	O
the	O
same	O
ratio	O
is	O
rep-n	O
.	O
</s>
<s>
However	O
,	O
a	O
regular	O
hexagon	B-Application
can	O
be	O
dissected	O
into	O
six	O
equilateral	O
triangles	O
,	O
each	O
of	O
which	O
can	O
be	O
dissected	O
into	O
a	O
regular	O
hexagon	B-Application
and	O
three	O
more	O
equilateral	O
triangles	O
.	O
</s>
<s>
This	O
is	O
the	O
basis	O
for	O
an	O
infinite	O
tiling	B-Algorithm
of	O
the	O
hexagon	B-Application
with	O
hexagons	B-Application
.	O
</s>
<s>
The	O
hexagon	B-Application
is	O
therefore	O
an	O
irrep-	O
∞	O
or	O
irrep-infinity	O
irreptile	O
.	O
</s>
