Puzzle connecting the golden ratio to mod 13

[fbunc]

Portfolio

https://www.youtube.com/watch?v=NY5tgKBtLQ0)

Consider the sequences.

0,8,3,11,6,1,9,4,12,7,2,10,5

and

0,5,10,2,7,12,4,9,1,6,11,3,8

Consider all intermediate sequences, with different offsets or lags.

Main lines included in OEIS A257961 & A025641 :

https://oeis.org/search?q=0%2C8%2C3%2C11%2C6%2C1%2C9%2C4%2C12%2C7%2C2%2C10%2C5&sort=&language=&go=Search

Other related docs, and inspiration:

In order :

N-QPSK - Rohde Schwarz:

https://www.youtube.com/watch?v=lDSzyEQKE6o

Golden ratio in two dimensions:

https://math.stackexchange.com/questions/1851698/imaginary-golden-ratio

All 3blue1brown , just all:

Eigen-functions essence

https://www.youtube.com/watch?v=bOXCLR3Wric

Fourier Transforms https://www.youtube.com/watch?v=spUNpyF58BY&t=702s

Hilbert games https://www.youtube.com/watch?v=3s7h2MHQtxc&t=916s

Ben Sparks lovely take on Mandelbrot and Phi in Numberphile:

https://www.youtube.com/watch?v=sj8Sg8qnjOg

https://www.youtube.com/watch?v=FFftmWSzgmk&t=365s

Of course master Mathologer was in familiar territory:

What comes next? https://www.youtube.com/watch?v=4AuV93LOPcE

FlowerPhi https://www.youtube.com/watch?v=_GkxCIW46to&t=484s

http://www.ams.org/publicoutreach/feature-column/fcarc-phyllotaxis#2

Reducible (Great channel!): FFT https://www.youtube.com/watch?v=h7apO7q16V0

SoME2 https://github.com/leios/SoME_Topics/issues

The unfinished game - Keith Devlin: https://www.youtube.com/watch?v=W5XbuQq65Xc

Veritasium was in the zeitgeist about closed loops, and his lovely documentary about complex numbers discovery historical context:

https://www.youtube.com/watch?v=iSNsgj1OCLA

https://www.youtube.com/watch?v=cUzklzVXJwo

Adam Kucharski giving perfect historical context :

https://www.youtube.com/watch?v=658xlubwnDc&t=825s

Context and plot machine:

https://github.com/fbunc/golden_ratio_mod_13_primes_disc/blob/master/Eigenfunction_mod_13_golden_wheel.ipynb

Work in progress in the context of SoME2:

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/README.md

Target medium

Make a better version of this puzzle, where you are asked how to derive the sequence using the hints.

https://www.youtube.com/watch?v=NY5tgKBtLQ0)

Contact details

dioni#6563 at discord

Additional context

CC-BY.)

[jakebman]

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/0_generating_functions.ipynb has been removed since this post went up. Is https://github.com/fbunc/ROOTS_OF_UNITY_AND_ROULETTE_AS_A_BIASED_DICE/blob/master/00_ROOTS_OF_UNITY_AND_GENERATING_FUNCTIONS.ipynb a good replacement, or am I misunderstanding?

[fbunc]

I didn't realize it was still in private settings. Please let me know if you can see the other github, more related to the puzzle

[jakebman]

It works now! I see that file and two repositories under your github account

[fbunc]

I made a cleaner version, to see the primes on the same disc: https://github.com/fbunc/golden_ratio_mod_13_primes_disc

[fbunc]

Another context of discovery for the sequences (as closed loops) :

0,8,3,11,6,1,9,4,12,7,2,10,5

and

0,5,10,2,7,12,4,9,1,6,11,3,8

Using eigenvectors consider all intermediate sequences for different lags.

Main lines included in OEIS A257961 & A025641 :

https://oeis.org/search?q=0%2C8%2C3%2C11%2C6%2C1%2C9%2C4%2C12%2C7%2C2%2C10%2C5&sort=&language=&go=Search

'Time' and 'Symbols' (other more convoluted solution to the same kind of puzzle) (Full solution to this puzzle at the end of this text to avoid spoilers)

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/mod_39_golden_ratio_positions.py

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/time_and_symbols.md

Other related docs, and inspiration:

In order :

N-QPSK - Rohde Schwarz:

https://www.youtube.com/watch?v=lDSzyEQKE6o

Smith chart to see a way you can move in the plane using impedances:

https://www.youtube.com/watch?v=rUDMo7hwihs

Golden ratio in two dimensions:

https://math.stackexchange.com/questions/1851698/imaginary-golden-ratio

All 3blue1brown , just all:

Eigen-functions essence

https://www.youtube.com/watch?v=bOXCLR3Wric

Fourier Transforms https://www.youtube.com/watch?v=spUNpyF58BY&t=702s

Hilbert games https://www.youtube.com/watch?v=3s7h2MHQtxc&t=916s

Ben Sparks lovely take on Mandelbrot and Phi in Numberphile:

https://www.youtube.com/watch?v=sj8Sg8qnjOg

https://www.youtube.com/watch?v=FFftmWSzgmk&t=365s

Of course master Mathologer was in familiar territory:

What comes next? https://www.youtube.com/watch?v=4AuV93LOPcE

FlowerPhi https://www.youtube.com/watch?v=_GkxCIW46to&t=484s

http://www.ams.org/publicoutreach/feature-column/fcarc-phyllotaxis#2

Couplings - Pattern formations in Biology: https://www.youtube.com/watch?v=AKZNv7wn1GU

Reducible (Great channel!): FFT https://www.youtube.com/watch?v=h7apO7q16V0

The unfinished game - Keith Devlin: https://www.youtube.com/watch?v=W5XbuQq65Xc

Veritasium was in the zeitgeist about closed loops, and his lovely documentary about complex numbers discovery historical context:

https://www.youtube.com/watch?v=iSNsgj1OCLA

https://www.youtube.com/watch?v=cUzklzVXJwo

Adam Kucharski giving perfect historical context :

https://www.youtube.com/watch?v=658xlubwnDc&t=825s

John Guttag Stochastic Thinking:

https://www.youtube.com/watch?v=-1BnXEwHUok

Context and plot machine (solution to this puzzle,spoiler alert):

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/time_and_symbols.md

https://github.com/fbunc/golden_ratio_mod_13_primes_disc/blob/master/Eigenfunction_mod_13_golden_wheel.ipynb

Work in progress in the context of SoME2:

https://github.com/fbunc/Golden_Ratio_and_13_structure/blob/master/README.md

SoME2 https://github.com/leios/SoME_Topics/issues