<s>
According	O
to	O
Benoit	O
Mandelbrot	O
,	O
"	O
A	O
fractal	B-Application
is	O
by	O
definition	O
a	O
set	O
for	O
which	O
the	O
Hausdorff-Besicovitch	O
dimension	O
strictly	O
exceeds	O
the	O
topological	O
dimension.	O
"	O
</s>
<s>
Presented	O
here	O
is	O
a	O
list	B-Library
of	I-Library
fractals	I-Library
,	O
ordered	O
by	O
increasing	O
Hausdorff	O
dimension	O
,	O
to	O
illustrate	O
what	O
it	O
means	O
for	O
a	O
fractal	B-Application
to	O
have	O
a	O
low	O
or	O
a	O
high	O
dimension	O
.	O
</s>
<s>
Name	O
Illustration	O
Remarks	O
Calculated	O
0.538	O
Feigenbaum	B-Algorithm
attractor	I-Algorithm
150px	O
The	O
Feigenbaum	B-Algorithm
attractor	I-Algorithm
(	O
see	O
between	O
arrows	O
)	O
is	O
the	O
set	O
of	O
points	O
generated	O
by	O
successive	O
iterations	O
of	O
the	O
logistic	O
function	O
for	O
the	O
critical	O
parameter	O
value	O
,	O
where	O
the	O
period	O
doubling	O
is	O
infinite	O
.	O
</s>
<s>
This	O
dimension	O
is	O
the	O
same	O
for	O
any	O
differentiable	O
and	O
unimodal	B-Algorithm
function	O
.	O
</s>
<s>
Varying	O
between	O
0	O
and	O
1	O
yields	O
any	O
fractal	B-Application
dimension	O
.	O
</s>
<s>
0.88137	O
Spectrum	O
of	O
Fibonacci	O
Hamiltonian	O
The	O
study	O
of	O
the	O
spectrum	O
of	O
the	O
Fibonacci	O
Hamiltonian	O
proves	O
upper	O
and	O
lower	O
bounds	O
for	O
its	O
fractal	B-Application
dimension	O
in	O
the	O
large	O
coupling	O
regime	O
.	O
</s>
<s>
1	O
Takagi	B-Algorithm
or	I-Algorithm
Blancmange	I-Algorithm
curve	I-Algorithm
150px	O
Defined	O
on	O
the	O
unit	O
interval	O
by	O
,	O
where	O
is	O
the	O
triangle	O
wave	O
function	O
.	O
</s>
<s>
Not	O
a	O
fractal	B-Application
under	O
Mandelbrot	O
's	O
definition	O
,	O
because	O
its	O
topological	O
dimension	O
is	O
also	O
.	O
</s>
<s>
Calculated	O
1.0812	O
Julia	B-Language
set	I-Language
z²	O
+	O
1/4	O
100px	O
Julia	B-Language
set	I-Language
for	O
c	O
=	O
1/4	O
.	O
</s>
<s>
Solution	O
s	O
of	O
1.0933	O
Boundary	O
of	O
the	O
Rauzy	O
fractal150px	O
Fractal	B-Application
representation	O
introduced	O
by	O
G.Rauzy	O
of	O
the	O
dynamics	O
associated	O
to	O
the	O
Tribonacci	O
morphism	O
:	O
,	O
and	O
.Messaoudi	O
,	O
Ali	O
.	O
</s>
<s>
The	O
Gosper	O
island	O
is	O
the	O
limit	O
of	O
the	O
Gosper	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
Measured	O
(	O
box	O
counting	O
)	O
1.2	O
Dendrite	O
Julia	B-Language
set	I-Language
150px	O
Julia	B-Language
set	I-Language
for	O
parameters	O
:	O
Real	O
=	O
0	O
and	O
Imaginary	O
=	O
1	O
.	O
</s>
<s>
1.2083	O
Fibonacci	B-Algorithm
word	I-Algorithm
fractal	I-Algorithm
60°	I-Algorithm
200px	O
Build	O
from	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
.	O
</s>
<s>
See	O
also	O
the	O
standard	O
Fibonacci	B-Algorithm
word	I-Algorithm
fractal	I-Algorithm
.	O
</s>
<s>
1.2108	O
Boundary	O
of	O
the	O
tame	O
twindragon	O
150px	O
One	O
of	O
the	O
six	O
2-rep-tiles	O
in	O
the	O
plane	O
(	O
can	O
be	O
tiled	O
by	O
two	O
copies	O
of	O
itself	O
,	O
of	O
equal	O
size	O
)	O
.Ngai	O
,	O
Sirvent	O
,	O
Veerman	O
,	O
and	O
Wang	O
(	O
October	O
2000	O
)	O
.	O
</s>
<s>
"	O
The	O
Boundary	O
of	O
Periodic	O
Iterated	B-Algorithm
Function	I-Algorithm
Systems	I-Algorithm
"	O
,	O
Wolfram.com	O
.	O
</s>
<s>
1.26	O
Hénon	B-Algorithm
map	I-Algorithm
100px	O
The	O
canonical	O
Hénon	B-Algorithm
map	I-Algorithm
(	O
with	O
parameters	O
a	O
=	O
1.4	O
and	O
b	O
=	O
0.3	O
)	O
has	O
Hausdorff	O
dimension	O
1.261	O
±	O
0.003	O
.	O
</s>
<s>
1.2619	O
Triflake	B-Algorithm
150px	O
Three	O
anti-snowflakes	O
arranged	O
in	O
a	O
way	O
that	O
a	O
koch-snowflake	O
forms	O
in	O
between	O
the	O
anti-snowflakes	O
.	O
</s>
<s>
1.2619	O
Koch	B-Algorithm
curve	I-Algorithm
200px	O
3	O
Koch	B-Algorithm
curves	I-Algorithm
form	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
or	O
the	O
anti-snowflake	O
.	O
</s>
<s>
1.2619	O
boundary	O
of	O
Terdragon	B-Algorithm
curve	I-Algorithm
150px	O
L-system	B-Language
:	O
same	O
as	O
dragon	B-Algorithm
curve	I-Algorithm
with	O
angle	O
=	O
30°	O
.	O
</s>
<s>
1.2619	O
2D	O
L-system	B-Language
branch	O
200px	O
L-Systems	B-Language
branching	O
pattern	O
having	O
4	O
new	O
pieces	O
scaled	O
by	O
1/3	O
.	O
</s>
<s>
Generating	O
the	O
pattern	O
using	O
statistical	O
instead	O
of	O
exact	O
self-similarity	O
yields	O
the	O
same	O
fractal	B-Application
dimension	O
.	O
</s>
<s>
Calculated	O
1.2683	O
Julia	B-Language
set	I-Language
z2	O
−	O
1	O
200px	O
Julia	B-Language
set	I-Language
for	O
c	O
=	O
−1	O
.	O
</s>
<s>
1.328	O
5	O
circles	O
inversion	O
fractal	B-Application
100px	O
The	O
limit	O
set	O
generated	O
by	O
iterated	O
inversions	O
with	O
respect	O
to	O
5	O
mutually	O
tangent	O
circles	O
(	O
in	O
red	O
)	O
.	O
</s>
<s>
The	O
Fractal	B-Application
Geometry	O
of	O
Nature	O
,	O
p.48	O
.	O
</s>
<s>
Cited	O
in	O
:	O
Quadratic	B-Algorithm
von	I-Algorithm
Koch	I-Algorithm
island	I-Algorithm
using	O
the	O
type	O
1	O
curve	O
as	O
generator150px	O
Also	O
known	O
as	O
the	O
Minkowski	B-Algorithm
Sausage	I-Algorithm
Calculated	O
1.3934	O
Douady	B-Algorithm
rabbit	I-Algorithm
150px	O
Julia	B-Language
set	I-Language
for	O
c	O
=	O
−	O
0,123	O
+	O
0.745i	O
.	O
</s>
<s>
1.4649	O
Vicsek	B-Algorithm
fractal	I-Algorithm
100px	O
Built	O
by	O
exchanging	O
iteratively	O
each	O
square	O
by	O
a	O
cross	O
of	O
5	O
squares	O
.	O
</s>
<s>
1.4649	O
Quadratic	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
(	O
type	O
1	O
)	O
150px	O
One	O
can	O
recognize	O
the	O
pattern	O
of	O
the	O
Vicsek	B-Algorithm
fractal	I-Algorithm
(	O
above	O
)	O
.	O
</s>
<s>
Images	O
generated	O
with	O
Fractal	B-Application
Generator	O
for	O
ImageJ	O
.	O
</s>
<s>
The	O
Hausdorff	O
dimension	O
of	O
the	O
graphs	O
of	O
fractal	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
1.5000	O
Quadratic	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
(	O
type	O
2	O
)	O
150px	O
Also	O
called	O
"	O
Minkowski	B-Algorithm
sausage	I-Algorithm
"	O
.	O
</s>
<s>
1.5236	O
Boundary	O
of	O
the	O
Dragon	B-Algorithm
curve	I-Algorithm
150px	O
cf	O
.	O
</s>
<s>
Chang	O
&	O
Zhang.Fractal	O
dimension	O
of	O
the	O
boundary	O
of	O
the	O
dragon	B-Algorithm
fractal	I-Algorithm
1.5236	O
Boundary	O
of	O
the	O
twindragon	O
curve150px	O
Can	O
be	O
built	O
with	O
two	O
dragon	B-Algorithm
curves	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
six	O
2-rep-tiles	O
in	O
the	O
plane	O
(	O
can	O
be	O
tiled	O
by	O
two	O
copies	O
of	O
itself	O
,	O
of	O
equal	O
size	O
)	O
.	O
</s>
<s>
The	O
fractal	B-Application
dimension	O
of	O
the	O
entire	O
tree	O
is	O
the	O
fractal	B-Application
dimension	O
of	O
the	O
terminal	O
branches	O
.	O
</s>
<s>
NB	O
:	O
the	O
2-branches	O
tree	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
only	O
1	O
.	O
</s>
<s>
1.5850	O
Boundary	O
of	O
the	O
T-square	B-Algorithm
fractal	I-Algorithm
200px	O
The	O
dimension	O
of	O
the	O
fractal	B-Application
itself	O
(	O
not	O
the	O
boundary	O
)	O
is	O
1.61803	O
a	O
golden	O
dragon	B-Algorithm
150px	O
Built	O
from	O
two	O
similarities	O
of	O
ratios	O
and	O
,	O
with	O
.	O
</s>
<s>
1.6309	O
Pascal	O
triangle	O
modulo	O
3	O
160px	O
For	O
a	O
triangle	O
modulo	O
k	O
,	O
if	O
k	O
is	O
prime	O
,	O
the	O
fractal	B-Application
dimension	O
is	O
(	O
cf	O
.	O
</s>
<s>
The	O
Koch	B-Algorithm
snowflake	I-Algorithm
is	O
present	O
at	O
all	O
scales	O
.	O
</s>
<s>
1.6379	O
Fibonacci	B-Algorithm
word	I-Algorithm
fractal	I-Algorithm
150px	O
Fractal	B-Application
based	O
on	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
(	O
or	O
Rabbit	B-Algorithm
sequence	I-Algorithm
)	O
Sloane	O
A005614	O
.	O
</s>
<s>
Solution	O
of	O
1.6402	O
Attractor	O
of	O
IFS	B-Algorithm
with	O
3	O
similarities	O
of	O
ratios	O
1/3	O
,	O
1/2	O
and	O
2/3	O
200px	O
Generalization	O
:	O
Providing	O
the	O
open	B-Algorithm
set	I-Algorithm
condition	I-Algorithm
holds	O
,	O
the	O
attractor	O
of	O
an	O
iterated	B-Algorithm
function	I-Algorithm
system	I-Algorithm
consisting	O
of	O
similarities	O
of	O
ratios	O
,	O
has	O
Hausdorff	O
dimension	O
,	O
solution	O
of	O
the	O
equation	O
coinciding	O
with	O
the	O
iteration	O
function	O
of	O
the	O
Euclidean	O
contraction	O
factor	O
:	O
.	O
</s>
<s>
1.6667	O
32-segment	O
quadric	O
fractal	B-Application
(	O
1/8	O
scaling	O
rule	O
)	O
frameless	O
see	O
also	O
:	O
:File:32	O
Segment	O
One	O
Eighth	O
Scale	O
Quadric	O
Fractal.jpgthumb	O
|Generator	O
for	O
32	O
segment	O
1/8	O
scale	O
quadric	O
fractal.Built	O
by	O
scaling	O
the	O
32	O
segment	O
generator	O
(	O
see	O
inset	O
)	O
by	O
1/8	O
for	O
each	O
iteration	O
,	O
and	O
replacing	O
each	O
segment	O
of	O
the	O
previous	O
structure	O
with	O
a	O
scaled	O
copy	O
of	O
the	O
entire	O
generator	O
.	O
</s>
<s>
The	O
fractal	B-Application
dimension	O
for	O
the	O
theoretical	O
structure	O
is	O
log	O
32/log	O
8	O
=	O
1.6667	O
.	O
</s>
<s>
Images	O
generated	O
with	O
Fractal	B-Application
Generator	O
for	O
ImageJ	O
.	O
</s>
<s>
1.6826	O
Pascal	O
triangle	O
modulo	O
5	O
160px	O
For	O
a	O
triangle	O
modulo	O
k	O
,	O
if	O
k	O
is	O
prime	O
,	O
the	O
fractal	B-Application
dimension	O
is	O
(	O
cf	O
.	O
</s>
<s>
Measured	O
(	O
box-counting	O
)	O
1.7	O
Ikeda	B-Algorithm
map	I-Algorithm
attractor	O
100px	O
For	O
parameters	O
a	O
=	O
1	O
,	O
b	O
=	O
0.9	O
,	O
k	O
=	O
0.4	O
and	O
p	O
=	O
6	O
in	O
the	O
Ikeda	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
1.6990	O
50	O
segment	O
quadric	O
fractal	B-Application
(	O
1/10	O
scaling	O
rule	O
)	O
150px	O
Built	O
by	O
scaling	O
the	O
50	O
segment	O
generator	O
(	O
see	O
inset	O
)	O
by	O
1/10	O
for	O
each	O
iteration	O
,	O
and	O
replacing	O
each	O
segment	O
of	O
the	O
previous	O
structure	O
with	O
a	O
scaled	O
copy	O
of	O
the	O
entire	O
generator	O
.	O
</s>
<s>
The	O
fractal	B-Application
dimension	O
for	O
the	O
theoretical	O
structure	O
is	O
log	O
50/log	O
10	O
=	O
1.6990	O
.	O
</s>
<s>
Images	O
generated	O
with	O
Fractal	B-Application
Generator	O
for	O
ImageJFractal	O
Generator	O
for	O
ImageJ	O
thumb	O
..	O
|Generator	O
for	O
50	O
Segment	O
Fractal	B-Application
.	O
</s>
<s>
1.7227	O
Pinwheel	O
fractal	B-Application
150px	O
Built	O
with	O
Conway	O
's	O
Pinwheel	O
tile	O
.	O
</s>
<s>
1.7712	O
Sphinx	O
fractal	B-Application
150px	O
Built	O
with	O
the	O
Sphinx	O
hexiamond	O
tiling	O
,	O
removing	O
two	O
of	O
the	O
nine	O
sub-sphinxes.W.	O
</s>
<s>
Its	O
boundary	O
is	O
the	O
von	O
Koch	O
flake	O
and	O
contains	O
an	O
infinity	O
of	O
Koch	B-Algorithm
snowflakes	I-Algorithm
(	O
black	O
or	O
white	O
)	O
.	O
</s>
<s>
1.7712	O
Fractal	B-Application
H-I	O
de	O
Rivera	O
100px	O
Starting	O
from	O
a	O
unit	O
square	O
dividing	O
its	O
dimensions	O
into	O
three	O
equal	O
parts	O
to	O
form	O
nine	O
self-similar	O
squares	O
with	O
the	O
first	O
square	O
,	O
two	O
middle	O
squares	O
(	O
the	O
one	O
that	O
is	O
above	O
and	O
the	O
one	O
below	O
the	O
central	O
square	O
)	O
are	O
removed	O
in	O
each	O
of	O
the	O
seven	O
squares	O
not	O
eliminated	O
the	O
process	O
is	O
repeated	O
,	O
so	O
it	O
continues	O
indefinitely	O
.	O
</s>
<s>
1.7848	O
Von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
85°	I-Algorithm
150px	O
Generalizing	O
the	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
with	O
an	O
angle	O
a	O
chosen	O
between	O
0	O
and	O
90°	O
.	O
</s>
<s>
The	O
fractal	B-Application
dimension	O
is	O
then	O
.	O
</s>
<s>
1.8272	O
A	O
self-affine	O
fractal	B-Application
set	O
200px	O
Build	O
iteratively	O
from	O
a	O
array	O
on	O
a	O
square	O
,	O
with	O
.	O
</s>
<s>
Unlike	O
self-similar	O
sets	O
,	O
the	O
Hausdorff	O
dimension	O
of	O
self-affine	O
sets	O
depends	O
on	O
the	O
position	O
of	O
the	O
iterated	O
elements	O
and	O
there	O
is	O
no	O
formula	O
,	O
so	O
far	O
,	O
for	O
the	O
general	O
case	O
.	O
</s>
<s>
solution	O
of	O
1.8687	O
Monkeys	O
tree	O
100px	O
This	O
curve	O
appeared	O
in	O
Benoit	O
Mandelbrot	O
's	O
"	O
Fractal	B-Application
geometry	O
of	O
Nature	O
"	O
(	O
1983	O
)	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
6	O
similarities	O
of	O
ratio	O
and	O
5	O
similarities	O
of	O
ratio	O
.Monkeys	O
tree	O
fractal	B-Algorithm
curve	I-Algorithm
1.8928	O
Sierpinski	O
carpet	O
100px	O
Each	O
face	O
of	O
the	O
Menger	B-Algorithm
sponge	I-Algorithm
is	O
a	O
Sierpinski	O
carpet	O
,	O
as	O
is	O
the	O
bottom	O
surface	O
of	O
the	O
3D	O
quadratic	O
Koch	O
surface	O
(	O
type	O
1	O
)	O
.	O
</s>
<s>
Cartesian	O
product	O
of	O
the	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
and	O
the	O
Cantor	O
set	O
150px	O
Generalization	O
:	O
Let	O
F×G	O
be	O
the	O
cartesian	O
product	O
of	O
two	O
fractals	B-Application
sets	O
F	O
and	O
G	O
.	O
Then	O
.	O
</s>
<s>
where	O
1.9340	O
Boundary	O
of	O
the	O
Lévy	B-Algorithm
C	I-Algorithm
curve	I-Algorithm
100px	O
Estimated	O
by	O
Duvall	O
and	O
Keesling	O
(	O
1999	O
)	O
.	O
</s>
<s>
The	O
curve	O
itself	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
2	O
.	O
</s>
<s>
2	O
Penrose	O
tiling	O
100px	O
See	O
Ramachandrarao	O
,	O
Sinha	O
&	O
Sanyal.Fractal	O
dimension	O
of	O
a	O
Penrose	O
tiling	O
2	O
Boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
100px	O
The	O
boundary	O
and	O
the	O
set	O
itself	O
have	O
the	O
same	O
Hausdorff	O
dimension	O
.	O
</s>
<s>
2	O
Julia	B-Language
set	I-Language
150px	O
For	O
determined	O
values	O
of	O
c	O
(	O
including	O
c	O
belonging	O
to	O
the	O
boundary	O
of	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
)	O
,	O
the	O
Julia	B-Language
set	I-Language
has	O
a	O
dimension	O
of	O
2	O
.	O
</s>
<s>
2	O
Sierpiński	B-Algorithm
curve	I-Algorithm
100px	O
Every	O
space-filling	B-Algorithm
curve	I-Algorithm
filling	O
the	O
plane	O
has	O
a	O
Hausdorff	O
dimension	O
of	O
2	O
.	O
</s>
<s>
2	O
Hilbert	B-Algorithm
curve	I-Algorithm
100px	O
2	O
Peano	B-Algorithm
curve	I-Algorithm
100px	O
And	O
a	O
family	O
of	O
curves	O
built	O
in	O
a	O
similar	O
way	O
,	O
such	O
as	O
the	O
Wunderlich	O
curves	O
.	O
</s>
<s>
2	O
Moore	B-Algorithm
curve	I-Algorithm
100px	O
Can	O
be	O
extended	O
in	O
3	O
dimensions	O
.	O
</s>
<s>
2	O
Lebesgue	B-Algorithm
curve	I-Algorithm
or	I-Algorithm
z-order	I-Algorithm
curve	I-Algorithm
100px	O
Unlike	O
the	O
previous	O
ones	O
this	O
space-filling	B-Algorithm
curve	I-Algorithm
is	O
differentiable	O
almost	O
everywhere	O
.	O
</s>
<s>
Like	O
the	O
Hilbert	B-Algorithm
Curve	I-Algorithm
it	O
can	O
be	O
extended	O
in	O
3D.Lebesgue	O
curve	O
variants	O
2	O
Dragon	B-Algorithm
curve	I-Algorithm
150px	O
And	O
its	O
boundary	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
1.5236270862	O
.	O
</s>
<s>
2	O
Terdragon	B-Algorithm
curve	I-Algorithm
150px	O
L-system	B-Language
:	O
F	O
→	O
F	O
+	O
F	O
–	O
F	O
,	O
angle	O
=	O
120°	O
.	O
</s>
<s>
2	O
Gosper	B-Algorithm
curve	I-Algorithm
100px	O
Its	O
boundary	O
is	O
the	O
Gosper	O
island	O
.	O
</s>
<s>
Solution	O
of	O
2	O
Curve	O
filling	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
100px	O
Proposed	O
by	O
Mandelbrot	O
in	O
1982	O
,	O
it	O
fills	O
the	O
Koch	B-Algorithm
snowflake	I-Algorithm
.	O
</s>
<s>
2	O
H-fractal	B-Algorithm
150px	O
Also	O
the	O
Mandelbrot	B-Algorithm
tree	I-Algorithm
which	O
has	O
a	O
similar	O
pattern	O
.	O
</s>
<s>
Pythagoras	B-Algorithm
tree	I-Algorithm
(	O
fractal	B-Application
)	O
150px	O
Every	O
square	O
generates	O
two	O
squares	O
with	O
a	O
reduction	O
ratio	O
of	O
.	O
</s>
<s>
2	O
2D	O
Greek	O
cross	O
fractal	B-Application
100px	O
Each	O
segment	O
is	O
replaced	O
by	O
a	O
cross	O
formed	O
by	O
4	O
segments	O
.	O
</s>
<s>
Measured	O
2.01	O
±	O
0.01	O
Rössler	B-Algorithm
attractor	I-Algorithm
100px	O
The	O
fractal	B-Application
dimension	O
of	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
is	O
slightly	O
above	O
2	O
.	O
</s>
<s>
For	O
a	O
=	O
0.1	O
,	O
b	O
=	O
0.1	O
and	O
c	O
=	O
14	O
it	O
has	O
been	O
estimated	O
between	O
2.01	O
and	O
2.02.Fractals	O
and	O
the	O
Rössler	B-Algorithm
attractor	I-Algorithm
Measured	O
2.06	O
±	O
0.01	O
Lorenz	B-Architecture
attractor	I-Architecture
100px	O
For	O
parameters	O
,	O
=	O
16	O
and	O
.	O
</s>
<s>
2.3219	O
Fractal	B-Application
pyramid	O
100px	O
Each	O
square	O
pyramid	O
is	O
replaced	O
by	O
5	O
half-size	O
square	O
pyramids	O
.	O
</s>
<s>
2.3347	O
3D	O
quadratic	O
Koch	O
surface	O
(	O
type	O
1	O
)	O
150px	O
Extension	O
in	O
3D	O
of	O
the	O
quadratic	O
Koch	B-Algorithm
curve	I-Algorithm
(	O
type	O
1	O
)	O
.	O
</s>
<s>
Dimension	O
calculated	O
by	O
M	O
.	O
Borkovec	O
,	O
W	O
.	O
De	O
Paris	O
,	O
and	O
R	O
.	O
Peikert.The	O
Fractal	B-Application
dimension	O
of	O
the	O
apollonian	O
sphere	O
packing	O
2.50	O
3D	O
quadratic	O
Koch	O
surface	O
(	O
type	O
2	O
)	O
150px	O
Extension	O
in	O
3D	O
of	O
the	O
quadratic	O
Koch	B-Algorithm
curve	I-Algorithm
(	O
type	O
2	O
)	O
.	O
</s>
<s>
2.5819	O
Icosahedron	O
fractal	B-Application
100px	O
Each	O
icosahedron	O
is	O
replaced	O
by	O
12	O
icosahedra	O
.	O
</s>
<s>
2.5849	O
3D	O
Greek	O
cross	O
fractal	B-Application
200px	O
Each	O
segment	O
is	O
replaced	O
by	O
a	O
cross	O
formed	O
by	O
6	O
segments	O
.	O
</s>
<s>
2.5849	O
Octahedron	O
fractal	B-Application
100px	O
Each	O
octahedron	O
is	O
replaced	O
by	O
6	O
octahedra	O
.	O
</s>
<s>
2.5849	O
von	B-Algorithm
Koch	I-Algorithm
surface	I-Algorithm
150px	O
Each	O
equilateral	O
triangular	O
face	O
is	O
cut	O
into	O
4	O
equal	O
triangles	O
.	O
</s>
<s>
2.7095	O
Von	B-Algorithm
Koch	I-Algorithm
in	I-Algorithm
3D	I-Algorithm
100px	O
Start	O
with	O
a	O
6-sided	O
polyhedron	O
whose	O
faces	O
are	O
isosceles	O
triangles	O
with	O
sides	O
of	O
ratio	O
2:2:3	O
.	O
</s>
<s>
2.7268	O
Menger	B-Algorithm
sponge	I-Algorithm
100px	O
And	O
its	O
surface	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
,	O
which	O
is	O
the	O
same	O
as	O
that	O
by	O
volume	O
.	O
</s>
<s>
3	O
3D	B-Algorithm
Hilbert	I-Algorithm
curve	I-Algorithm
100px	O
A	O
Hilbert	B-Algorithm
curve	I-Algorithm
extended	O
to	O
3	O
dimensions	O
.	O
</s>
<s>
3	O
3D	B-Algorithm
Lebesgue	I-Algorithm
curve	I-Algorithm
100px	O
A	O
Lebesgue	O
curve	O
extended	O
to	O
3	O
dimensions	O
.	O
</s>
<s>
3	O
3D	B-Algorithm
Moore	I-Algorithm
curve	I-Algorithm
100px	O
A	O
Moore	B-Algorithm
curve	I-Algorithm
extended	O
to	O
3	O
dimensions	O
.	O
</s>
<s>
3	O
3D	O
H-fractal	B-Algorithm
120px	O
A	O
H-fractal	B-Algorithm
extended	O
to	O
3	O
dimensions	O
.	O
</s>
<s>
Name	O
Illustration	O
Remarks1/2	O
0.5	O
Zeros	O
of	O
a	O
Wiener	O
process	O
150px	O
The	O
zeros	O
of	O
a	O
Wiener	O
process	O
(	O
Brownian	O
motion	O
)	O
are	O
a	O
nowhere	O
dense	O
set	O
of	O
Lebesgue	O
measure	O
0	O
with	O
a	O
fractal	B-Application
structure.Peter	O
Mörters	O
,	O
Yuval	O
Peres	O
,	O
"	O
Brownian	O
Motion	O
"	O
,	O
Cambridge	O
University	O
Press	O
,	O
2010	O
Solution	O
of	O
where	O
and	O
0.7499	O
a	O
random	O
Cantor	O
set	O
with	O
50%	O
-	O
30%	O
150px	O
Generalization	O
:	O
at	O
each	O
iteration	O
,	O
the	O
length	O
of	O
the	O
left	O
interval	O
is	O
defined	O
with	O
a	O
random	O
variable	O
,	O
a	O
variable	O
percentage	O
of	O
the	O
length	O
of	O
the	O
original	O
interval	O
.	O
</s>
<s>
Its	O
Hausdorff	O
Dimension	O
satisfies	O
:	O
(	O
where	O
is	O
the	O
expected	O
value	O
of	O
)	O
.Solution	O
of	O
144	O
...	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
with	O
random	O
interval	O
200px	O
The	O
length	O
of	O
the	O
middle	O
interval	O
is	O
a	O
random	O
variable	O
with	O
uniform	O
distribution	O
on	O
the	O
interval	O
(	O
0	O
,	O
1/3	O
)	O
.Measured1.22	O
±	O
0.02Coastline	O
of	O
Ireland	O
150px	O
Values	O
for	O
the	O
fractal	B-Application
dimension	O
of	O
the	O
entire	O
coast	O
of	O
Ireland	O
were	O
determined	O
by	O
McCartney	O
,	O
Abernethy	O
and	O
Gault	O
at	O
the	O
University	O
of	O
Ulster	O
and	O
Theoretical	O
Physics	O
students	O
at	O
Trinity	O
College	O
,	O
Dublin	O
,	O
under	O
the	O
supervision	O
of	O
S	O
.	O
</s>
<s>
Note	O
that	O
there	O
are	O
marked	O
differences	O
between	O
Ireland	O
's	O
ragged	O
west	O
coast	O
(	O
fractal	B-Application
dimension	O
of	O
about	O
1.26	O
)	O
and	O
the	O
much	O
smoother	O
east	O
coast	O
(	O
fractal	B-Application
dimension	O
1.10	O
)	O
Measured1.25Coastline	O
of	O
Great	O
Britain	O
200px	O
Fractal	B-Application
dimension	O
of	O
the	O
west	O
coast	O
of	O
Great	O
Britain	O
,	O
as	O
measured	O
by	O
Lewis	O
Fry	O
Richardson	O
and	O
cited	O
by	O
Benoît	O
Mandelbrot.How	O
long	O
is	O
the	O
coast	O
of	O
Britain	O
?	O
</s>
<s>
Statistical	O
self-similarity	O
and	O
fractional	O
dimension	O
,	O
B	O
.	O
Mandelbrot	O
1.2619	O
von	B-Algorithm
Koch	I-Algorithm
curve	I-Algorithm
with	O
random	O
orientation	O
200px	O
One	O
introduces	O
here	O
an	O
element	O
of	O
randomness	O
which	O
does	O
not	O
affect	O
the	O
dimension	O
,	O
by	O
choosing	O
,	O
at	O
each	O
iteration	O
,	O
to	O
place	O
the	O
equilateral	O
triangle	O
above	O
or	O
below	O
the	O
curve	O
.	O
</s>
<s>
1.333	O
Polymer	B-Language
in	O
2D	O
Similar	O
to	O
the	O
Brownian	O
motion	O
in	O
2D	O
with	O
non-self-intersection	O
.	O
</s>
<s>
1.333	O
Percolation	O
front	O
in	O
2D	O
,	O
Corrosion	O
front	O
in	O
2D	O
150px	O
Fractal	B-Application
dimension	O
of	O
the	O
percolation-by-invasion	O
front	O
(	O
accessible	O
perimeter	O
)	O
,	O
at	O
the	O
percolation	O
threshold	O
(	O
59.3	O
%	O
)	O
.	O
</s>
<s>
It	O
's	O
also	O
the	O
fractal	B-Application
dimension	O
of	O
a	O
stopped	O
corrosion	O
front	O
.	O
</s>
<s>
Measured	O
1.52	O
Coastline	O
of	O
Norway	O
100px	O
See	O
J	O
.	O
Feder.Feder	O
,	O
J.	O
,	O
"	O
Fractals	B-Application
,	O
"	O
,	O
Plenum	O
Press	O
,	O
New	O
York	O
,	O
(	O
1988	O
)	O
.	O
</s>
<s>
1.66	O
Polymer	B-Language
in	O
3D	O
Similar	O
to	O
the	O
Brownian	O
motion	O
in	O
a	O
cubic	O
lattice	O
,	O
but	O
without	O
self-intersection	O
.	O
</s>
<s>
1.70	O
2D	O
DLA	O
Cluster	O
150px	O
In	O
2	O
dimensions	O
,	O
clusters	O
formed	O
by	O
diffusion-limited	O
aggregation	O
,	O
have	O
a	O
fractal	B-Application
dimension	O
of	O
around	O
1.70	O
.	O
</s>
<s>
1.7381	O
Fractal	B-Application
percolation	O
with	O
75%	O
probability150px	O
The	O
fractal	B-Application
percolation	O
model	O
is	O
constructed	O
by	O
the	O
progressive	O
replacement	O
of	O
each	O
square	O
by	O
a	O
grid	O
in	O
which	O
is	O
placed	O
a	O
random	O
collection	O
of	O
sub-squares	O
,	O
each	O
sub-square	O
being	O
retained	O
with	O
probability	O
p	O
.	O
The	O
"	O
almost	O
sure	O
"	O
Hausdorff	O
dimension	O
equals	O
.	O
</s>
<s>
1.8958	O
2D	O
percolation	O
cluster	O
150px	O
In	O
a	O
square	O
lattice	O
,	O
under	O
the	O
site	O
percolation	O
threshold	O
(	O
59.3	O
%	O
)	O
the	O
percolation-by-invasion	O
cluster	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
91/48	O
.	O
</s>
<s>
Beyond	O
that	O
threshold	O
,	O
the	O
cluster	O
is	O
infinite	O
and	O
91/48	O
becomes	O
the	O
fractal	B-Application
dimension	O
of	O
the	O
"	O
clearings	O
"	O
.	O
</s>
<s>
The	O
Hausdorff	O
dimensions	O
equals	O
2	O
in	O
2D	O
,	O
in	O
3D	O
and	O
in	O
all	O
greater	O
dimensions	O
(	O
K.Falconer	O
"	O
The	O
geometry	O
of	O
fractal	B-Application
sets	O
"	O
)	O
.	O
</s>
<s>
2.50	O
3D	O
DLA	O
Cluster	O
150px	O
In	O
3	O
dimensions	O
,	O
clusters	O
formed	O
by	O
diffusion-limited	O
aggregation	O
,	O
have	O
a	O
fractal	B-Application
dimension	O
of	O
around	O
2.50	O
.	O
</s>
<s>
2.5	O
regular	O
Brownian	B-Algorithm
surface	I-Algorithm
150px	O
A	O
function	O
,	O
gives	O
the	O
height	O
of	O
a	O
point	O
such	O
that	O
,	O
for	O
two	O
given	O
positive	O
increments	O
and	O
,	O
then	O
has	O
a	O
centered	O
Gaussian	O
distribution	O
with	O
variance	O
=	O
.	O
</s>
<s>
Generalization	O
:	O
the	O
fractional	O
Brownian	B-Algorithm
surface	I-Algorithm
of	O
index	O
follows	O
the	O
same	O
definition	O
but	O
with	O
a	O
variance	O
,	O
in	O
that	O
case	O
its	O
Hausdorff	O
dimension	O
.	O
</s>
<s>
Measured	O
2.52	O
3D	O
percolation	O
cluster	O
225px	O
In	O
a	O
cubic	O
lattice	O
,	O
at	O
the	O
site	O
percolation	O
threshold	O
(	O
31.1	O
%	O
)	O
,	O
the	O
3D	O
percolation-by-invasion	O
cluster	O
has	O
a	O
fractal	B-Application
dimension	O
of	O
around	O
2.52	O
.	O
</s>
<s>
Beyond	O
that	O
threshold	O
,	O
the	O
cluster	O
is	O
infinite.Measured	O
and	O
calculated	O
~	O
2.7	O
The	O
surface	O
of	O
Broccoli	O
100px	O
San-Hoon	O
Kim	O
used	O
a	O
direct	O
scanning	O
method	O
and	O
a	O
cross	O
section	O
analysis	O
of	O
a	O
broccoli	O
to	O
conclude	O
that	O
the	O
fractal	B-Application
dimension	O
of	O
it	O
is	O
~	O
2.7	O
.	O
</s>
<s>
2.79	O
Surface	O
of	O
human	O
brain	O
100px	O
Fractal	B-Application
dimension	O
of	O
the	O
surface	O
of	O
the	O
human	O
brainMeasured	O
and	O
calculated	O
~	O
2.8	O
Cauliflower	O
100pxSan-Hoon	O
Kim	O
used	O
a	O
direct	O
scanning	O
method	O
and	O
a	O
mathematical	O
analysis	O
of	O
the	O
cross	O
section	O
of	O
a	O
cauliflower	O
to	O
conclude	O
that	O
the	O
fractal	B-Application
dimension	O
of	O
it	O
is	O
~	O
2.8	O
.	O
</s>
<s>
2.97	O
Lung	O
surface	O
100px	O
The	O
alveoli	O
of	O
a	O
lung	O
form	O
a	O
fractal	B-Application
surface	O
close	O
to	O
3	O
.	O
</s>
<s>
Calculated	O
Multiplicative	B-Algorithm
cascade	I-Algorithm
150px	O
This	O
is	O
an	O
example	O
of	O
a	O
multifractal	B-Algorithm
distribution	O
.	O
</s>
