Matt Downey @SciFiGameDev@mastodon.gamedev.place

Finished my 6-second game trailer for my Asteroids (1979)-clone in virtual reality:

https://youtu.be/tGoc2T6ABUc

Hmmmm,

(pi/ln(2) -1.13309)/(1.13309) ~= 3 (probably exact)

max(1/(3 - 2 sqrt(2) cos(n log(2))), (3 - 2 sqrt(2) cos(n log(2))))-min(1/(3 - 2 sqrt(2) cos(n log(2))), (3 - 2 sqrt(2) cos(n log(2)))), n = 1.13308 to 1.1331

The graph has a lot more to do with 2 than 3, so this is surprising/unexpected to me.

I don't know why I was so curious about this, but:

max(1/(3 - 2 sqrt(2) cos(n log(2))), (3 - 2 sqrt(2) cos(n log(2))))-min(1/(3 - 2 sqrt(2) cos(n log(2))), (3 - 2 sqrt(2) cos(n log(2)))), n = 0 to pi/ln(2)

I finally figured out something I wanted to know in bifurcation theory: the length of the first two bifurcation paths

arc length 1 - 1/r, r= 1 to 3 --> 2.1466222089377...
arc length 0*r, r = -1 to +1 --> 2 (duh)

I already knew this bifurcation had a 135 degree angle (in the eyes of a layperson, although it may be far more complicated).

(3 - sqrt(8) cos(t log(2))) has an infinite set of roots, so the hypothesis that 1/(3 - sqrt(8) cos(t log(2))) has imaginary roots isn't far-fetched.

Oh wait, derp, these aren't the same because the other one uses +/- 0.5i (this one is real, not imaginary).

(Didn't immediately drop it.)

I wanted to figure out when the imaginary roots matched so I calculated:

roots of (ζ(1/2 - i n, 1/2) ζ(1/2 + i n, 1/2)) * cos^2(n*(pi/(2*ln(1 + sqrt(2)))*2) * sinh^(-1)(1))

Which Wolfram tells me is simply

roots of (ζ(1/2 - i n, 1/2) ζ(1/2 + i n, 1/2)) * cos^2(pi*n)

I think this is unrelated or I'm applying the skew with the wrong coefficient.

Dropping it to work on something else.

You can rewrite my fancy equations to just be:

cos^2(n*sinh(1)/x) according to Wolfram Alpha, which is pretty useful.

1st zeta ~7.931x
2nd zeta ~11.795x
3rd zeta ~14.034x
4th zeta ~17.071x

So clearly it's not nearly that simple.

I imagine the zeta values that are extremely close together are explained away by there being two https://en.m.wikipedia.org/wiki/Grothendieck_inequality or something.

Which is incidentally 3* https://en.m.wikipedia.org/wiki/Grothendieck_inequality or so.

The next zero after ~ https://en.m.wikipedia.org/wiki/Grothendieck_inequality is ~5.34664

I'm not even sure anymore whether the ~5 constant I'm looking for exists or is a single constant, but it might as well be a running gag that this must be the one.

You can actually multiply by both skews for better zeroes, meaning I sort of react the "half zeroes" thing I said, although they could be energy quanta.

I believe:

plot (zeta(1/2 + n * I, 1/2) * zeta(1/2 - n * I, 1/2)) * (1 + sin(n*ln(1 + sqrt(2))))*(1 - sin(n*ln(1 + sqrt(2))))

Where the next shift quanta level next is n ~= +/- 18.2

Oh wait, the pi/(2*ln(1 + sqrt(2))) number on Wolfram is misleading since it states equality but later says "hey this person was wrong".

It does seem really close, though.

Which actually gives me a vague (or better) understanding of what I need to do.

I hypothesize there are "hidden zeta half-zeroes", and the first one is ~ -1.782214

It seems to be a https://en.m.wikipedia.org/wiki/Grothendieck_inequality

Although floating-point imprecision means I think it's within 1e-7 (but even that could be off because floats suck to deal with).

I believe

plot (zeta(1/2 + n * I, 1/2) * zeta(1/2 - n * I, 1/2)) * (1 + sin(n*ln(1 + sqrt(2))/1)) contains at least some of the zeta zeroes and more unique ones to the function.

A better skew function yet is:

roots of (zeta(1/2 + n * I, 1/2) * zeta(1/2 - n * I, 1/2)) * (1 + i*sin(n*ln(1 + sqrt(2))/x))

Although it's an imaginary skew and makes the "zeta waves" decoherent.

If you want to skew the function right by "x" you can use:

roots of (zeta(1/2 + n * I, 1/2) * zeta(1/2 - n * I, 1/2)) * (1 - sin(n*pi/2/x))

Notably I made a copy error from my laptop. All three of the above should use sine.

So roots of (zeta(1/2 + n * I, 1/2) * zeta(1/2 - n * I, 1/2)) * (1 - sin(n*pi/2))^(1)

Is even better. It seems to skew the zeta function in one direction.