<s>
In	O
mathematics	O
,	O
a	O
finite	B-Algorithm
subdivision	I-Algorithm
rule	I-Algorithm
is	O
a	O
recursive	O
way	O
of	O
dividing	O
a	O
polygon	B-General_Concept
or	O
other	O
two-dimensional	O
shape	O
into	O
smaller	O
and	O
smaller	O
pieces	O
.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
in	O
a	O
sense	O
are	O
generalizations	O
of	O
regular	O
geometric	O
fractals	B-Application
.	O
</s>
<s>
Instead	O
of	O
repeating	O
exactly	O
the	O
same	O
design	O
over	O
and	O
over	O
,	O
they	O
have	O
slight	O
variations	O
in	O
each	O
stage	O
,	O
allowing	O
a	O
richer	O
structure	O
while	O
maintaining	O
the	O
elegant	O
style	O
of	O
fractals	B-Application
.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
have	O
been	O
used	O
in	O
architecture	O
,	O
biology	O
,	O
and	O
computer	O
science	O
,	O
as	O
well	O
as	O
in	O
the	O
study	O
of	O
hyperbolic	O
manifolds	O
.	O
</s>
<s>
Substitution	O
tilings	O
are	O
a	O
well-studied	O
type	O
of	O
subdivision	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
A	O
subdivision	B-Algorithm
rule	I-Algorithm
takes	O
a	O
tiling	O
of	O
the	O
plane	O
by	O
polygons	B-General_Concept
and	O
turns	O
it	O
into	O
a	O
new	O
tiling	O
by	O
subdividing	O
each	O
polygon	B-General_Concept
into	O
smaller	O
polygons	B-General_Concept
.	O
</s>
<s>
It	O
is	O
finite	O
if	O
there	O
are	O
only	O
finitely	O
many	O
ways	O
that	O
every	O
polygon	B-General_Concept
can	O
subdivide	O
.	O
</s>
<s>
Finite	B-Algorithm
subdivision	I-Algorithm
rules	I-Algorithm
can	O
only	O
subdivide	O
tilings	O
that	O
are	O
made	O
up	O
of	O
polygons	B-General_Concept
labelled	O
by	O
tile	O
types	O
.	O
</s>
<s>
Such	O
tilings	O
are	O
called	O
subdivision	O
complexes	O
for	O
the	O
subdivision	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Given	O
any	O
subdivision	O
complex	O
for	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
,	O
we	O
can	O
subdivide	O
it	O
over	O
and	O
over	O
again	O
to	O
get	O
a	O
sequence	O
of	O
tilings	O
.	O
</s>
<s>
Barycentric	O
subdivision	O
is	O
an	O
example	O
of	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
with	O
one	O
edge	O
type	O
(	O
that	O
gets	O
subdivided	O
into	O
two	O
edges	O
)	O
and	O
one	O
tile	O
type	O
(	O
a	O
triangle	O
that	O
gets	O
subdivided	O
into	O
6	O
smaller	O
triangles	O
)	O
.	O
</s>
<s>
The	O
Penrose	O
tiling	O
can	O
be	O
generated	O
by	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
on	O
a	O
set	O
of	O
four	O
tile	O
types	O
(	O
the	O
curved	O
lines	O
in	O
the	O
table	O
below	O
only	O
help	O
to	O
show	O
how	O
the	O
tiles	O
fit	O
together	O
)	O
:	O
</s>
<s>
Certain	O
rational	O
maps	O
give	O
rise	O
to	O
finite	B-Algorithm
subdivision	I-Algorithm
rules	I-Algorithm
.	O
</s>
<s>
Every	O
prime	O
,	O
non-split	O
alternating	O
knot	O
or	O
link	O
complement	O
has	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
,	O
with	O
some	O
tiles	O
that	O
do	O
not	O
subdivide	O
,	O
corresponding	O
to	O
the	O
boundary	O
of	O
the	O
link	O
complement	O
.	O
</s>
<s>
The	O
subdivision	B-Algorithm
rules	I-Algorithm
show	O
what	O
the	O
night	O
sky	O
would	O
look	O
like	O
to	O
someone	O
living	O
in	O
a	O
knot	O
complement	O
;	O
because	O
the	O
universe	O
wraps	O
around	O
itself	O
(	O
i.e.	O
</s>
<s>
The	O
subdivision	B-Algorithm
rule	I-Algorithm
describes	O
that	O
pattern	O
.	O
</s>
<s>
The	O
subdivision	B-Algorithm
rule	I-Algorithm
looks	O
different	O
for	O
different	O
geometries	O
.	O
</s>
<s>
This	O
is	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
for	O
the	O
trefoil	O
knot	O
,	O
which	O
is	O
not	O
a	O
hyperbolic	O
knot	O
:	O
</s>
<s>
And	O
this	O
is	O
the	O
subdivision	B-Algorithm
rule	I-Algorithm
for	O
the	O
Borromean	O
rings	O
,	O
which	O
is	O
hyperbolic	O
:	O
</s>
<s>
In	O
each	O
case	O
,	O
the	O
subdivision	B-Algorithm
rule	I-Algorithm
would	O
act	O
on	O
some	O
tiling	O
of	O
a	O
sphere	O
(	O
i.e.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
can	O
easily	O
be	O
generalized	O
to	O
other	O
dimensions	O
.	O
</s>
<s>
Also	O
,	O
binary	O
subdivision	O
can	O
be	O
generalized	O
to	O
other	O
dimensions	O
(	O
where	O
hypercubes	B-Operating_System
get	O
divided	O
by	O
every	O
midplane	O
)	O
,	O
as	O
in	O
the	O
proof	O
of	O
the	O
Heine	O
–	O
Borel	O
theorem	O
.	O
</s>
<s>
A	O
finite	B-Algorithm
subdivision	I-Algorithm
rule	I-Algorithm
consists	O
of	O
the	O
following	O
.	O
</s>
<s>
An	O
-complex	O
for	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
is	O
a	O
2-dimensional	O
CW	O
complex	O
which	O
is	O
the	O
union	O
of	O
its	O
closed	O
2-cells	O
,	O
together	O
with	O
a	O
continuous	O
cellular	O
map	O
whose	O
restriction	O
to	O
each	O
open	O
cell	O
is	O
a	O
homeomorphism	O
.	O
</s>
<s>
The	O
plane	O
,	O
tiled	O
by	O
squares	O
,	O
is	O
a	O
subdivision	O
complex	O
for	O
this	O
subdivision	B-Algorithm
rule	I-Algorithm
,	O
with	O
the	O
structure	O
map	O
given	O
by	O
the	O
standard	O
covering	O
map	O
.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
can	O
be	O
used	O
to	O
study	O
the	O
quasi-isometry	O
properties	O
of	O
certain	O
spaces	O
.	O
</s>
<s>
Given	O
a	O
subdivision	B-Algorithm
rule	I-Algorithm
and	O
subdivision	O
complex	O
,	O
we	O
can	O
construct	O
a	O
graph	O
called	O
the	O
history	O
graph	O
that	O
records	O
the	O
action	O
of	O
the	O
subdivision	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
The	O
quasi-isometry	O
properties	O
of	O
the	O
history	O
graph	O
can	O
be	O
studied	O
using	O
subdivision	B-Algorithm
rules	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
history	O
graph	O
is	O
quasi-isometric	O
to	O
hyperbolic	O
space	O
exactly	O
when	O
the	O
subdivision	B-Algorithm
rule	I-Algorithm
is	O
conformal	O
,	O
as	O
described	O
in	O
the	O
combinatorial	B-Algorithm
Riemann	I-Algorithm
mapping	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
Islamic	O
Girih	O
tiles	O
in	O
Islamic	O
architecture	O
are	O
self-similar	O
tilings	O
that	O
can	O
be	O
modeled	O
with	O
finite	B-Algorithm
subdivision	I-Algorithm
rules	I-Algorithm
.	O
</s>
<s>
In	O
2007	O
,	O
Peter	O
J	O
.	O
Lu	O
of	O
Harvard	O
University	O
and	O
Professor	O
Paul	O
J	O
.	O
Steinhardt	O
of	O
Princeton	O
University	O
published	O
a	O
paper	O
in	O
the	O
journal	O
Science	O
suggesting	O
that	O
girih	O
tilings	O
possessed	O
properties	O
consistent	O
with	O
self-similar	O
fractal	B-Application
quasicrystalline	O
tilings	O
such	O
as	O
Penrose	O
tilings	O
(	O
presentation	O
1974	O
,	O
predecessor	O
works	O
starting	O
in	O
about	O
1964	O
)	O
predating	O
them	O
by	O
five	O
centuries	O
.	O
</s>
<s>
Subdivision	O
surfaces	O
in	O
computer	O
graphics	O
use	O
subdivision	B-Algorithm
rules	I-Algorithm
to	O
refine	O
a	O
surface	O
to	O
any	O
given	O
level	O
of	O
precision	O
.	O
</s>
<s>
These	O
subdivision	O
surfaces	O
(	O
such	O
as	O
the	O
Catmull-Clark	O
subdivision	O
surface	O
)	O
take	O
a	O
polygon	B-Algorithm
mesh	I-Algorithm
(	O
the	O
kind	O
used	O
in	O
3D	O
animated	O
movies	O
)	O
and	O
refines	O
it	O
to	O
a	O
mesh	B-Algorithm
with	O
more	O
polygons	B-General_Concept
by	O
adding	O
and	O
shifting	O
points	O
according	O
to	O
different	O
recursive	O
formulas	O
.	O
</s>
<s>
Although	O
many	O
points	O
get	O
shifted	O
in	O
this	O
process	O
,	O
each	O
new	O
mesh	B-Algorithm
is	O
combinatorially	O
a	O
subdivision	O
of	O
the	O
old	O
mesh	B-Algorithm
(	O
meaning	O
that	O
for	O
every	O
edge	O
and	O
vertex	O
of	O
the	O
old	O
mesh	B-Algorithm
,	O
you	O
can	O
identify	O
a	O
corresponding	O
edge	O
and	O
vertex	O
in	O
the	O
new	O
one	O
,	O
plus	O
several	O
more	O
edges	O
and	O
vertices	O
)	O
.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
were	O
applied	O
by	O
Cannon	O
,	O
Floyd	O
and	O
Parry	O
(	O
2000	O
)	O
to	O
the	O
study	O
of	O
large-scale	O
growth	O
patterns	O
of	O
biological	O
organisms	O
.	O
</s>
<s>
Cannon	O
,	O
Floyd	O
and	O
Parry	O
produced	O
a	O
mathematical	O
growth	O
model	O
which	O
demonstrated	O
that	O
some	O
systems	O
determined	O
by	O
simple	O
finite	B-Algorithm
subdivision	I-Algorithm
rules	I-Algorithm
can	O
results	O
in	O
objects	O
(	O
in	O
their	O
example	O
,	O
a	O
tree	O
trunk	O
)	O
whose	O
large-scale	O
form	O
oscillates	O
wildly	O
over	O
time	O
even	O
though	O
the	O
local	O
subdivision	O
laws	O
remain	O
the	O
same	O
.	O
</s>
<s>
They	O
suggested	O
that	O
the	O
"	O
negatively	O
curved	O
"	O
(	O
or	O
non-euclidean	O
)	O
nature	O
of	O
microscopic	O
growth	O
patterns	O
of	O
biological	O
organisms	O
is	O
one	O
of	O
the	O
key	O
reasons	O
why	O
large-scale	O
organisms	O
do	O
not	O
look	O
like	O
crystals	O
or	O
polyhedral	O
shapes	O
but	O
in	O
fact	O
in	O
many	O
cases	O
resemble	O
self-similar	O
fractals	B-Application
.	O
</s>
<s>
Cannon	O
,	O
Floyd	O
,	O
and	O
Parry	O
first	O
studied	O
finite	B-Algorithm
subdivision	I-Algorithm
rules	I-Algorithm
in	O
an	O
attempt	O
to	O
prove	O
the	O
following	O
conjecture	O
:	O
</s>
<s>
Cannon	O
and	O
Swenson	O
showed	O
that	O
a	O
hyperbolic	O
group	O
with	O
a	O
2-sphere	O
at	O
infinity	O
has	O
an	O
associated	O
subdivision	B-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
If	O
this	O
subdivision	B-Algorithm
rule	I-Algorithm
is	O
conformal	O
in	O
a	O
certain	O
sense	O
,	O
the	O
group	O
will	O
be	O
a	O
3-manifold	O
group	O
with	O
the	O
geometry	O
of	O
hyperbolic	O
3-space	O
.	O
</s>
<s>
Subdivision	B-Algorithm
rules	I-Algorithm
give	O
a	O
sequence	O
of	O
tilings	O
of	O
a	O
surface	O
,	O
and	O
tilings	O
give	O
an	O
idea	O
of	O
distance	O
,	O
length	O
,	O
and	O
area	O
(	O
by	O
letting	O
each	O
tile	O
have	O
length	O
and	O
area	O
1	O
)	O
.	O
</s>
<s>
The	O
Combinatorial	B-Algorithm
Riemann	I-Algorithm
Mapping	I-Algorithm
Theorem	I-Algorithm
gives	O
necessary	O
and	O
sufficient	O
conditions	O
for	O
this	O
to	O
occur	O
.	O
</s>
<s>
A	O
sequence	O
of	O
tilings	O
is	O
conformal	O
(	O
)	O
if	O
mesh	B-Algorithm
approaches	O
0	O
and	O
:	O
</s>
<s>
The	O
Combinatorial	B-Algorithm
Riemann	I-Algorithm
Mapping	I-Algorithm
Theorem	I-Algorithm
implies	O
that	O
a	O
group	O
acts	O
geometrically	O
on	O
if	O
and	O
only	O
if	O
it	O
is	O
Gromov	O
hyperbolic	O
,	O
it	O
has	O
a	O
sphere	O
at	O
infinity	O
,	O
and	O
the	O
natural	O
subdivision	B-Algorithm
rule	I-Algorithm
on	O
the	O
sphere	O
gives	O
rise	O
to	O
a	O
sequence	O
of	O
tilings	O
that	O
is	O
conformal	O
in	O
the	O
sense	O
above	O
.	O
</s>
<s>
Thus	O
,	O
Cannon	O
's	O
conjecture	O
would	O
be	O
true	O
if	O
all	O
such	O
subdivision	B-Algorithm
rules	I-Algorithm
were	O
conformal	O
.	O
</s>
