A Complex Way to Graph

[@frendlyleaf6187]

Man, this is asmr for my ears, eyes and mind at the same time...

[@theneongoomba]

This is a great visualization technique, it really gets you thinking about complex roots of polynomials. I like it when I'm able to "package up" mathematical concepts and "carry them around in my pocket," and this definitely scratches that itch.

[@thepro4805]

this is so greately visualised and super intuitive! love it!

[@Faroshkas]

Criminally underrated

[@ajs1998]

This was so interesting. Love your channel

[@tcaDNAp]

8:05 I am beyond excited that 105 was included because it's the first cyclotomic polynomial with a coefficient as large as 2! I'm sure there was a lot of toying with conjectures on special cases like 105=3*5*7

[@tcaDNAp]

Also, I never knew about inverse cyclotomics as a casual that reads Wikipedia instead of papers, so it's time for a rabbit hole!

[@TheGrayCuber]

kzbin.info/www/bejne/i3inmGyEmahonq8 this video I made also includes discussion of them - though I called them 'reciprocal cyclotomics' instead of inverse

[@applimu7992]

Is the last animation made with a linear interpolation between each polynomial? I would be curious to see these animations with different methods of interpolation! like maybe exponential interpolation (t -> a(z)^(1-t) * b(z)^t )

[@TheGrayCuber]

Yes it is linear. That is a great idea, I'm going to look at other methods! It is also intersting to use a differnt sequence, such as square numbers, or numbers with all the same totient

[@icosahedrondodecahedraldual]

@@TheGrayCuber Since the interpolations are linear, could we theoretically make a 5.5th cyclotomic by freezing on the frame right in the middle of the 5th and 6th cyclotomics?

[@sachs6]

How do you interpolate between the polynomials at the end? What is the eth polynomial?

[@kdicus]

Makes me wonder if cyclotomic shapes could be a way to identify large primes…

[@fr4781]

So glad i found this. The animation's absolutely gorgeous, how did you make it?

[@TheGrayCuber]

I used p5js. It is one of my favorite tools for making visualizations

[@fr4781]

@@TheGrayCuber Would you be willing to provide any leftover code snippets I could take a peek at. Absolutely love ur style and can't see how one could make anything similar

[@TheGrayCuber]

A lot of the code I wrote is on openprocessing here: openprocessing.org/user/448907?o=7&view=sketches

[@brandonklein1]

Always totient of n roots to the nth cyclotonic. Very cool

[@Peccomment]

I love your graph, could you open what is special about x8 + x4 + x3 + x + 1, since it is used in SHA512. Do you insight for that special case?

[@davecorry7723]

Very nice.

[@CodecYT-w4n]

Could you do this for the zeta function?

[@jakeaustria5445]

Thank You

[@BirdsAreVeryCool]

Is there a way to calculate the length of these weird shapes? With contour integrals or soemthing

[@escher4401]

Why people keep saying that we need 4d to graph C to C functions? There's already a lot of tools that can plot these functions using color schemes where hue is angle and brightness is intensity at each point.

[@TheGrayCuber]

Yes, there are other great ways to graph C to C in fewer than 4 dimensions! I only intend to say that we would need 4 dimensions to graph in the same manner as the 'typical' real graph using separate axes for input and output.

[@galoomba5559]

And why do people keep saying that the fact that we need 4 dimensions makes it impossible to do?

[@thelocalsage]

we’ll always need many ways of representing mathematical objects so the more ways the better, but hue and intensity is a uniquely terrible one because it doesn’t map onto human color perception well and is less accessible to the colorblind (which includes me). stark visual changes are not linear with hue.

[@pendragon7600]

Hue and brightness are the third and forth dimensions. Nobody said 4 spatial dimensions. The graph of a function C to C can only be faithfully embedded in 4 or more dimensions. The representation is your choice.

[@thelocalsage]

@@pendragon7600 to be fair to the original comment, when talking about representations there’s no reason to assume a bijective embedding is the perfect criteria. In all representation we sacrifice something, and if you’re willing to make that sacrificed thing precise information then it’s fine as representation. It depends on what you care about, sometimes you want to ignore some information-that’s one of the powers of topology or graph theory. But yes for the graph to be lossless you’d need 4D representations.

[@elijahberegovsky8957]

How do you interpolate between cyclotomics in your animation?

[@TheGrayCuber]

I take Phi_n(x)*(1-t) + Phi_n+1(x)*t and move t from 0 to 1

[@Abraccuda]

how are computed the intermediate curves between two cyclotomic curves in the ending animation?

[@TheGrayCuber]

I use a weighted sum of coefficients for each term, using (cos + 1)/2 and (1 - cos)/2 as weights

[@Abraccuda]

@@TheGrayCuber Thank you for your answer!

[@lumpyspaceprincess6335]

Is this an entertainment video or an education video?

[@TheGrayCuber]

Yes!

[@yoavboaz1078]

is the title are reference to "a better way to count"

[@TheGrayCuber]

Not intentionally, although I am a fan of jan Misali. sitelen tawa ona li pona mute!

[@oKrybia]

Why didn't you graph other functions? Polynomials aren't that cool...