hi all. there are many ways to visualize the collatz problem. not sure when stumbled across a particular very nice visualization [possibly as one of the top results in a google search!]. was thinking about different directions & reminded of it this week and then found it again via google.

its by jason davies, and shows a remarkable symmetry, via a spherical attractive/repulsive force plotting of the reverse-order tree. the tree is built in reverse order of trajectories, starting from 1.[1] alas, presumably not symmetric enough to turn into a proof. but it does show the basic idea of using visualization and datamining to find exploitable patterns/structure. featured on oreilly “visualization of the week” oct7 2011.

this gave me the idea for this new very simple program. its similar to the davies visualization. it starts at 1 and goes in reverse for the smallest value in the current list. it outputs new “resolved” integers as they are found. there is a strong connection to the davies graph and quite a bit of structure.

l = [1] seen = {} c = 0 loop \ { l.sort! # p(l) x = l.shift l2 = [x * 2] l2 << (x - 1) / 3 if (x != 1 && x != 4 && (x - 1) % 3 == 0) l2.each \ { |y| next if (seen.member?(y)) p(y) l << y seen[y] = nil c += 1 exit if (c == 5000) } }

know how clouds look like shapes when you’re a kid? or rorschach tests? hmmm, it looks kinda like a fence with semiregular bumpy/wobbly fenceposts going off to infinity. another slight variation, advancing the coordinate only when a new coordinate maximum is found, “straightens out the fenceposts” and leads to this graph.

l = [1] seen = {} c = 0 mx = 0 z = 0 loop \ { l.sort! # p(l) x = l.shift l2 = [x * 2] l2 << (x - 1) / 3 if (x != 1 && x != 4 && (x - 1) % 3 == 0) l2.each \ { |y| next if (seen.member?(y)) puts([z, y].join("\t")) l << y seen[y] = nil if (y > mx) then mx = y z += 1 end c += 1 exit if (c == 5000) } }

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there are so many ways to visualize the collatz conjecture, it would be very interesting to create a directory of different visualizations. actually, riffing off that a little, a glossy/colorful coffee table book on “number theoretic fractals” would really be *way cool* eh?

another neat recent development/highlight, the TCS question by reyzin, “a simple problem whose decidability is not known”[4] with a remarkable/striking 37 votes as of writing & some contributions by yours truly [more great blog fodder, maybe more on that some other time]. new TCS.se questions with high votes are really rare these days, it seems one gets over 10-20 only rarely, like once a month at best. discovering a new question with high votes seems like a real event. also, Turkistany cited the collatz problem in a comment that got 19 upvotes. but then he wrote it up as an answer, and it currently has *one* vote by (dramatically pulling down the mask) yours truly!

stackexchange voting can be really crazily capricious sometimes…. with most questions in the sub-5 vote range, it does tend to show theres a significant mostly dormant audience there that is “mostly bored” with the questions asked. also, there appears to be a “winner takes all” power-law like distribution of votes. it would be very fun to do datamining on that data. [which reminds me, stackexchange did announce a cool new dataminer position/competition, intend to write about that some time.]

as I mentioned recently the collatz problem seems to naturally relate to fractals, although am not really aware of papers on the subject linking the two, but havent searched too much [so far].

elsewhere, via the TCS aggregator ran across the neat new paper finding and analyzing fractal properties in TM computational tableaus[2], something [an idea/concept/angle] that seems to have been pioneered by Wolfram but would be hard pressed to cite a paper of his. [3] is a brief recorded musing on the subject/connection that was not well-received at the time. as is sometimes said *the pioneers are the ones with the arrows in their backs…*

- [1] Collatz Graph: All Numbers Lead to One, jason davies
- [2] Fractal dimension versus computational complexity Joost J. Joosten, Fernando Soler-Toscano, Hector Zenil
- [3] Important work in fractals—fractals defined as computational tableaus, tcs.se
- [4] A simple problem whose decidability is unknown, tcs.se
- [5] algorithms as fractals, tcs.se

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