Jatsko's YouTube and other media ventures

[Vortex]

I seemingly missed that video. It's pretty cool, I didn't know about these patterns

I can't sleep so I made a program to make the symbol for any rational number XD

prog.png (23.51 KiB) Viewed 4080 times

Also the similarities you noticed occur at the even and odd Lucas numbers:
2, 3, 7, 18, 47, 123, 322,...
1, 4, 11, 29, 76, 199, 521,...

The symbols for these numbers end up converging to a fractal-like figure.

[Anteroinen]

The reason for the same symbols coming up every second, third and so on symbols in different charts would be that you essentially multiplied every term in the Fibonacci series by two and then three and so on. So we have that.

As for the halfs and thirds and so on thing... The thing is that in the world of modulo n, one half is not really a thing, one half is actually just the number which, when multiplied by two, gives you 1 modulo n. These fractions do not necessarily exist for all integer values of n either. They do for all prime numbers though.

For example, if n=5, 1/2 mod 5=3, because 2*3 mod 5=1 mod 5. On the other hand, if n=4, 1/2 mod 4 is not defined, since 0*2 mod 4 = 0 mod 4, 1*2 mod 4 = 2 mod 4, 2*2 mod 4 = 0 mod 4 and 3*2 mod 4 = 2 mod 4.

Notice, though, that 3*3 mod 4 = 9 mod 4 = 1 mod 4, so 1/3 mod 4 = 3 mod 4. So the fractions can exist even if n is not prime.

Not sure if I can see any patterns related to that with my bare eye, but still. That's how fractions work in modular arithmetic. Numbers are not strictly speaking numbers. as we usually understand them; they're classes of equidistant integers with the same remainder over n.

[Vortex]

The reason for the same symbols coming up every second, third and so on symbols in different charts would be that you essentially multiplied every term in the Fibonacci series by two and then three and so on. So we have that.

As for the halfs and thirds and so on thing... The thing is that in the world of modulo n, one half is not really a thing, one half is actually just the number which, when multiplied by two, gives you 1 modulo n. These fractions do not necessarily exist for all integer values of n either. They do for all prime numbers though.

For example, if n=5, 1/2 mod 5=3, because 2*3 mod 5=1 mod 5. On the other hand, if n=4, 1/2 mod 4 is not defined, since 0*2 mod 4 = 0 mod 4, 1*2 mod 4 = 2 mod 4, 2*2 mod 4 = 0 mod 4 and 3*2 mod 4 = 2 mod 4.

Notice, though, that 3*3 mod 4 = 9 mod 4 = 1 mod 4, so 1/3 mod 4 = 3 mod 4. So the fractions can exist even if n is not prime.

Not sure if I can see any patterns related to that with my bare eye, but still. That's how fractions work in modular arithmetic. Numbers are not strictly speaking numbers. as we usually understand them; they're classes of equidistant integers with the same remainder over n.

Yeah, if I'm not mistaken the general rule is that every non-zero-divisor (which means not a divisor of n) is invertible in the ring of integers modulo n. On the other hand you can still take the projective line over the ring and make all elements "invertible" if you wanted I like projective lines

Anyways, the symbol patterns Jatsko noticed are not strictly only modular arithmetic (for example, the pattern 1-2-3 in a circle with 3 marked points is identified with the pattern 2-4-6 in a circle with 6 marked points). I would best describe them as the image of the Fibonacci sequence under the map fib_n -> exp(2*pi*i/k *fib_n) into the complex circle, this obviously works for any rational number k and does not run into issues of invertibility. It's what I used in the small program above.

[Anteroinen]

Yeah, if I'm not mistaken the general rule is that every non-zero-divisor (which means not a divisor of n) is invertible in the ring of integers modulo n. On the other hand you can still take the projective line over the ring and make all elements "invertible" if you wanted I like projective lines

That sounds about right, yeah. It's been a while since I took a course in this stuff, my whole last year was spent doing a dozen physics courses to give me "enough ECTSs" even though it turned out some of that wasn't strictly speaking necessary.

Anyways, the symbol patterns Jatsko noticed are not strictly only modular arithmetic (for example, the pattern 1-2-3 in a circle with 3 marked points is identified with the pattern 2-4-6 in a circle with 6 marked points). I would best describe them as the image of the Fibonacci sequence under the map fib_n -> exp(2*pi*i/k *fib_n) into the complex circle, this obviously works for any rational number k and does not run into issues of invertibility. It's what I used in the small program above.

Yeah, that is true, there's no problem there. It's not like there's any problem with just deciding to associate the rationals with the symbols a priori either. It was the first thing that came to mind at 6:45 am though. XD At least it looks the symbols have the same shapes for 2/4 = 1/2 and so on. I mean, there is the obvious problem that 5/7=5, for example, but that's... uhh...

I would need to think about it more to see the pattern.

EDIT: If one exists, since if I recall correctly Pisano periods aren't entirely regular or anything, even though individual patterns seem to be arising.

[Jatsko]

Yeah, that is true, there's no problem there. It's not like there's any problem with just deciding to associate the rationals with the symbols a priori either. It was the first thing that came to mind at 6:45 am though. XD

That's something I probably should have made clearer, the fact that modulo doesn't really deal with integers by itself and the numbers are more of an association tactic.

Vortex do you have an easily accessible way to share your programming results so others can generate any symbol they want?

[Vortex]

Vortex do you have an easily accessible way to share your programming results so others can generate any symbol they want?

The complete code is in the image I posted, it works in Mathematica. You only have to change the value of n to the rational number you want.

The problem is that Mathematica software costs money, but it's easy to translate the code to another programming language. Which ones do you know?

EDIT: If you want to try it yourself, the idea is to make a line plot of the points with coordinates ( sin(2*pi*f_k/n), cos(2*pi*f_k/n) ), where f_k is the usual sequence of Fibonacci numbers and n is the rational number you want to make a symbol of. You have to draw enough points from the Fibonacci sequence for the graph to close, I chose to take a number of points equal to the squared numerator of n but probably there is a better bound.

[Jatsko]

The only one I've had experience with is Matlab, and I don't have an activated account on that anymore (it's been a while anyway).

Maybe towerofnix can configure it to work in Khan Academy and whatever program they use.

[Vortex]

Hmm, in Matlab the code would be something like this

n=5; d=1;
plot(sin(2*pi*fibonacci(1:n^2)/n*d), cos(2*pi*fibonacci(1:n^2)/n*d))

where n is the numerator and d the denominator of the rational number. I haven't tested it though. I don't know how Khan Academy works but it probably won't be very difficult.

[Clbsfn]

Before anything else, I have to say that I never wanted to have anything to do with the drama that has transpired recently involving gemini and jatsko. I do not want to take a side on an issue like this, and I have only come here to post this because a tumblr account has messaged me four imgur links to parts of a text post (which the account has posted all four parts of on their own blog).

I want absolutely no part in this, and frankly, I'm frustrated that simply by having a pastel forum account (having never even mentioned gemini on this forum) and commenting on jatsko's vids (ones that aren't related to gemini) have I been roped into this situation by whoever "screechingbananaland" is.

I want to make this clear: The only reason I am posting this here is because it directly relates to jatsko and his youtube channel. In posting this, know that 1) I have not written the message, I'm only putting links to it here so people are aware of its existence 2) I do not necessarily agree or disagree with anything stated in the message, and 3) I do not condone sending hate to, or threatening, anyone involved in this situation, be it jatsko, gemini, or ko9.

The message was sent to me in the form of four imgur links, and is the only content on the blog https://screechingbananaland.tumblr.com at the time of my writing this.

I'll screenshot them in case they get deleted, but for now, here are the four links to imgur that contain this message

1) http://i.imgur.com/rLPf0a1.png

2) http://i.imgur.com/chjG7qR.png

3) http://i.imgur.com/DQOrzCP.png

4) http://i.imgur.com/R2fEGkA.png

Again, I did not write that post (I'm not gemini), I'm only posting links to it here because it directly involves jatsko and because links to the message were DMed to me on tumblr. The account that sent it is https://screechingbananaland.tumblr.com, and if the imgur posts get deleted, you can check to see if that tumblr account still has the posts up. If not, then I can post the screenshotted images here if requested.

I'm sorry if this continues any drama - I want it to end just as much as everyone here does. However, I feel like the right thing to do is to bring this to the community's attention, as this message is highly relevant to both past and recent events that have occurred in the community.

[Jatsko]

You haven't done anything wrong. A lot of the forum is already aware of the messages and pictures. You are basically helping to confirm that Gemini is tracking down anyone who's associated with me on any social media site that he can and is trying to spread these messages and his opinions.

These messages are already under intense scrutiny and you can be assured that you haven't done anything wrong and no one suspects you of anything, Clbsfn.