<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
(	O
also	O
known	O
as	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
,	O
Koch	B-Algorithm
star	I-Algorithm
,	O
or	O
Koch	B-Algorithm
island	I-Algorithm
)	O
is	O
a	O
fractal	B-Algorithm
curve	I-Algorithm
and	O
one	O
of	O
the	O
earliest	O
fractals	B-Application
to	O
have	O
been	O
described	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
,	O
which	O
appeared	O
in	O
a	O
1904	O
paper	O
titled	O
"	O
On	O
a	O
Continuous	O
Curve	O
Without	O
Tangents	O
,	O
Constructible	O
from	O
Elementary	O
Geometry	O
"	O
by	O
the	O
Swedish	O
mathematician	O
Helge	O
von	O
Koch	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
can	O
be	O
built	O
up	O
iteratively	O
,	O
in	O
a	O
sequence	O
of	O
stages	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
can	O
be	O
constructed	O
by	O
starting	O
with	O
an	O
equilateral	O
triangle	O
,	O
then	O
recursively	O
altering	O
each	O
line	O
segment	O
as	O
follows	O
:	O
</s>
<s>
The	O
first	O
iteration	B-Algorithm
of	O
this	O
process	O
produces	O
the	O
outline	O
of	O
a	O
hexagram	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
is	O
the	O
limit	O
approached	O
as	O
the	O
above	O
steps	O
are	O
followed	O
indefinitely	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
curve	I-Algorithm
originally	O
described	O
by	O
Helge	O
von	O
Koch	O
is	O
constructed	O
using	O
only	O
one	O
of	O
the	O
three	O
sides	O
of	O
the	O
original	O
triangle	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
three	O
Koch	B-Algorithm
curves	I-Algorithm
make	O
a	O
Koch	B-Algorithm
snowflake	I-Algorithm
.	O
</s>
<s>
A	O
Koch	B-Algorithm
curve	I-Algorithm
–	O
based	O
representation	O
of	O
a	O
nominally	O
flat	O
surface	O
can	O
similarly	O
be	O
created	O
by	O
repeatedly	O
segmenting	O
each	O
line	O
in	O
a	O
sawtooth	O
pattern	O
of	O
segments	O
with	O
a	O
given	O
angle	O
.	O
</s>
<s>
Each	O
iteration	B-Algorithm
multiplies	O
the	O
number	O
of	O
sides	O
in	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
by	O
four	O
,	O
so	O
the	O
number	O
of	O
sides	O
after	O
iterations	B-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
If	O
the	O
original	O
equilateral	O
triangle	O
has	O
sides	O
of	O
length	O
,	O
the	O
length	O
of	O
each	O
side	O
of	O
the	O
snowflake	O
after	O
iterations	B-Algorithm
is	O
:	O
</s>
<s>
The	O
perimeter	O
of	O
the	O
snowflake	O
after	O
iterations	B-Algorithm
is	O
:	O
</s>
<s>
The	O
Koch	B-Algorithm
curve	I-Algorithm
has	O
an	O
infinite	O
length	O
,	O
because	O
the	O
total	O
length	O
of	O
the	O
curve	O
increases	O
by	O
a	O
factor	O
of	O
with	O
each	O
iteration	B-Algorithm
.	O
</s>
<s>
Each	O
iteration	B-Algorithm
creates	O
four	O
times	O
as	O
many	O
line	O
segments	O
as	O
in	O
the	O
previous	O
iteration	B-Algorithm
,	O
with	O
the	O
length	O
of	O
each	O
one	O
being	O
the	O
length	O
of	O
the	O
segments	O
in	O
the	O
previous	O
stage	O
.	O
</s>
<s>
Hence	O
,	O
the	O
length	O
of	O
the	O
curve	O
after	O
iterations	B-Algorithm
will	O
be	O
times	O
the	O
original	O
triangle	O
perimeter	O
and	O
is	O
unbounded	O
,	O
as	O
tends	O
to	O
infinity	O
.	O
</s>
<s>
As	O
the	O
number	O
of	O
iterations	B-Algorithm
tends	O
to	O
infinity	O
,	O
the	O
limit	O
of	O
the	O
perimeter	O
is	O
:	O
</s>
<s>
In	O
each	O
iteration	B-Algorithm
a	O
new	O
triangle	O
is	O
added	O
on	O
each	O
side	O
of	O
the	O
previous	O
iteration	B-Algorithm
,	O
so	O
the	O
number	O
of	O
new	O
triangles	O
added	O
in	O
iteration	B-Algorithm
is	O
:	O
</s>
<s>
The	O
area	O
of	O
each	O
new	O
triangle	O
added	O
in	O
an	O
iteration	B-Algorithm
is	O
of	O
the	O
area	O
of	O
each	O
triangle	O
added	O
in	O
the	O
previous	O
iteration	B-Algorithm
,	O
so	O
the	O
area	O
of	O
each	O
triangle	O
added	O
in	O
iteration	B-Algorithm
is	O
:	O
</s>
<s>
The	O
total	O
new	O
area	O
added	O
in	O
iteration	B-Algorithm
is	O
therefore	O
:	O
</s>
<s>
The	O
total	O
area	O
of	O
the	O
snowflake	O
after	O
iterations	B-Algorithm
is	O
:	O
</s>
<s>
Thus	O
,	O
the	O
area	O
of	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
is	O
of	O
the	O
area	O
of	O
the	O
original	O
triangle	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
is	O
self-replicating	O
with	O
six	O
smaller	O
copies	O
surrounding	O
one	O
larger	O
copy	O
at	O
the	O
center	O
.	O
</s>
<s>
Hence	O
,	O
it	O
is	O
an	O
irrep-7	O
irrep-tile	O
(	O
see	O
Rep-tile	B-Algorithm
for	O
discussion	O
)	O
.	O
</s>
<s>
The	O
fractal	B-Application
dimension	O
of	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
is	O
.	O
</s>
<s>
This	O
is	O
greater	O
than	O
that	O
of	O
a	O
line	O
(	O
)	O
but	O
less	O
than	O
that	O
of	O
Peano	O
's	O
space-filling	B-Algorithm
curve	I-Algorithm
(	O
)	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
curve	I-Algorithm
is	O
continuous	O
everywhere	O
,	O
but	O
differentiable	O
nowhere	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
tessellate	B-Algorithm
the	O
plane	O
by	O
copies	O
of	O
Koch	B-Algorithm
snowflakes	I-Algorithm
in	O
two	O
different	O
sizes	O
.	O
</s>
<s>
However	O
,	O
such	O
a	O
tessellation	B-Algorithm
is	O
not	O
possible	O
using	O
only	O
snowflakes	O
of	O
one	O
size	O
.	O
</s>
<s>
Since	O
each	O
Koch	B-Algorithm
snowflake	I-Algorithm
in	O
the	O
tessellation	B-Algorithm
can	O
be	O
subdivided	O
into	O
seven	O
smaller	O
snowflakes	O
of	O
two	O
different	O
sizes	O
,	O
it	O
is	O
also	O
possible	O
to	O
find	O
tessellations	B-Algorithm
that	O
use	O
more	O
than	O
two	O
sizes	O
at	O
once	O
.	O
</s>
<s>
Koch	B-Algorithm
snowflakes	I-Algorithm
and	O
Koch	B-Algorithm
antisnowflakes	I-Algorithm
of	O
the	O
same	O
size	O
may	O
be	O
used	O
to	O
tile	O
the	O
plane	O
.	O
</s>
<s>
A	O
turtle	B-Application
graphic	I-Application
is	O
the	O
curve	O
that	O
is	O
generated	O
if	O
an	O
automaton	O
is	O
programmed	O
with	O
a	O
sequence	O
.	O
</s>
<s>
the	O
resulting	O
curve	O
converges	O
to	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
.	O
</s>
<s>
The	O
Koch	B-Algorithm
curve	I-Algorithm
can	O
be	O
expressed	O
by	O
the	O
following	O
rewrite	O
system	O
(	O
Lindenmayer	B-Language
system	I-Language
)	O
:	O
</s>
<s>
To	O
create	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
,	O
one	O
would	O
use	O
F--F--F	O
(	O
an	O
equilateral	O
triangle	O
)	O
as	O
the	O
axiom	O
.	O
</s>
<s>
Variant	O
(	O
dimension	O
,	O
angle	O
)	O
Illustration	O
Construction	O
≤1D	O
,	O
60-90	O
°	O
angle	O
150px	O
Cesàro	O
fractal	B-Application
(	O
85°	O
)	O
The	O
Cesàro	O
fractal	B-Application
is	O
a	O
variant	O
of	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
with	O
an	O
angle	O
between	O
60°	O
and	O
90°	O
.	O
</s>
<s>
450px	O
First	O
four	O
iterations	B-Algorithm
of	O
a	O
Cesàro	O
antisnowflake	O
(	O
four	O
60°	O
curves	O
arranged	O
in	O
a	O
90°	O
square	O
)	O
≈	O
1.46D	O
,	O
90°	O
angle	O
150px	O
Quadratic	O
type	O
1	O
curve	O
450px	O
First	O
two	O
iterations	B-Algorithm
1.5D	O
,	O
90°	O
angle	O
150px	O
Quadratic	O
type	O
2	O
curve	O
Minkowski	O
SausagePaul	O
S	O
.	O
Addison	O
,	O
Fractals	B-Application
and	O
Chaos	O
:	O
An	O
illustrated	O
course	O
,	O
p	O
.	O
19	O
,	O
CRC	O
Press	O
,	O
1997	O
.	O
</s>
<s>
450px	O
First	O
two	O
iterations	B-Algorithm
.	O
</s>
<s>
Its	O
fractal	B-Application
dimension	O
equals	O
and	O
is	O
exactly	O
half-way	O
between	O
dimension	O
1	O
and	O
2	O
.	O
</s>
<s>
It	O
is	O
therefore	O
often	O
chosen	O
when	O
studying	O
the	O
physical	O
properties	O
of	O
non-integer	O
fractal	B-Application
objects	O
.	O
</s>
<s>
≤2D	O
,	O
90°	O
angle	O
150px	O
Third	O
iteration	B-Algorithm
Minkowski	B-Algorithm
Island	I-Algorithm
450px	O
Four	O
quadratic	O
type	O
2	O
curves	O
arranged	O
in	O
a	O
square	O
≈	O
1.37D	O
,	O
90°	O
angle	O
150px	O
Quadratic	O
flake	O
450px	O
4	O
quadratic	O
type	O
1	O
curves	O
arranged	O
in	O
a	O
polygon	O
:	O
First	O
two	O
iterations	B-Algorithm
.	O
</s>
<s>
Known	O
as	O
the	O
"	O
Minkowski	B-Algorithm
Sausage	I-Algorithm
"	O
,	O
Weisstein	O
,	O
Eric	O
W	O
.	O
(	O
1999	O
)	O
.	O
</s>
<s>
"	O
Minkowski	B-Algorithm
Sausage	I-Algorithm
"	O
,	O
archive.lib.msu.edu	O
.	O
</s>
<s>
"	O
Minkowski	B-Algorithm
Sausage	I-Algorithm
"	O
,	O
user.math.uoc.gr/~pamfilos/	O
.	O
</s>
<s>
Accessed	O
:	O
21	O
September	O
2019	O
.	O
its	O
fractal	B-Application
dimension	O
equals	O
.Mandelbrot	O
,	O
B	O
.	O
</s>
<s>
The	O
Fractal	B-Application
Geometry	O
of	O
Nature	O
,	O
p.48	O
.	O
</s>
<s>
≤2D	O
,	O
90°	O
angle	O
150px	O
Quadratic	O
antiflake	O
Anticross-stitch	B-Algorithm
curve	I-Algorithm
,	O
the	O
quadratic	O
flake	O
type	O
1	O
,	O
with	O
the	O
curves	O
facing	O
inwards	O
instead	O
of	O
outwards	O
(	O
Vicsek	B-Algorithm
fractal	I-Algorithm
)	O
≈	O
1.49D	O
,	O
90°	O
angle	O
150px	O
Quadratic	O
Cross	O
Another	O
variation	O
.	O
</s>
<s>
Its	O
fractal	B-Application
dimension	O
equals	O
.	O
</s>
<s>
Introducing	O
Fractal	B-Application
Geometry	O
.	O
</s>
<s>
450px	O
Quadratic	O
curve	O
,	O
iterations	B-Algorithm
0	O
,	O
1	O
,	O
and	O
2	O
;	O
dimension	O
of	O
≤2D	O
,	O
60°	O
angle	O
150px	O
von	O
Koch	O
surface	O
450px	O
First	O
three	O
iterations	B-Algorithm
of	O
a	O
natural	O
extension	O
of	O
the	O
Koch	B-Algorithm
curve	I-Algorithm
in	O
two	O
dimensions	O
.	O
</s>
<s>
≤2D	O
,	O
90°	O
angle	O
150px	O
First	O
(	O
blue	O
block	O
)	O
,	O
second	O
(	O
plus	O
green	O
blocks	O
)	O
,	O
third	O
(	O
plus	O
yellow	O
blocks	O
)	O
and	O
fourth	O
(	O
plus	O
transparent	O
blocks	O
)	O
iterations	B-Algorithm
of	O
the	O
type	O
1	O
3D	O
Koch	O
quadratic	O
fractal	B-Application
Extension	O
of	O
the	O
quadratic	O
type	O
1	O
curve	O
.	O
</s>
<s>
The	O
illustration	O
at	O
left	O
shows	O
the	O
fractal	B-Application
after	O
the	O
second	O
iteration	B-Algorithm
300px	O
Animation	O
quadratic	O
surface	O
≤3D	O
,	O
any	O
150px	O
Koch	B-Algorithm
curve	I-Algorithm
in	O
3D	O
A	O
three-dimensional	O
fractal	B-Application
constructed	O
from	O
Koch	B-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
The	O
shape	O
can	O
be	O
considered	O
a	O
three-dimensional	O
extension	O
of	O
the	O
curve	O
in	O
the	O
same	O
sense	O
that	O
the	O
Sierpiński	O
pyramid	O
and	O
Menger	B-Algorithm
sponge	I-Algorithm
can	O
be	O
considered	O
extensions	O
of	O
the	O
Sierpinski	O
triangle	O
and	O
Sierpinski	O
carpet	O
.	O
</s>
<s>
Squares	O
can	O
be	O
used	O
to	O
generate	O
similar	O
fractal	B-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
Starting	O
with	O
a	O
unit	O
square	O
and	O
adding	O
to	O
each	O
side	O
at	O
each	O
iteration	B-Algorithm
a	O
square	O
with	O
dimension	O
one	O
third	O
of	O
the	O
squares	O
in	O
the	O
previous	O
iteration	B-Algorithm
,	O
it	O
can	O
be	O
shown	O
that	O
both	O
the	O
length	O
of	O
the	O
perimeter	O
and	O
the	O
total	O
area	O
are	O
determined	O
by	O
geometric	O
progressions	O
.	O
</s>
<s>
The	O
progression	O
for	O
the	O
area	O
converges	O
to	O
while	O
the	O
progression	O
for	O
the	O
perimeter	O
diverges	O
to	O
infinity	O
,	O
so	O
as	O
in	O
the	O
case	O
of	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
,	O
we	O
have	O
a	O
finite	O
area	O
bounded	O
by	O
an	O
infinite	O
fractal	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
total	O
area	O
covered	O
at	O
the	O
th	O
iteration	B-Algorithm
is	O
:	O
</s>
