Interesting Specimens

[katestange]

For specimens that are not necessarily for the gallery, or for anything in particular, right now, but that are fun and interesting and we shouldn't forget them.

Example, mousing-over this one is fun:
http://localhost:5173/?name=FactorFence&viz=FactorFence&seq=OEIS+A001764

[katestange]

The beatty sequence for golden ratio shows the slowness with which Virahanka-Fibonacci moduli fill out:
http://localhost:5173/?name=Formula&viz=ModFill&modDimension=200&seq=Formula&formula=floor%28n*%281%2Bsqrt%285%29%29%2F2%29

[katestange]

And here's a very simple modfill that is lovely to watch the animation of:
http://localhost:5173/?name=Formula&viz=ModFill&modDimension=700&seq=Formula&formula=311*n

[gwhitney]

Here's a Turtle visualization of the sum of the terms of the eta-expansion of the Riemann zeta function, along the critical line as the imaginary part increases. I have spot-checked numerically, and it really is computing the zeta function in that the star is bouncing all around the correct value, up to a reflection about the real line which may either stem from p5.js coordinates being reversed vertically or in my borking a sign somewhere, not certain but I don't think it's too crucial.

It's not so pretty though because this alternating-series version converges well on the critical line in the sense of quickly starting to oscillate around the true value (it doesn't actually converge quickly in the sense of finite partial sums with few terms being almost equal to the true value -- the oscillations go on and on at sqrt(n) magnitude, which is not conducive to "landing", so to speak).

http://localhost:5173/?name=Critical+Line+Zeta+via+Eta&viz=Turtle&speed=10&ruleMode=1&turnFormula=%28n+%3F+log1p%281%2Fn%29%281%2B0.01f%29+%3A+arg%281-2%5E%281%2F2-%281%2B0.01f%29i%29%29%29+*+360%2Ftau&stepFormula=200%28-1%29%5En%2F%28sqrt%28n%2B1%29*abs%281-2%5E%281%2F2-%281%2B0.01f%29i%29%29%29&colorFormula=rainbow%282b%29&seq=Formula&last=99&length=100

[gwhitney]

From Kate: We should be able to create a turtle preset that shows the space filling curve here: https://open.substack.com/pub/apieceofthepi/p/constructing-space-filling-curves?r=semnj&utm_campaign=post&utm_medium=email (see the colourful one at the bottom)

[gwhitney]

I agree. It would be sort of nice if the turtle traced out the first iteration, and then drew the second iteration over it in a different color, and so on. (Maybe that's what the colorful one at the bottom looks like; I can't see it because I am not a substack subscriber.)

[gwhitney]

From Kate: Also fun: http://localhost:5173/?name=Countdown&viz=Turtle&bgColor=22274e&speed=10&ruleMode=1&turnFormula=30%2B15*a&stepFormula=50&widthFormula=2&colorFormula=chroma%28%27skyblue%27%29.set%28%27hsl.h%27%2C+n%25360%29&seq=Formula&formula=%28n%5E2-n%2B2%29%2543