Golden ratio in natural binary.

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Golden ratio in natural binary.

Postby JamesLogan on April 9th, 2014, 6:22 pm 

Halo,

Turns out the golden spiral numbers can be found in binary core concepts.

I have outlined a simple procedure for you to check it out yourself if interested. It has been tested and it proves accurate up to 26 terms of the fibo sequence. (26 : 121393). I have manually written out and sorted 7 terms below.

A method to test is:

i. assign the binary symbols 0 and 1 the values of 1 and 2 respectively.
ii. sum the components of binary-strings (0 to n) with the alternately assigned values.

like:

6 = 110 >> 2 + 2 + 1 = 5
9 = 1001 >> 2 + 1 + 1 + 2 = 6

iii. sort similar sums when counting all sequantial integers.
iv. calculate the frequency of the sums occurrances.
v. say wow! :B


results (manual count and sort up to 7 : 13):

0 = 0 ........................1........................1
----------------------------------------------------> 0!
*1 = 1 .......................2.........................1
----------------------------------------------------> 1
*2 = 10 ....................2 + 1 ................... 3
----------------------------------------------------> 1
3 = 11 ....................2 + 2 .....................4
*4 = 100 ..................2 + 1 + 1................4
----------------------------------------------------> 2
5 = 101 ...................2 + 1 + 2 .............5
6 = 110 ..................2 + 2 + 1 .............5
*8 = 1000 .................1 + 1 + 1 + 2 ........5
----------------------------------------------------> 3
7 = 111 ..................2 + 2 + 2...............6
9 = 1001 ................2 + 1 + 1 + 2.........6
10 = 1010 ...............2 + 1 + 2 + 1 ........6
12 = 1100 ...............2 + 2 + 1 + 1........6
*16 = 10000 ............2 + 1 + 1 + 1 + 1 ...6
----------------------------------------------------> 5
11 = 1011 ...............2 + 1 + 2 + 2.........7
13 = 1101 ...............2 + 2 + 1 + 2.........7
14 = 1110 ............2 + 2 + 2 + 1.............7
17 = 10001 ...... 2 + 1 + 1 + 1 + 2..........7
18 = 10010 ...... 2 + 1 + 1 + 2 + 1.........7
20 = 10100 ........ 2 + 1 + 2 + 1 + 1 .......7
24 = 11000 .........2 + 2 + 1 + 1 +1.........7
*32 = 100000 ......2 + 1 + 1+ 1 + 1 + 1.....7
----------------------------------------------------> 8
15 = 1111 ...........2 + 2 + 2 + 2 ............8
19 = 10011 ........ 2 + 1 + 1 + 2 + 2 .... ..8
21 = 10101 ........2 + 1 + 2 + 1 + 2.........8
22 = 10110 ......2 + 1 + 2 + 2 + 1............8
25 = 11001 .....2 + 2 + 1 + 1 + 2............8
26 = 11010 .....2 + 2 + 1 + 2 + 1............8
28 = 11100 ......2 + 2 + 2 + 1 + 1...........8
33 = 100001 ....2 + 1 + 1 + 1 + 1 + 2......8
34 = 100010 ....2 + 1 + 1 + 1 + 2 + 1 .....8
36 = 100100 ...2 + 1 + 1 + 2 + 1 + 1......8
40 = 101000 ...2 + 1 + 2 + 1 + 1 + 1......8
48 = 110000 ....2 + 2 + 1 + 1 + 1 + 1......8
*64 = 1000000 ...2+1+1 +1 +1+1+ 1+ 1....8
----------------------------------------------------> 13



I have a pretty basic python script to calculate it out if anyone is interested I would be more than happy to share it if you want to see it (please email or pm) me. (im still working on a bit tho cuz im brand new to code)

Please feel free to share this around, Im in no way taking credit for something found in nature, it was already there. I dont care much for the practice of attaching peoples names to principles that deserve and require the greater attention.

Cheers.
Last edited by mtbturtle on April 9th, 2014, 6:26 pm, edited 1 time in total.
Reason: removed email
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JamesLogan
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Re: Golden ratio in natural binary.

Postby DigitalReality on April 16th, 2015, 10:45 am 

COOL!!

Have you seen that guy who used a video card to render a set of concentric circles along a number line, and then checking if/where theses overlap, in order to calculate the two largest known primes?

http://articles.sun-sentinel.com/2012-10-07/news/fl-prime-discoveries-20121005_1_prime-numbers-riemann-broward-college

It takes someone with a GREAT understanding to come up with these visualizations of numbers. Ulam's prime spiral is another.
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