[ *October 22, 2001* ]

Have you ever.... palindromes in the Golden String !

from John McNamara (email)

What is the Golden String ?

The Golden String is a fascinating sequence of 0's and 1's that

is very closely related to the Fibonacci Numbers.

Other names for it are The Fibonacci Rabbit sequence (Dr Knott)

or The infinite Fibonacci word (N. Sloane).

The easiest algorithm to construct this Golden String

is in my opinion the following one :

Start with juxtaposition of 1 and 0 or 1__0__ and set the pointer at the 0.

Now apply the next rule :

If digit pointed at is a 0 then append '1' else append '10'

Move the pointer one place to the right. Iterate.

We arrive thus at sequence 10__1__ and already we moved

our pointer one digit further to the right (see underline).

Apply the rule again and we get 101__1__0.

Continue and the Golden String gradually takes shape 1011__0__10.

The first 100 'bits' of the string looks like this

10110101101101011010110110101101101011010110110101

10101101101011011010110101101101011011010110101101

two ways

Remember we started with 10 which is binary for 2 !

Now, John McNamara from Sydney Australia, discovered also

2 ways to see palindromes in this curious Golden String.

Firstly break the sequence up in real Fibonacci style :

1

0

11

010

11011

01011010

1101101011011

010110101101101011010

1101101011011010110101101101011011

0101101011011010110101101101011011010110101101101011010

etc.

All are palindromes with each string the length of the previous two,

formed by stringing together the previous two and changing the __last__

element from 0 to 1 or vice versa.

Well have you ever... amazing !

Secondly break the sequence up like this :

10

1

101

0110

1101011

01011011010

110110101101011011

01011010110110101101101011010

11011010110110101101011011010110101101101011011

etc.

all are palindromic and, with the exception of the artifice

at the start (which is in fact our start situation 10),

from the second line on contains the Lucas series numbers of bits.

1, 3, 4, 7, 11, 18, 29, ...

Have you ever... strikingly beautiful !

A065353

Decimal representation of palindromes

extracted from the Golden String using

ever increasing Fibonacci-style subdivisions.

A065354

Decimal representation of palindromes

extracted from the Golden String using

ever increasing Lucas-style subdivisions.

Website: mistermac

See also related WONplate 123