Worms

by in Boundary Behavior, Mystery, Numbers, Science

“What’s worse than biting into an apple and finding a worm?…  Finding half a worm.”

Considering the question quantitatively, however, finding something less than half of a worm is even more disgusting… a quarter of a worm, or a tenth, or one-hundredth, or some infinitesimal speck visible only by close inspection. The less one finds, the greater the “yuck” factor. So it would seem that the worst, worst possible case must be at the mathematical limit, which is zero. So the maximally-worst possible case would be finding… no worm at all!

Hmmm…

My university research project predecessors had discovered a worm in their apple. An unpredictable pattern had entered their measurements, and it seemed to appear in random places. It was as if something in the process that was so subtle as to be indiscernible was causing sudden departures into utter chaos.

After months of work, they had quantitatively concluded that there was simply no practicable way to assure a reliable measurement.  I was sitting in a mathematics class when it suddenly occurred to me that they were wrong.

It’s not that the tiniest bits of worms can’t eat the guts out of things. There was a time when I worked with machines that spun like washing-machines-from-Hell, far faster than jet-engines, and precipitously near their structural limits. Merely keeping them from flying explosively to pieces amounted to an entire discipline in itself. And yet, there was an incentive to get them spinning ever faster. A new type of titanium metallic-glass seemed like a promising route.

One early test exploded with such violence as to destroy the instruments that were monitoring it. Others both worked and failed without any apparent pattern. One test would look good for awhile, and then make a sudden “departure”. Another would simply shatter long before reaching working speeds.

It took months for engineers to determine that microscopic scratches in the material were causing the failures, with many of the flaws visible only under a very powerful microscope. Smaller imperfections would simply result in a higher velocity before a more spectacular failure. Only a flawlessly perfect part would have no worm at all.

Bristol University Physics Professor, Micheal Berry, discussed such worms (or in his case, “maggots”) as mathematically “discontinuous limits”. These are cases where the mathematics simply don’t work out to the value of the function at its limit. That’s to say that the worm ceases to exist at the point where it reaches the extreme.

A real-world example of this is found in the measurement of “viscosity” (the “gooey-ness” of a fluid) and the amount of “turbulence” (chaotic movement) that it will exhibit. The lower the viscosity of a fluid, the higher the turbulence.  Put simply, this just means that water moving over rocks will be more splashy than the same flow of thick Maple syrup.  Mmm…

However, when viscosity reaches zero (as in “superfluid” helium), the turbulence does not rise to some super-splashy maximum. Rather, it disappears entirely.

Likewise, what my predecessors had actually discovered was how to determine exactly what they were measuring by finding the point at which it became wormy.   Below this limit, the worm disappeared entirely.  And then, all they had really needed was to adjust to a measure appropriate to that kind of non-wormy produce.

To be fair, the problem was a little more complicated than I’m making it out, and it took me more than a year to work out its solutions.  But the misinterpretation of a discontinuous limit had left me with a whole bunch of delicious, low-hanging fruit.

Images from, Clipart Library:
Worm!
Fruit Cocktail Tree.

8 thoughts on “Worms”

  1. Hmm.

    Still guessing, but I don’t mind.

    I only had a couple of semesters of control theory in school, but weird dynamics with multiple steady solutions were the things that the instructors were really interested in at the time. Never really followed up on that much, but it is certainly interesting for living systems.

    As oft happens to me your intro brought to mind a musical piece ( Maggot Brain (1971), which has kind of a proto Pink Floyd sound).

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    • I was thinking about you when I wrote this. I need to be a little careful…

      This was based in finding the tipping-points between two “chaotic complex non-linear dynamic systems”, one of which could be applied to a flow measurement. That was, in fact, the mathematics class I was sitting in. The tipping-point identified the refinement of the material, after which the device could be adjusted to make a reliable measurement. The original (frankly sloppy) research came out of UCSD (which I thought was odd). Since I could just barely get through the mathematics myself, and yet still found errors in the original work, I hired two mathematicians and relegated myself to mostly engineering and data-collection. Most of my machinery, which would be constantly rebuilt according to a schedule of subtle variations, was fabricated at Cal Poly Pomona, and installed at a third facility for data collection. Brute-force approach, but it allowed me to patent the means to vary the measurement without rebuilding the apparatus.

      Maggot Brain… yes, Eddie Hazel! For whatever reason, that psychedelic/acid sound has always appealed to me. Probably gives away a divergent/flow aspect to my thinking… or maybe just from hearing my parents’ jazz going in the background.

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      • Sounds more interesting. Have folks followed up on refinements? Or was your solution enough for the time being?

        The control problem that we played with was diabetes, and I’m sure the literature on that is vast nowadays.

        Yes, I really like Maggot Brain. I’m trying not to listen to it too often.

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        • It’s a long sordid story that crosses borders. All I’ll say is that I never had any false pretense regarding my US “business partner”, but it was… rewarding. I’m sure things have been improved upon; however, I no longer have any connection to either the US versions or their applications. If you know where to look, however, you can find my leverage on some things in Japan.

          Yes indeed… at some point, I came to understand that I had to rise above it all, or drown in… well, you know. 😉

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      • Got subsumed by Maggot Brain again after maybe my last drive back from TN. Just does that to one. What was interesting to me is that what was once considered radical, fringe art now features on NPR. Time filters. Memories too. Pop culture just evolves.

        Emptied out the last of TN and brought it back South. Somewhat exhausting.

        In pop culture, George Clinton told Eddie to play it like someone told him that his mother just died. I think Eddie did that.

        Interested in how one can go through life and not travel at all. In some ways envious of huge family legacies, but I appreciate the benefit of just leaving and finding one’s own way.

        Been interested in turbulence for a while. I would like to see spectra of the time scales for fluctuations in various situations. Aeronautical types probably have that covered.

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        • It was years after my dad died before I could re-enter the US. In response to my urgings, his things ended up packed densely into about 2,000 cubic feet of storage. I spent years making long trips, picking through it all.

          For a long time, I found it very difficult to part with anything because it all seemed to connect with some memory. We exist in the moment, but see ourselves as a story.
          “Maggots in the mind of the universe.”

          After I bought my first house here, I gave some of his collections to a couple of relatives who were able to benefit from selling them, moved his things into about 1,000 cubic-feet of local storage, and transferred all his books into my own library. Ironically, it was one of those books that eventually resolved things for me:
          Meditations of Marcus Aurelius.
          “…to rise above it all.”
          Aside from the books, I kept just three things that most remind me of who I am.

          You can laugh if you want… (long ago) a late night sitting in a hot tub with a girlfriend, I was trying to encourage her to pursue an ambition. She said it wasn’t worth it anymore because she’d be in her forties before she finished. I reminded her that she’d be in her forties regardless. The world changes around us. We all travel. The only difference is whether or not it’s in ways that reflect our own volition. In that regard, only the moments we experience really matter. Amor fati.

          Peace, and good wishes to you.

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          • My Mom just saved all sorts of flotsam mostly. Sort of a chore. A disorganized, but otherwise remarkable brain.

            I do enjoy reading the careful thoughts put out by her and my father and their parents. Suspect that one’s ramblings today are simply too much.

            Grandpa was involved in educating rural farmers to figure out a way to make it clear from the property owners. I think it was a fine line that he walked.

            The only paper that I have kept is a notebook used in the mid 80’s for doing symbolic algebra…..just away of simplifying PDE’s along the lines of GI Taylor dispersion theory. It is one of the few things that I took care to do precisely and with a minor amount of flair. Macsyma was just becoming available to us, but I still had to document things by hand. I still like the notes, but not anyone else would. Look like the rantings of John Nash. But it is the only script in my life that I took care to make neat. and tidy.

            Anyway, I have a dining room table to mend. The pedestal got trashed in transit, but the top is fine. For me, the top is most of it.

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            • I was just writing elsewhere in this collective consciousness that my last ritual meditation of the day is usually to write in a journal. (After that, I’m free to drink a glass of wine… or surf WP.) Several tens are stacked neatly at the bottom of a bookshelf. I can’t imagine anyone would ever want to read them… truly meditations. At least neat drafting-print in mechanical pencil. But the daily entries are a rather Stoic pattern: improvement, accomplishment, observation, idea. Usually a couple of pages, sometimes a map, technical sketch or flow-chart, or some mathematics.

              It was a struggle to read any of my father’s journals. He wrote in either Japanese cursive (personal), or in German (technical). Just an odd note, but someone (SE Asian) just recently said to me that “Germans are the ‘Asians’ of Europe.” Hmm…

              Two boxes in the garage are the sorted-down remains of my college work. There are some pages of mathematics that I’ve kept simply because they’re cool. I’ve never been a good mathematician… I make too many stupid mistakes. But the feeling of suddenly seeing a real pattern emerge from the numbers or plots is almost religious… maybe is? Middle school, plotting trajectories… Every time I could add some form of drag to the process was like a rite-of-passage.

              Nash… You must be able to see into my drafts! “Shapley, Nash, and Peyton Young”… I’ve lately become somewhat interested in Game Theory.

              In the midst of another purge, this time it’s bigger items. I moved some resources several years back, kind of anticipating a few of the things we’re seeing now. Lately thinking I might want to be able to pull up stakes easily.

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