<s>
An	O
n-flake	B-Algorithm
,	O
polyflake	O
,	O
or	O
Sierpinski	B-Algorithm
n-gon	I-Algorithm
,	O
is	O
a	O
fractal	B-Application
constructed	O
starting	O
from	O
an	O
n-gon	O
.	O
</s>
<s>
This	O
process	O
is	O
repeated	O
recursively	O
to	O
result	O
in	O
the	O
fractal	B-Application
.	O
</s>
<s>
The	O
most	O
common	O
variety	O
of	O
n-flake	B-Algorithm
is	O
two-dimensional	O
(	O
in	O
terms	O
of	O
its	O
topological	O
dimension	O
)	O
and	O
is	O
formed	O
of	O
polygons	O
.	O
</s>
<s>
Its	O
boundary	O
is	O
the	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
of	O
varying	O
types	O
depending	O
on	O
the	O
n-gon	O
and	O
infinitely	O
many	O
Koch	B-Algorithm
curves	I-Algorithm
are	O
contained	O
within	O
.	O
</s>
<s>
The	O
fractals	B-Application
occupy	O
zero	O
area	O
yet	O
have	O
an	O
infinite	O
perimeter	O
.	O
</s>
<s>
The	O
formula	O
of	O
the	O
scale	B-Algorithm
factor	I-Algorithm
r	O
for	O
any	O
n-flake	B-Algorithm
is	O
:	O
</s>
<s>
The	O
Hausdorff	O
dimension	O
of	O
a	O
n-flake	B-Algorithm
is	O
,	O
where	O
m	O
is	O
the	O
number	O
of	O
polygons	O
in	O
each	O
individual	O
flake	O
and	O
r	O
is	O
the	O
scale	B-Algorithm
factor	I-Algorithm
.	O
</s>
<s>
The	O
Sierpinski	O
triangle	O
is	O
an	O
n-flake	B-Algorithm
formed	O
by	O
successive	O
flakes	O
of	O
three	O
triangles	O
.	O
</s>
<s>
If	O
a	O
sierpinski	O
4-gon	O
were	O
constructed	O
from	O
the	O
given	O
definition	O
,	O
the	O
scale	B-Algorithm
factor	I-Algorithm
would	O
be	O
1/2	O
and	O
the	O
fractal	B-Application
would	O
simply	O
be	O
a	O
square	O
.	O
</s>
<s>
A	O
more	O
interesting	O
alternative	O
,	O
the	O
Vicsek	B-Algorithm
fractal	I-Algorithm
,	O
rarely	O
called	O
a	O
quadraflake	O
,	O
is	O
formed	O
by	O
successive	O
flakes	O
of	O
five	O
squares	O
scaled	O
by	O
1/3	O
.	O
</s>
<s>
The	O
boundary	O
of	O
the	O
Vicsek	B-Algorithm
Fractal	I-Algorithm
is	O
a	O
Type	O
1	O
quadratic	O
Koch	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
boundary	O
of	O
a	O
pentaflake	O
is	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
of	O
72	O
degrees	O
.	O
</s>
<s>
This	O
variation	O
still	O
contains	O
infinitely	O
many	O
Koch	B-Algorithm
curves	I-Algorithm
,	O
but	O
they	O
are	O
somewhat	O
more	O
visible	O
.	O
</s>
<s>
Concentric	O
patterns	O
of	O
pentaflake	O
boundary	O
shaped	O
tiles	O
can	O
cover	O
the	O
plane	O
,	O
with	O
the	O
central	O
point	O
being	O
covered	O
by	O
a	O
third	O
shape	O
formed	O
of	O
segments	O
of	O
72-degree	O
Koch	B-Algorithm
curve	I-Algorithm
,	O
also	O
with	O
5-fold	O
rotational	O
and	O
reflective	O
symmetry	O
.	O
</s>
<s>
The	O
boundary	O
of	O
a	O
hexaflake	O
is	O
the	O
standard	O
Koch	B-Algorithm
curve	I-Algorithm
of	O
60	O
degrees	O
and	O
infinitely	O
many	O
Koch	B-Algorithm
snowflakes	I-Algorithm
are	O
contained	O
within	O
.	O
</s>
<s>
The	O
hexaflake	O
has	O
been	O
applied	O
in	O
the	O
design	O
of	O
antennas	O
and	O
optical	B-Architecture
fibers	I-Architecture
.	O
</s>
<s>
This	O
variation	O
still	O
contains	O
infinitely	O
many	O
Koch	B-Algorithm
curves	I-Algorithm
of	O
60	O
degrees	O
.	O
</s>
<s>
n-flakes	B-Algorithm
of	O
higher	O
polygons	O
also	O
exist	O
,	O
though	O
they	O
are	O
less	O
common	O
and	O
do	O
n't	O
usually	O
have	O
a	O
central	O
polygon	O
.	O
</s>
<s>
While	O
it	O
may	O
not	O
be	O
obvious	O
,	O
these	O
higher	O
polyflakes	O
still	O
contain	O
infinitely	O
many	O
Koch	B-Algorithm
curves	I-Algorithm
,	O
but	O
the	O
angle	O
of	O
the	O
Koch	B-Algorithm
curves	I-Algorithm
decreases	O
as	O
n	O
increases	O
.	O
</s>
<s>
Their	O
Hausdorff	O
dimensions	O
are	O
slightly	O
more	O
difficult	O
to	O
calculate	O
than	O
lower	O
n-flakes	B-Algorithm
because	O
their	O
scale	B-Algorithm
factor	I-Algorithm
is	O
less	O
obvious	O
.	O
</s>
<s>
An	O
interesting	O
n-flake	B-Algorithm
is	O
the	O
∞	O
-flake	O
,	O
because	O
as	O
the	O
value	O
of	O
n	O
increases	O
,	O
an	O
n-flake	B-Algorithm
'	O
s	O
Hausdorff	O
dimension	O
approaches	O
1	O
,	O
</s>
<s>
n-flakes	B-Algorithm
can	O
generalized	O
to	O
higher	O
dimensions	O
,	O
in	O
particular	O
to	O
a	O
topological	O
dimension	O
of	O
three	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
three-dimensional	O
n-flakes	B-Algorithm
are	O
also	O
called	O
platonic	O
solid	O
fractals	B-Application
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
the	O
fractals	B-Application
 '	O
volume	O
is	O
zero	O
.	O
</s>
<s>
A	O
hexahedron	O
,	O
or	O
cube	O
,	O
flake	O
defined	O
in	O
the	O
same	O
way	O
as	O
the	O
Sierpinski	O
tetrahedron	O
is	O
simply	O
a	O
cube	O
and	O
is	O
not	O
interesting	O
as	O
a	O
fractal	B-Application
.	O
</s>
<s>
One	O
is	O
the	O
Menger	B-Algorithm
Sponge	I-Algorithm
,	O
where	O
every	O
cube	O
is	O
replaced	O
by	O
a	O
three	O
dimensional	O
ring	O
of	O
cubes	O
.	O
</s>
<s>
Another	O
hexahedron	O
flake	O
can	O
be	O
produced	O
in	O
a	O
manner	O
similar	O
to	O
the	O
Vicsek	B-Algorithm
fractal	I-Algorithm
extended	O
to	O
three	O
dimensions	O
.	O
</s>
<s>
Every	O
cube	O
is	O
divided	O
into	O
27	O
smaller	O
cubes	O
and	O
the	O
center	O
cross	O
is	O
retained	O
,	O
which	O
is	O
the	O
opposite	O
of	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
where	O
the	O
cross	O
is	O
removed	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
not	O
the	O
Menger	B-Algorithm
Sponge	I-Algorithm
complement	O
.	O
</s>
