Exploring symmetries using 'monads': How 'counting/labeling' becomes easy. Even inside virtual 'atoms' or 'worm-holes'

[fbunc]

About the author

Math student that worked as a teacher (Math became a passion many years after graduation as electronic-engineer ).

Trying to explain non-trivial topics related to complex numbers, in a simple visual and intuitive way, was the reason why I started doing some animations to make some ideas more manageable.

Doing this animations and watching 3b1b was the spark that gave me some research ideas I started exploring in my free time for the last years.

Quick Summary

Introduction draft (sorry for the fuzzy language):

Exploring symmetry using 4D symbol-strips: https://www.youtube.com/watch?v=xHIXNLHVCHs

How 'counting/labeling' becomes easy, even inside virtual 'atoms' or 'worm-holes' , when you start exploring symmetries with 'monads'

This kind of game was proposed by 3b1b in this video about Hilbert Curves (See with your ears):

https://youtu.be/3s7h2MHQtxc?t=42

My personal 'game/long-term-puzzle' started trying to connect some sequences-of-interest, with colors(h,s,v),positions (x,y,z) , music(A,B,...F...) and brainwaves (alpha,beta...).

I just mention 'monads' because for me reading that book was an inspiration to start organizing my mind when I started studying hard again after many years more focused in literature than math.

Please consider that the 'quotes' mean that this is just recreational-math exercise and maybe is useful as introduction to some other topics.

So I made two new videos trying to show that using loosely the term 'monad' maybe you can see that it might be easy to 'do-some-counting' inside a numbers 'worm-hole' or inside a numbers 'atom' .

Puzzle A: Counting inside a 'worm-hole' is easy with 'monads' #SoME2

https://youtu.be/B-zmg52fh-c

Puzzle B : Counting inside an 'atom' is easy with 'monads' . #SoME2

https://youtu.be/qU06_hx1Sgc

(For both puzzles read the description for more information)

Target Medium

Solutions, formulas and explanations are being corrected, but can be shared if someone want's them RAW.

I'd like to see a better version of this kind of ideas in other channels videos, maybe in another context, and with explanations and formulas making it comprehensible for everyone.

3D animations skills would be awesome to navigate inside the structures.

Also if someone using manim could help great.

Musicians that can guide us on how to relate synth-instruments in the same timeline following a group of correlated number-sequences . (I have some work-in-progress on this, but not a musician and I have a hard time choosing ensembles and harmonies)

More details

Formulas and basic code for animations is provided. Also any further explanations can be provided by email, or added to the github if someone is interested.

A github with some basic explanations (work-in-progress):

https://github.com/fbunc

Contact details

fbatrouni at mi.unc.edu.ar

Long summary TLDR

From another perspective (summary of previous posts):

This is a way to draw a map of prime numbers in 3D, and get the feeling they follow orbits in a pattern as if they had some kind of bigger/smaller "mass" than the others. Notice also how these "orbits" concentrate prime numbers in specific areas of the plot.

Starting with a 4D spaghetti :

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

Adding then some more 'matter/numbers/colors' in 8 extending wheels, mirroring the first spaghetti and using diamonds for all prime-positions and stars for the main 'primes-observer' :

Adding gray numbers:

https://www.youtube.com/watch?v=OEYpuI8MvxU

Or collision and 'Correlated-colors'

https://www.youtube.com/watch?v=tzrLZQ6y9ZE

A (corrected) open-script-story between two main fictional characters called 'Time' and 'Symbols' :

https://github.com/fbunc/golden_ratio_mod_13_structure/blob/master/time_and_symbols.md

Original proposal:

https://www.youtube.com/watch?v=vlELOLepxi0

First introductory puzzle describing relation between 'golden-ratio' and 13 :

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

Some references:

Gottfried Wilhelm Leibniz philosophy: https://plato-philosophy.org/wp-content/uploads/2016/07/The-Monadology-1714-by-Gottfried-Wilhelm-LEIBNIZ-1646-1716.pdf

Playing with $\phi$ Math is Fun dot com: https://www.mathsisfun.com/numbers/golden-ratio.html

Some open questions for experts (please bring us some more light on this topics):

What is a 'monad' in Leibiniz terms? (When reading the anagram of Dog in his book ,I just imagine it meaning love, or beauty)

And in Haskell?

What was the first-issue between the real 'Pangloss' and Newton? (Voltaire/Candide)

What is a Frobenioid?

https://mathoverflow.net/questions/195353/what-is-a-frobenioid https://en.wikipedia.org/wiki/Frobenioid

Thank you!