Euler's Phi Function
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- čas přidán 17. 12. 2021
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This is my video series about Advent of Mathematical Symbols. I hope that it will help everyone who wants to learn about it.
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This is #Day19 in the series.
(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
An application of Euler's phi function: For a lattice to have n-fold rotational symmetry, the lattice must have at least φ(n) dimensions. Thus in two and three dimensions, lattices only have one-, two-, three-, four- and sixfold rotations. For fivefold rotations, the lattice has to have at least four dimensions, etc.
By any chance, are you referring to the crystallographic restriction theorem?
en.wikipedia.org/wiki/Crystallographic_restriction_theorem
I didn't have any knowledge of it before reading your comment and looking it up, but it seems fascinating!
@@PunmasterSTP Yes, that's what I was referring to.
Quasicrystals can have rotational (quasi)-symmetry of order other than 2, 3, 4 or 6 because they can be described as a three-dimensional section through an object periodic in higher-dimensional space. For example there are quasicrystals with dodecagonal symmetry.
@@sulfosalt3451 Thanks for getting back to me, and that's really interesting!
Einfach perfekt erklärt. Danke dir!
Thank you for this lovely video!
Would be interested in seeing a video on the group theory derivation/motivation for this function
I mean I guess you can say that phi(n) is the order of the multiplicative group of integers modulo n (written (Z/nZ)^x, or Z_n^x), but that doesn't shed any light on what the phi function is or its properties - the phi function just counts the number of positive integers
If you'd like to see more about the totient function or group theory, I'd recommend checking out some of Michael Penn's videos, including his playlists on number theory and group theory:
czcams.com/play/PL22w63XsKjqwAgBzVFVqZNMcVKpOOAA7c.html
czcams.com/play/PL22w63XsKjqxaZ-v5N4AprggFkQXgkNoP.html
Euler's phi? More like "I'm going to try"...to watch more of these amazing videos. Thanks so much for making and sharing them!