<s>
A	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
a	O
specific	O
sequence	O
of	O
binary	O
digits	O
(	O
or	O
symbols	O
from	O
any	O
two-letter	O
alphabet	O
)	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
formed	O
by	O
repeated	O
concatenation	O
in	O
the	O
same	O
way	O
that	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
are	O
formed	O
by	O
repeated	O
addition	O
.	O
</s>
<s>
The	O
name	O
"	O
Fibonacci	B-Algorithm
word	I-Algorithm
"	O
has	O
also	O
been	O
used	O
to	O
refer	O
to	O
the	O
members	O
of	O
a	O
formal	O
language	O
L	O
consisting	O
of	O
strings	O
of	O
zeros	O
and	O
ones	O
with	O
no	O
two	O
repeated	O
ones	O
.	O
</s>
<s>
Any	O
prefix	O
of	O
the	O
specific	O
Fibonacci	B-Algorithm
word	I-Algorithm
belongs	O
to	O
L	O
,	O
but	O
so	O
do	O
many	O
other	O
strings	O
.	O
</s>
<s>
L	O
has	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
of	O
members	O
of	O
each	O
possible	O
length	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
the	O
limit	O
,	O
that	O
is	O
,	O
the	O
(	O
unique	O
)	O
infinite	O
sequence	O
that	O
contains	O
each	O
,	O
for	O
finite	O
,	O
as	O
a	O
prefix	O
.	O
</s>
<s>
The	O
first	O
few	O
elements	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
are	O
:	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
can	O
be	O
characterized	O
by	O
a	O
cutting	O
sequence	O
of	O
a	O
line	O
of	O
slope	O
or	O
.	O
</s>
<s>
Alternatively	O
,	O
one	O
can	O
imagine	O
directly	O
generating	O
the	O
entire	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
by	O
the	O
following	O
process	O
:	O
start	O
with	O
a	O
cursor	O
pointing	O
to	O
the	O
single	O
digit	O
0	O
.	O
</s>
<s>
A	O
similar	O
infinite	O
word	O
,	O
sometimes	O
called	O
the	O
rabbit	B-Algorithm
sequence	I-Algorithm
,	O
is	O
generated	O
by	O
a	O
similar	O
infinite	O
process	O
with	O
a	O
different	O
replacement	O
rule	O
:	O
whenever	O
the	O
cursor	O
is	O
pointing	O
to	O
a	O
0	O
,	O
append	O
1	O
,	O
and	O
whenever	O
the	O
cursor	O
is	O
pointing	O
to	O
a	O
1	O
,	O
append	O
0	O
,	O
1	O
.	O
</s>
<s>
However	O
this	O
sequence	O
differs	O
from	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
only	O
trivially	O
,	O
by	O
swapping	O
0s	O
for	O
1s	O
and	O
shifting	O
the	O
positions	O
by	O
one	O
.	O
</s>
<s>
A	O
closed	O
form	O
expression	O
for	O
the	O
so-called	O
rabbit	B-Algorithm
sequence	I-Algorithm
:	O
</s>
<s>
The	O
word	O
is	O
related	O
to	O
the	O
famous	O
sequence	O
of	O
the	O
same	O
name	O
(	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
)	O
in	O
the	O
sense	O
that	O
addition	O
of	O
integers	O
in	O
the	O
inductive	O
definition	O
is	O
replaced	O
with	O
string	O
concatenation	O
.	O
</s>
<s>
This	O
causes	O
the	O
length	O
of	O
Sn	O
to	O
be	O
Fn	O
+2	O
,	O
the	O
(	O
n	O
+2	O
)	O
nd	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
not	O
periodic	O
and	O
not	O
ultimately	O
periodic	O
.	O
</s>
<s>
The	O
last	O
two	O
letters	O
of	O
a	O
Fibonacci	B-Algorithm
word	I-Algorithm
are	O
alternately	O
"	O
01	O
"	O
and	O
"	O
10	O
"	O
.	O
</s>
<s>
Suppressing	O
the	O
last	O
two	O
letters	O
of	O
a	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
or	O
prefixing	O
the	O
complement	O
of	O
the	O
last	O
two	O
letters	O
,	O
creates	O
a	O
palindrome	O
.	O
</s>
<s>
The	O
palindromic	O
density	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
thus	O
1/φ	O
,	O
where	O
φ	O
is	O
the	O
golden	O
ratio	O
:	O
this	O
is	O
the	O
largest	O
possible	O
value	O
for	O
aperiodic	O
words	O
.	O
</s>
<s>
In	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
the	O
ratio	O
(	O
number	O
of	O
letters	O
)	O
/	O
( number	O
of	O
zeroes	O
)	O
is	O
φ	O
,	O
as	O
is	O
the	O
ratio	O
of	O
zeroes	O
to	O
ones	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
a	O
balanced	O
sequence	O
:	O
Take	O
two	O
factors	O
of	O
the	O
same	O
length	O
anywhere	O
in	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
.	O
</s>
<s>
The	O
complexity	O
function	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
n+1	O
:	O
it	O
contains	O
n+1	O
distinct	O
subwords	O
of	O
length	O
n	O
.	O
Example	O
:	O
There	O
are	O
4	O
distinct	O
subwords	O
of	O
length	O
3	O
:	O
"	O
001	O
"	O
,	O
"	O
010	O
"	O
,	O
"	O
100	O
"	O
and	O
"	O
101	O
"	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
the	O
standard	O
word	O
generated	O
by	O
the	O
directive	O
sequence	O
(	O
1	O
,	O
1	O
,	O
1	O
,....	O
)	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
recurrent	O
;	O
that	O
is	O
,	O
every	O
subword	O
occurs	O
infinitely	O
often	O
.	O
</s>
<s>
If	O
is	O
a	O
subword	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
then	O
so	O
is	O
its	O
reversal	O
,	O
denoted	O
.	O
</s>
<s>
If	O
is	O
a	O
subword	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
then	O
the	O
least	O
period	O
of	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
The	O
concatenation	O
of	O
two	O
successive	O
Fibonacci	B-Algorithm
words	I-Algorithm
is	O
"	O
almost	O
commutative	O
"	O
.	O
</s>
<s>
The	O
number	O
010010100	O
...	O
,	O
whose	O
digits	O
are	O
built	O
with	O
the	O
digits	O
of	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
is	O
transcendental	O
.	O
</s>
<s>
Although	O
the	O
above	O
generating	O
process	O
of	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
does	O
not	O
correspond	O
directly	O
to	O
the	O
successive	O
division	O
of	O
circle	O
segments	O
,	O
this	O
pattern	O
is	O
if	O
the	O
pattern	O
starts	O
at	O
the	O
point	O
nearest	O
to	O
the	O
first	O
point	O
in	O
clockwise	O
direction	O
,	O
whereupon	O
0	O
corresponds	O
to	O
the	O
long	O
distance	O
and	O
1	O
to	O
the	O
short	O
distance	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
contains	O
repetitions	O
of	O
3	O
successive	O
identical	O
subwords	O
,	O
but	O
none	O
of	O
4	O
.	O
</s>
<s>
The	O
critical	O
exponent	O
for	O
the	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
often	O
cited	O
as	O
the	O
worst	B-General_Concept
case	I-General_Concept
for	O
algorithms	O
detecting	O
repetitions	O
in	O
a	O
string	O
.	O
</s>
<s>
The	O
infinite	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
a	O
morphic	O
word	O
,	O
generated	O
in	O
{0,1}*	O
by	O
the	O
endomorphism	O
0	O
→	O
01	O
,	O
1	O
→	O
0	O
.	O
</s>
<s>
The	O
nth	O
element	O
of	O
a	O
Fibonacci	B-Algorithm
word	I-Algorithm
,	O
,	O
is	O
1	O
if	O
the	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
(	O
the	O
sum	O
of	O
a	O
specific	O
set	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
)	O
of	O
n	O
includes	O
a	O
1	O
,	O
and	O
0	O
if	O
it	O
does	O
not	O
include	O
a	O
1	O
.	O
</s>
<s>
The	O
digits	O
of	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
may	O
be	O
obtained	O
by	O
taking	O
the	O
sequence	O
of	O
fibbinary	O
numbers	O
modulo	O
2	O
.	O
</s>
<s>
Fibonacci	O
based	O
constructions	O
are	O
currently	O
used	O
to	O
model	O
physical	O
systems	O
with	O
aperiodic	O
order	O
such	O
as	O
quasicrystals	O
,	O
and	O
in	O
this	O
context	O
the	O
Fibonacci	B-Algorithm
word	I-Algorithm
is	O
also	O
called	O
the	O
Fibonacci	B-Algorithm
quasicrystal	I-Algorithm
.	O
</s>
