Interesting Specimens #477
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The beatty sequence for golden ratio shows the slowness with which Virahanka-Fibonacci moduli fill out: |
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And here's a very simple modfill that is lovely to watch the animation of: |
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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). |
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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) |
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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.) |
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I posted this one on illustrating math because I am not really sure why it closes; replace 729 with 727 (or most other integers) and of course you get a periodic pattern but it rarely seems to close, whereas it closes with all the powers of 3 that I tried...: |
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P.S. Oh I did actually figure out why it closes. The pattern has a multiplier a (here 2) and a denominator d (here 729=3^6). If a is a primitive root mod p, then if the denominator is p the turtle's path sums all primitive pth roots of unity, which sum to -1, so the path does not close; whereas if the denominator in p^k the path sums all primitive (p^k)th roots of unity, which sum to the negative of all (p^(k-1))th roots of unity, which sum to 0, so the path closes. |
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Some thoughts:
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I think with pan and rotate the initial positioning isn't too crucial , and there's nothing we can do about the order that this curve visits points. But if you feel we should implement an optional start position for Turtle, please do file an issue for it. |
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Ooh, this one looks sort of like a triangular-grid dragon curve. |
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This doesn't produce anything... but it fails silently: http://localhost:5173/?name=OEIS+A000521&viz=Turtle&domain=0+1&steps=2+3&strokeColor=%235e8d85&bgColor=000000&speed=100&ruleMode=1&angleFormula=a+%25+30&stepFormula=3&widthFormula=6&colorFormula=chroma%28%7Bh%3A+%28n%2F1.5%29+%25+360%2C+s%3A0.65%2C+l%3A+0.75%7D%29&seq=OEIS+A000521 |
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Oh, interesting that you didn't just turn by ka and always step 5, as opposed to always turn k and step by al. Both wander, albeit differently looking.. Oh wait, you're not turning by a constant amount, you're turning left, about face, turning right, not turning in a cycle. Hmmm. You also get a wandering by always turning k. Any theoretical reason to prefer one of these wanderings over another, or is it purely the aesthetics of the results? |
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This just zigzags off in a uniform direction periodically forever, or so it appears... is that what you expected? |
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Here's the trinomial equivalent of the usual arithmetic triangle coloring by residue; this one is mod 3. It's a nice example because it naturally uses the triangle shape (we didn't have any other example of that) and it uses a custom fill order. Also, it's familiar yet a bit different. |
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Here's an attractive Ulam spiral devised by @Vectornaut with a "background" that highlights the spiral path: |
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An interesting cubical-looking texture resulting from the overlapping hexagons: |
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Here's a pretty variation on the usual chaos game: |
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Very nice! |
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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
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